A021823 -id:A021823

A212336

Expansion of 1/(1 - 23*x + 23*x^2 - x^3).

+10
21

1, 23, 506, 11110, 243915, 5355021, 117566548, 2581109036, 56666832245, 1244089200355, 27313295575566, 599648413462098, 13164951800590591, 289029291199530905, 6345479454589089320, 139311518709760434136, 3058507932160140461673

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Partial sums of A077421.

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..200

Robert K. Moniot, A Family of Integer Sequences, 2021.

Index entries for linear recurrences with constant coefficients, signature (23,-23,1).

FORMULA

G.f.: 1/((1-x)*(1 - 22*x + x^2)).

a(n) = (((6+sqrt(30))^(2*n+3) + (6-sqrt(30))^(2*n+3))/6^(n+1) - 12)/240.

a(n) = a(-n-3) = 23*a(n-1) - 23*a(n-2) + a(n-3).

a(n)*a(n+2) = a(n+1)*(a(n+1)-1).

a(n+1) - 11*a(n) = A133285(n+2).

11*a(n+1) - a(n) = (1/5)*A157096(n+2).

a(n) = (1/20)*(-1 + 21*ChebyshevU(n, 11) - ChebyshevU(n-1, 11)). - G. C. Greubel, Feb 07 2022

MAPLE

a:= n-> (<<0|1|0>, <0|0|1>, <1|-23|23>>^n. <<1, 23, 506>>)[1, 1]:

seq(a(n), n=0..20); # Alois P. Heinz, Jun 15 2012

MATHEMATICA

CoefficientList[Series[1/(1 - 23 x + 23 x^2 - x^3), {x, 0, 16}], x]

LinearRecurrence[{23, -23, 1}, {1, 23, 506}, 20] (* Vincenzo Librandi, Aug 18 2013 *)

PROG

(PARI) Vec(1/(1-23*x+23*x^2-x^3)+O(x^17))

(Maxima) makelist(coeff(taylor(1/(1-23*x+23*x^2-x^3), x, 0, n), x, n), n, 0, 16);

(Magma) m:=17; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-23*x+23*x^2-x^3)));

(Magma) I:=[1, 23, 506]; [n le 3 select I[n] else 23*Self(n-1)-23*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Aug 18 2013

(Sage) [(1/20)*(-1 +21*chebyshev_U(n, 11) -chebyshev_U(n-1, 11)) for n in (0..30)] # G. C. Greubel, Feb 07 2022

CROSSREFS

Sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3): A334673 (k=24), A212336 (k=23), A212335 (k=22), A097833 (k=21), A097832 (k=20), A049664 (k=19), A097831-A097829 (k=18,17,16), A076139 (k=15), A097828-A097826 (k=14,13,12), A097784 (k=11), A092420 (k=10), A076765 (k=9), A092521 (k=8), A053142 (k=7), A089817(k=6), A061278 (k=5), A027941 (k=4), A000217 (k=3), A021823 (k=2), A133872 (k=1), A079978 (k=0).

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jun 08 2012

STATUS

approved

A131026

Periodic sequence (2, 2, 1, 0, 0, 1).

+10
12

2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Second column of triangular array T defined in A131022.

LINKS

Table of n, a(n) for n=1..105.

Index entries for linear recurrences with constant coefficients, signature (2, -2, 1).

FORMULA

a(1) = a(2) = 2, a(3) = 1, a(4) = a(5) = 0, a(6) = 1; for n > 6, a(n) = a(n-6).

G.f.: (2-2*x+x^2)/((1-x)*(1-x+x^2)).

a(n) = A021823(n+2).

a(n) = floor(((n+3) mod 6)/4)+floor(((n+2) mod 3)/2). - Gary Detlefs, Oct 02 2013

a(n) = 1+2/sqrt(3)*sin(Pi/3*n). - Werner Schulte, Jul 21 2017

MATHEMATICA

PadRight[{}, 120, {2, 2, 1, 0, 0, 1}] (* or *) LinearRecurrence[{2, -2, 1}, {2, 2, 1}, 120] (* Harvey P. Dale, Jul 16 2012 *)

PROG

(PARI) {m=105; for(n=1, m, r=(n-1)%6; print1(if(r<2, 2, if(r==2||r==5, 1, 0)), ", "))}

(Magma) m:=105; [ [2, 2, 1, 0, 0, 1][(n-1) mod 6 + 1]: n in [1..m] ];

CROSSREFS

Cf. A131022, A021823. Other columns of T are in A088911, A131027, A131028, A131029, A131030.

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

STATUS

approved

A341383

Numbers m such that the largest digit in the decimal expansion of 1/m is 2.

+10
9

5, 45, 50, 450, 495, 500, 819, 825, 4500, 4545, 4950, 4995, 5000, 8190, 8250, 8325, 45000, 45045, 45450, 47619, 49500, 49950, 49995, 50000, 81819, 81900, 82500, 83250, 83325, 89109, 450000, 450045, 450450, 454500, 454545, 476190, 495000, 499500, 499950, 499995, 500000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If m is a term, 10*m is also a term.

5 is the only prime up to 2.6*10^8 (comments in A333237).

Some subsequences: {45, 4545, 454545, ...}, {45045, 45045045, 45045045045, ...}, {45, 495, 4995, 49995, ...}, {819, 81819, 8181819, ...}, {825, 8325, 83325, 833325...}, ...

The subsequence of terms where 1/m has only digits {0,2} is m = 5*A333402 = 5, 45, 50, etc. A333402 is those t where 1/t has only digits {0,1}, so that 1/(5*t) = 2*(1/t)*(1/10) has digits {0,2}, starting from 1/5 = 0.2. These m are also A333402/2 of the even terms from A333402, since A333402 (like here) is self-similar in that the multiples of 10, divided by 10, are the sequence itself. - Kevin Ryde, Feb 13 2021

LINKS

Table of n, a(n) for n=1..41.

Index entries for sequences related to decimal expansion of 1/n

EXAMPLE

As 1/45 = 0.0202020202..., 45 is a term.

As 1/825 = 0.0012121212121212...., 825 is a term.

As 1/47619 = 0.000021000021000021..., 47619 is a term.

As 1/4545045 = 0.000000220019824..., 4545045 is not a term.

MATHEMATICA

Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 2 &] (* Amiram Eldar, Feb 10 2021 *)

PROG

(Python)

from itertools import count, islice

from sympy import n_order, multiplicity

def A341383_gen(startvalue=1): # generator of terms >= startvalue

for m in count(max(startvalue, 1)):

m2, m5 = multiplicity(2, m), multiplicity(5, m)

if max(str(10**(max(m2, m5)+n_order(10, m//2**m2//5**m5))//m)) == '2':

yield m

A341383_list = list(islice(A341383_gen(), 10)) # Chai Wah Wu, Feb 07 2022

CROSSREFS

Cf. A333236.

Similar with largest digit k: A333402 (k=1), A333237 (k=9).

Subsequence: A093143 \ {1}.

Decimal expansion: A021499 (1/495), A021823 (1/819).

KEYWORD

nonn,base

AUTHOR

Bernard Schott, Feb 10 2021

EXTENSIONS

Missing terms added by Amiram Eldar, Feb 10 2021

STATUS

approved

A164965

Cumulative sums of A010892.

+10
8

1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..83.

Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

FORMULA

G.f.: 1/((1 - x)*(1 - x + x^2)). - Philippe Deléham, Oct 11 2011

a(n) = a(n-1) - a(n-2) + 1. - Arkadiusz Wesolowski, Jun 08 2013

a(n) = -floor((n - 4)/6) - floor((n - 3)/6) + floor((n - 1)/6) + floor(n/6). - John M. Campbell, Dec 23 2016

E.g.f.: cosh(x) + 2*exp(x/2)*sin(sqrt(3)*x/2)/sqrt(3) + sinh(x). - Stefano Spezia, Feb 20 2023

MATHEMATICA

Table[-Floor[1/6 (-4 + n)] - Floor[1/6 (-3 + n)] + Floor[1/6 (-1 + n)] + Floor[n/6], {n, 0, 100}] (* John M. Campbell, Dec 23 2016 *)

LinearRecurrence[{2, -2, 1}, {1, 2, 2}, 100] (* Harvey P. Dale, Jul 17 2020 *)

CROSSREFS

Cf. A010892, A021823, A164925, A108299.

KEYWORD

easy,nonn

AUTHOR

Mark Dols, Sep 02 2009

EXTENSIONS

Offset corrected by John M. Campbell, Dec 23 2016

STATUS

approved

A077859

Expansion of g.f. (1 - x)^(-1)/(1 - 2*x + 2*x^2 + x^3).

+10
7

1, 3, 5, 6, 6, 6, 7, 9, 11, 12, 12, 12, 13, 15, 17, 18, 18, 18, 19, 21, 23, 24, 24, 24, 25, 27, 29, 30, 30, 30, 31, 33, 35, 36, 36, 36, 37, 39, 41, 42, 42, 42, 43, 45, 47, 48, 48, 48, 49, 51, 53, 54, 54, 54, 55, 57, 59, 60, 60, 60, 61, 63, 65, 66, 66, 66, 67, 69, 71, 72, 72, 72, 73, 75

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Partial sums of A021823. Second partial sums of A010892. - Paul Barry, Jun 06 2003

Row sums of A144083. - Gary W. Adamson, Sep 10 2008

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-4,3,-1).

FORMULA

G.f.: 1/((1-x)^2*(1-x+x^2)).

a(n) = Sum_{k=0..n} (k+1)*2*sin(Pi(n-k)/3 + Pi/3)/sqrt(3). - Paul Barry, May 18 2004

a(n) = Sum_{k=0..n} binomial(n-2k, n-k-1). - Paul Barry, Jan 15 2005

a(n) = n + 2 + (-1 + n - 3*floor(n/3))*(-1)^floor(n/3). - Tani Akinari, Jun 27 2013

a(n) = n + 1 + a(n-1) - a(n-2), with a(-1) = a(-2) = 0. - Richard R. Forberg, Jul 11 2013

a(n) = 3*a(n-1) - 4*a(n-2) + 3*a(n-3) - 1*a(n-4). - Joerg Arndt, Jul 12 2013

a(n) = Sum_{k=0..n} (-1)^k*(n+1-k)*b(k), where b(n) = A049347(n). - Mircea Merca, Feb 04 2014

E.g.f.: exp(x)*(2 + x) + exp(x/2)*(sqrt(3)*sin(sqrt(3)*x/2) - 3*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Feb 11 2023

Sum_{n>=0} (-1)^n/a(n) = log(2)/3 + log(3)/2 = log(108)/6. - Amiram Eldar, Feb 14 2023

MAPLE

A010892 := proc(n) op(1+(n mod 6), [1, 1, 0, -1, -1, 0]) ; end proc:

A077859 := proc(n) n+2+A010892(n+4) ; end proc:

seq(A077859(n), n=0..50) ; # R. J. Mathar, Mar 22 2011

MATHEMATICA

CoefficientList[Series[(1 - x)^(-1)/(1 - 2 x + 2 x^2 - x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 04 2014 *)

LinearRecurrence[{3, -4, 3, -1}, {1, 3, 5, 6}, 80] (* Harvey P. Dale, Apr 21 2023 *)

PROG

(PARI) Vec(1/(1-x)/(1-2*x+2*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

(Magma) I:=[1, 3, 5, 6]; [n le 4 select I[n] else 3*Self(n-1)-4*Self(n-2)+3*Self(n-3)-Self(n-4): n in [1..100]]

CROSSREFS

Cf. A010892, A021823, A049347, A144083.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 17 2002

STATUS

approved

A128494

Coefficient table for sums of Chebyshev's S-Polynomials.

+10
4

1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, -1, -2, 1, 1, 1, 2, -2, -3, 1, 1, 0, 2, 4, -3, -4, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, -2, -6, 7, 11, -5, -6, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, 3, 9, -13, -24, 16, 22, -7, -8, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, -3, -12, 22, 46, -40, -62, 29, 37, -9, -10, 1, 1, 1, 4, -12

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,13

COMMENTS

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k) = a(2*p+1,2*k) = A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1) = a(2*p+2,2*k+1) = A059260(p+k+1,2*k+1)*(-1)^(p+k), k >= 0.

LINKS

Table of n, a(n) for n=0..93.

Wolfdieter Lang, First 15 rows

Per Alexandersson, Luis Angel González-Serrano, and Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables, arXiv:1912.12725 [math.CO], 2019.

Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, and Mario Alberto Moctezuma-Salazar, Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables z_j = x_j + x_j^(-1), The Elec. J. of Combinatorics (2021) Vol. 28, No. 1, #P1.56.

FORMULA

S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.

a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).

G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).

From Wolfdieter Lang, Oct 16 2012: (Start)

a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.

a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)

EXAMPLE

The triangle a(n,m) begins:

n\m 0 1 2 3 4 5 6 7 8 9 10

0: 1

1: 1 1

2: 0 1 1

3: 0 -1 1 1

4: 1 -1 -2 1 1

5: 1 2 -2 -3 1 1

6: 0 2 4 -3 -4 1 1

7: 0 -2 4 7 -4 -5 1 1

8: 1 -2 -6 7 11 -5 -6 1 1

9: 1 3 -6 -13 11 16 -6 -7 1 1

10: 0 3 9 -13 -24 16 22 -7 -8 1 1

... reformatted by Wolfdieter Lang, Oct 16 2012

Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).

S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).

From Wolfdieter Lang, Oct 16 2012: (Start)

S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).

S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)

CROSSREFS

Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).

Cf. A128495 for S(2; n, x) coefficient table.

The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.

For m >= 1 the column sequences (without leading zeros) are of the form a(m, 2*k) = a(m, 2*k+1) = ((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.

KEYWORD

sign,tabl,easy

AUTHOR

Wolfdieter Lang, Apr 04 2007

STATUS

approved

A077784

Numbers k such that (10^k - 1)/3 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

+10
2

3, 5, 35, 159, 237, 325, 355, 371, 481, 1649, 3641, 4709, 269623

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Prime versus probable prime status and proofs are given in the author's table.

a(13) > 2*10^5. - Robert Price, Apr 03 2016

REFERENCES

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

LINKS

Table of n, a(n) for n=1..13.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)

Makoto Kamada, Prime numbers of the form 33...33533...33

Index entries for primes involving repunits.

FORMULA

a(n) = 2*A183175(n) + 1.

EXAMPLE

5 is a term because (10^5 - 1)/3 + 2*10^2 = 33533.

MATHEMATICA

Do[ If[ PrimeQ[(10^n + 6*10^Floor[n/2] - 1)/3], Print[n]], {n, 3, 4800, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

CROSSREFS

Partial sums of S(n, x), for x=1...9: A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420.

Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

KEYWORD

more,nonn,base

AUTHOR

Patrick De Geest, Nov 16 2002

EXTENSIONS

Name corrected by Jon E. Schoenfield, Oct 31 2018

a(13) from Robert Price, Aug 03 2024

STATUS

approved

A077829

Expansion of 1/(1-3*x-3*x^2-2*x^3).

+10
2

1, 3, 12, 47, 183, 714, 2785, 10863, 42372, 165275, 644667, 2514570, 9808261, 38257827, 149227404, 582072215, 2270414511, 8855914986, 34543132921, 134737972743, 525555146964, 2049965624963, 7996038261267, 31189121952618, 121655411891581, 474525678055131

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (3,3,2).

FORMULA

G.f.: 1/(1-3*x-3*x^2-2*x^3).

a(n) = 3*a(n-1) + 3*a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Jan 20 2024

MATHEMATICA

CoefficientList[Series[1/(1 - 3*x - 3*x^2 - 2*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 20 2024 *)

LinearRecurrence[{3, 3, 2}, {1, 3, 12}, 30] (* Harvey P. Dale, Dec 20 2024 *)

PROG

(PARI) Vec(1/(1-3*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

CROSSREFS

Partial sums of S(n, x), for x=1...14, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 17 2002

STATUS

approved

A077826

Expansion of (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).

+10
1

1, 3, 10, 32, 101, 319, 1006, 3172, 10001, 31531, 99410, 313416, 988125, 3115319, 9821846, 30965900, 97627977, 307797347, 970410426, 3059468848, 9645763669, 30410754735, 95877738174, 302279267892, 953013259777, 3004619799579, 9472837914274, 29865561746840

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..27.

Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-2).

FORMULA

From Wesley Ivan Hurt, Jun 26 2022: (Start)

G.f.: (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).

a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4). (End)

MATHEMATICA

LinearRecurrence[{3, 1, -1, -2}, {1, 3, 10, 32}, 30] (* Harvey P. Dale, May 12 2024 *)

PROG

(PARI) Vec((1-x)^(-1)/(1-2*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

CROSSREFS

Partial sums of S(n, x), for x=1...10, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784.

Partial sums of A077833.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 17 2002

STATUS

approved

A077827

Expansion of (1-x)^(-1)/(1-2*x-2*x^2-2*x^3).

+10
1

1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16803, 49059, 143235, 418195, 1220979, 3564819, 10407987, 30387571, 88720755, 259032627, 756281907, 2208070579, 6446770227, 18822245427, 54954172467, 160446376243, 468445588275, 1367692273971, 3993168476979, 11658612678451

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..28.

Index entries for linear recurrences with constant coefficients, signature (3,0,0,-2).

MATHEMATICA

CoefficientList[Series[(1-x)^(-1)/(1-2x-2x^2-2x^3), {x, 0, 40}], x] (* Harvey P. Dale, Mar 27 2011 *)

PROG

(PARI) Vec((1-x)^(-1)/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

CROSSREFS

Partial sums of S(n, x), for x=1...11, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 17 2002, Jun 05 2007

STATUS

approved