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

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Keywords

• nonn
• easy

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).

Crossrefs

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

Cf. A132240 (primes).

Programs

Haskell

a175887 n = a175887_list !! (n-1)
a175887_list = 1 : 14 : map (+ 15) a175887_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

[n: n in [1..450] | n mod 15 in [1,14]];

Magma

[(30*n+11*(-1)^n-15)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013

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 *)

PARI

a(n)=(30*n+11*(-1)^n-15)/4 \\ Charles R Greathouse IV, Sep 28 2015

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

From Amiram Eldar, Nov 25 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = (Pi/15)*cosec(Pi/15).

Product_{n>=2} (1 + (-1)^n/a(n)) = 2*cos(Pi/15). (End)