A175887 Numbers that are congruent to {1, 14} mod 15.

1, 14, 16, 29, 31, 44, 46, 59, 61, 74, 76, 89, 91, 104, 106, 119, 121, 134, 136, 149, 151, 164, 166, 179, 181, 194, 196, 209, 211, 224, 226, 239, 241, 254, 256, 269, 271, 284, 286, 299, 301, 314, 316, 329, 331, 344, 346, 359, 361, 374, 376, 389, 391, 404

Offset

1,2

Comments

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 15).

Links

Bruno Berselli, Table of n, a(n) for n = 1..10000.

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

Formula

G.f.: x*(1+13*x+x^2)/((1+x)*(1-x)^2).

a(n) = (30*n+11*(-1)^n-15)/4.

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

a(n) = 15*A000217(n-1) -2*sum(a(i), i=1..n-1) +1 for n>1.

a(n) = A047209(A047225(n+1)).

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/15)*cot(Pi/15) = A019693 * A019976 / 10. - Amiram Eldar, Dec 04 2021

E.g.f.: 1 + ((30*x - 15)*exp(x) + 11*exp(-x))/4. - David Lovler, Sep 05 2022

Mathematica

Select[Range[1, 450], MemberQ[{1, 14}, Mod[#, 15]]&]
CoefficientList[Series[(1 + 13 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)

Prog

(Magma) [n: n in [1..450] | n mod 15 in [1, 14]];
(Haskell)
a175887 n = a175887_list !! (n-1)
a175887_list = 1 : 14 : map (+ 15) a175887_list
-- Reinhard Zumkeller, Jan 07 2012
(Magma) [(30*n+11*(-1)^n-15)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013
(PARI) a(n)=(30*n+11*(-1)^n-15)/4 \\ Charles R Greathouse IV, Sep 28 2015

Crossrefs

Cf. A000217, A019693, A019976, A113801, A175886, A091998, A175885, A090771, A056020, A047522, A047336, A007310, A047209, A005408, A001651.

Cf. A132240 (primes).

Sequence in context: A076023 A163629 A228207 * A305885 A224402 A067844

Adjacent sequences: A175884 A175885 A175886 * A175888 A175889 A175890

Keyword

nonn,easy

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Status

approved