A027693 -id:A027693

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A322595

a(n) = (n^3 + 9*n + 14*n + 9)/3.

+10
4

3, 11, 21, 35, 55, 83, 121, 171, 235, 315, 413, 531, 671, 835, 1025, 1243, 1491, 1771, 2085, 2435, 2823, 3251, 3721, 4235, 4795, 5403, 6061, 6771, 7535, 8355, 9233, 10171, 11171, 12235, 13365, 14563, 15831, 17171, 18585, 20075, 21643, 23291, 25021, 26835

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

OFFSET

0,1

COMMENTS

For n >= 6, a(n) is the number of evaluating points on the hypersphere in R^n in Stoyanovas's degree 7 cubature rule.

LINKS

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

Ronald Cools, Encyclopaedia of Cubature Formulas

Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.

Srebra B. Stoyanova, Cubature of the seventh degree of accuracy for the hypersphere, Journal of Computational and Applied Mathematics Vol. 84 (1997), 15-21.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.

a(n) = 2*binomial(n + 1, 3) + 6*binomial(n + 1, 2) + 2*binomial(n + 1, 1) + 1.

G.f.: (3 - x - 5*x^2 + 5*x^3)/(1 - x)^4. [Corrected by Georg Fischer, May 23 2019]

E.g.f.: (1/3)*(9 + 24*x + 12*x^2 + x^3)*exp(x).

MATHEMATICA

Table[(n^3 + 9*n + 14*n + 9)/3, {n, 0, 50}]

LinearRecurrence[{4, -6, 4, -1}, {3, 11, 21, 35}, 50] (* Harvey P. Dale, Aug 19 2020 *)

PROG

(Maxima) makelist((n^3 + 9*n + 14*n + 9)/3, n, 0, 50);

(Magma) [(n^3 + 9*n + 14*n + 9)/3: n in [0..45]]; // Vincenzo Librandi, Jun 05 2019

CROSSREFS

First differences: A027693.

Cf. A000292, A161680, A174794, A321124, A322594.

KEYWORD

nonn,easy

AUTHOR

Franck Maminirina Ramaharo, Dec 19 2018

STATUS

approved

A027724

Numbers k such that k^2+k+8 is a palindrome.

+10
3

0, 29, 202, 285, 2925, 2935, 20377, 29570, 297119, 2834699, 2837875, 2990390, 2997334, 287010920, 2926428849, 202542945597, 295431039629, 21495814697072, 21614586653852

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..19.

P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

MATHEMATICA

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 8; Select[Range[0, 3*10^5], palQ@ f@ # &] (* Giovanni Resta, Aug 29 2018 *)

CROSSREFS

Cf. A027725, A027693, A027722, A027726.

KEYWORD

nonn,base,more

AUTHOR

Patrick De Geest

EXTENSIONS

a(14)-a(19) from Giovanni Resta, Aug 29 2018

STATUS

approved

A027725

Palindromes of form k^2 + k + 8.

+10
3

8, 878, 41014, 81518, 8558558, 8617168, 415242514, 874414478, 88279997288, 8035521255308, 8053537353508, 8942435342498, 8984014104898, 82375268486257328, 8563985811185893658, 41023644811311844632014, 87279499176567199497278, 462070049490878094940070264, 467190356216898612653091764

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

OFFSET

1,1

COMMENTS

Palindromes h such that 4*h - 31 is a square. - Bruno Berselli, Aug 29 2018

LINKS

Table of n, a(n) for n=1..19.

P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

MATHEMATICA

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 8; Select[f@ Range[0, 3*10^5], palQ] (* Giovanni Resta, Aug 29 2018 *)

CROSSREFS

Cf. A027724, A027693, A027723, A027727.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest

EXTENSIONS

a(14)-a(19) from Giovanni Resta, Aug 29 2018

STATUS

approved

A281699

Sierpinski stellated octahedron numbers: a(n) = 2*(-3*2^(n+1) + 2^(2n+3) + 5).

+10
2

14, 50, 218, 938, 3914, 16010, 64778, 260618, 1045514, 4188170, 16764938, 67084298, 268386314, 1073643530, 4294770698, 17179475978, 68718690314, 274876334090, 1099508482058, 4398040219658, 17592173461514, 70368719011850, 281474926379018, 1125899806179338, 4503599426043914, 18014398106828810

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

OFFSET

0,1

COMMENTS

Stella octangula with Sierpinski recursion.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Wikipedia, Sierpinski triangle, see section on higher dimensional analogs.

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

FORMULA

a(n) = 8*(2^(2*n+1)+2) - 6*(2^(n+1)+1).

From Colin Barker, Jan 28 2017: (Start)

a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.

G.f.: 2*(7 - 24*x + 32*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).

(End)

MATHEMATICA

Table[8 (2^(2 n + 1) + 2) - 6 (2^(n + 1) + 1), {n, 0, 25}] (* or *)

LinearRecurrence[{7, -14, 8}, {14, 50, 218}, 26] (* or *)

CoefficientList[Series[2 (7 - 24 x + 32 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Jan 28 2017 *)

PROG

(PARI) Vec(2*(7 - 24*x + 32*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 28 2017

(PARI) a(n) = 16*4^n - 12*2^n + 10 \\ Charles R Greathouse IV, Jan 29 2017

CROSSREFS

Cf. A007588, A027693, A052539, A052548, A067771, A178789, A233774, A279511.

KEYWORD

nonn,easy

AUTHOR

Steven Beard, Jan 27 2017

STATUS

approved

A214870

Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

+10
1

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

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

OFFSET

1,2

COMMENTS

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).

Enumeration table T(n,k) layer by layer. The order of the list:

T(1,1)=1;

T(1,2), T(2,1), T(2,2);

. . .

T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);

. . .

LINKS

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Eric Weisstein's World of Mathematics, Pairing functions

Index entries for sequences that are permutations of the natural numbers

FORMULA

As table

T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;

T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.

As linear sequence

a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;

a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

EXAMPLE

The start of the sequence as table:

1 2 5 10 17 26 ...

3 4 9 16 25 36 ...

7 8 6 11 18 27 ...

13 14 12 15 24 35 ...

21 22 20 23 19 28 ...

31 32 30 33 29 34 ...

...

The start of the sequence as triangle array read by rows:

2, 3;

5, 4, 7;

10, 9, 8, 13;

17, 16, 6, 14, 21;

26, 25, 11, 12, 22, 31;

...

PROG

(Python)

t=int((math.sqrt(8*n-7) - 1)/ 2)

i=n-t*(t+1)/2

j=(t*t+3*t+4)/2-n

if i > j:

result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)

else:

result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

CROSSREFS

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

KEYWORD

nonn,tabl,changed

AUTHOR

Boris Putievskiy, Mar 11 2013

STATUS

approved

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