A146512 Numbers congruent to {1, 3} mod 12.
1, 3, 13, 15, 25, 27, 37, 39, 49, 51, 61, 63, 73, 75, 85, 87, 97, 99, 109, 111, 121, 123, 133, 135, 145, 147, 157, 159, 169, 171, 181, 183, 193, 195, 205, 207, 217, 219, 229, 231, 241, 243, 253, 255, 265, 267, 277, 279, 289, 291, 301, 303, 313, 315, 325, 327
Offset: 1
Examples
G.f. = x + 3*x^2 + 13*x^3 + 15*x^4 + 25*x^5 + 27*x^6 + 37*x^7 + 39*x^8 + ...
Links
- David Lovler, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Programs
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Mathematica
Select[Range[300],MemberQ[{1,3},Mod[#,12]]&] (* Ray Chandler, Dec 06 2016 *) -
PARI
{a(n) = 6*n - 9 + n%2*4}; /* Michael Somos, Dec 06 2016 */
Formula
a(2k-1) = 12*(k-1)+1, a(2k) = 12*(k-1)+3, where k>0.
With offset 0, a(n) = 8*floor(n/2) + 2*n + 1, or a(n) = 6*n - 1 + 2*(-1)^n. - Gary Detlefs, Mar 13 2010
a(n) = 12*n-a(n-1)-20 (with a(1)=1). - Vincenzo Librandi, Nov 26 2010
G.f.: x * (1 + 2*x + 9*x^2) / (1 - x - x^2 + x^3). - Michael Somos, Dec 06 2016
a(n) = a(n-1)+a(n-2)-a(n-3). - Wesley Ivan Hurt, May 03 2021
E.g.f.: 9 + (6*x - 7)*exp(x) - 2*exp(-x). - David Lovler, Sep 07 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)+1)*(2*Pi + 2*arccosh(26) - 4*sqrt(3)*arccoth(sqrt(3)) + 3*(sqrt(3)-1)*log(3))/48. - Amiram Eldar, Sep 26 2022
Extensions
Formula and crossrefs corrected by Ray Chandler, Dec 06 2016
Comments