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Search: a047478 -id:a047478
Displaying 1-3 of 3 results found.
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A047529
Numbers that are congruent to {1, 3, 7} mod 8.
+10
12
1, 3, 7, 9, 11, 15, 17, 19, 23, 25, 27, 31, 33, 35, 39, 41, 43, 47, 49, 51, 55, 57, 59, 63, 65, 67, 71, 73, 75, 79, 81, 83, 87, 89, 91, 95, 97, 99, 103, 105, 107, 111, 113, 115, 119, 121, 123, 127, 129, 131, 135, 137, 139, 143, 145, 147, 151, 153, 155, 159
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Terms that occur on the first two rows of array A257852. Odd numbers that are not of the form 4k+1, where k is an odd number. - Antti Karttunen, Jun 06 2024
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
FORMULA
a(n) = (24*n+2*sqrt(3)*sin(2*Pi*n/3)+6*cos(2*Pi*n/3)-15)/9. - Fred Daniel Kline, Nov 12 2015
From Colin Barker, Nov 12 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
G.f.: x*(x^3+4*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)). (End)
a(n+3) = a(n) + 8 for all n in Z. - Michael Somos, Nov 15 2015
a(3k) = 8k-1, a(3k-1) = 8k-5, a(3k-2) = 8k-7. - Wesley Ivan Hurt, Jun 13 2016
a(n) = 8 * floor((n-1) / 3) + 2^(((n-1) mod 3) + 1) - 1. - Fred Daniel Kline, Aug 09 2016
a(n) = 2*(n + floor(n/3)) - 1. - Wolfdieter Lang, Sep 10 2021
EXAMPLE
G.f. = x + 3*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 15*x^6 + 17*x^7 + 19*x^8 + 23*x^9 + ...
MAPLE
A047529:=n->(24*n+2*sqrt(3)*sin(2*Pi*n/3)+6*cos(2*Pi*n/3)-15)/9: seq(A047529(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016
MATHEMATICA
Select[Range[150], MemberQ[{1, 3, 7}, Mod[#, 8]]&] (* Harvey P. Dale, May 02 2011 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 3, 7, 9}, 100] (* Vincenzo Librandi, Jun 14 2016 *)
PROG
(PARI) Vec(x*(x^3+4*x^2+2*x+1)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 12 2015
(PARI) {a(n) = n\3 * 8 + [-1, 1, 3][n%3 + 1]}; /* Michael Somos, Nov 15 2015 */
(Magma) [n : n in [0..150] | n mod 8 in [1, 3, 7]]; // Wesley Ivan Hurt, Jun 13 2016
CROSSREFS
Cf. A047478, A047484, A047623.
Setwise difference A005408 \ A004770.
Disjoint union of A004767 and A017077; see A257852.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047484
Numbers that are congruent to {3, 5, 7} mod 8.
+10
5
3, 5, 7, 11, 13, 15, 19, 21, 23, 27, 29, 31, 35, 37, 39, 43, 45, 47, 51, 53, 55, 59, 61, 63, 67, 69, 71, 75, 77, 79, 83, 85, 87, 91, 93, 95, 99, 101, 103, 107, 109, 111, 115, 117, 119, 123, 125, 127, 131, 133, 135, 139, 141, 143, 147, 149, 151, 155, 157, 159
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
FORMULA
G.f.: x*(3+2*x+2*x^2+x^3)/((1-x)^2*(1+x+x^2)). [Colin Barker, May 14 2012]
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. - Vincenzo Librandi, May 17 2012
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = (24*n-3-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-1, a(3k-1) = 8k-3, a(3k-2) = 8k-5. (End)
a(n) = 3*n - floor((n-1)/3) - ((n-1) mod 3). - Wesley Ivan Hurt, Sep 26 2017
a(n) = 2*(n + floor((n-1)/3)) + 1. - Wolfdieter Lang, Sep 11 2021
MAPLE
A047484:=n->(24*n-3-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047484(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016
MATHEMATICA
Select[Range[0, 300], MemberQ[{3, 5, 7}, Mod[#, 8]]&] (* Vincenzo Librandi, May 17 2012 *)
PROG
(Magma) I:=[3, 5, 7, 11]; [n le 4 select I[n] else Self(n-1)+Self(n-3) -Self(n-4): n in [1..70]]; // Vincenzo Librandi, May 17 2012
CROSSREFS
Cf. A047478, A047529, A047623.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047623
Numbers that are congruent to {1, 3, 5} mod 8.
+10
4
1, 3, 5, 9, 11, 13, 17, 19, 21, 25, 27, 29, 33, 35, 37, 41, 43, 45, 49, 51, 53, 57, 59, 61, 65, 67, 69, 73, 75, 77, 81, 83, 85, 89, 91, 93, 97, 99, 101, 105, 107, 109, 113, 115, 117, 121, 123, 125, 129, 131, 133, 137, 139, 141, 145, 147, 149, 153, 155, 157
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
FORMULA
a(n) = 2*floor((n-1)/3) + 2*n - 1. - Gary Detlefs, Mar 18 2010
From Colin Barker, Feb 03 2012: (Start)
G.f.: x*(1+2*x+2*x^2+3*x^3)/(1-x-x^3+x^4).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. (End)
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = (24*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-3, a(3k-1) = 8k-5, a(3k-2) = 8k-7. (End)
MAPLE
A047623:=n->(24*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047623(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016
MATHEMATICA
Select[Range[0, 150], MemberQ[{1, 3, 5}, Mod[#, 8]]&] (* Vincenzo Librandi, Apr 27 2012 *)
PROG
(Magma) I:=[1, 3, 5, 9]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Apr 27 2012
CROSSREFS
Cf. A047478, A047484, A047529.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
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