A195045 Concentric 13-gonal numbers.

0, 1, 13, 27, 52, 79, 117, 157, 208, 261, 325, 391, 468, 547, 637, 729, 832, 937, 1053, 1171, 1300, 1431, 1573, 1717, 1872, 2029, 2197, 2367, 2548, 2731, 2925, 3121, 3328, 3537, 3757, 3979, 4212, 4447, 4693, 4941, 5200, 5461, 5733, 6007, 6292, 6579, 6877, 7177, 7488, 7801, 8125

Offset: 0

Author

Omar E. Pol, Sep 27 2011

Keywords

• nonn
• easy

Comments

Also concentric tridecagonal numbers or concentric triskaidecagonal numbers.

Partial sums of A175886. - Reinhard Zumkeller, Jan 07 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Column 13 of A195040.

Cf. A032527, A032528, A069126, A175886, A195043, A195046, A195143, A195145.

Programs

Haskell

a195045 n = a195045_list !! n
a195045_list = scanl (+) 0 a175886_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

[13*n^2/4+9*((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

Maple

A195045:=n->13*n^2/4+9*((-1)^n-1)/8: seq(A195045(n), n=0..70); # Wesley Ivan Hurt, Nov 22 2015

Mathematica

Table[13 n^2/4 + 9 ((-1)^n - 1)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Nov 22 2015 *)

PARI

a(n)=13*n^2/4+9*((-1)^n-1)/8 \\ Charles R Greathouse IV, Oct 07 2015

PARI

concat(0, Vec(-x*(1+11*x+x^2)/((1+x)*(x-1)^3) + O(x^50))) \\ Altug Alkan, Nov 22 2015

Formula

a(n) = 13*n^2/4+9*((-1)^n-1)/8.

From R. J. Mathar, Sep 28 2011: (Start)

G.f.: -x*(1+11*x+x^2) / ( (1+x)*(x-1)^3 ).

a(n)+a(n+1) = A069126(n+1). (End)

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3. - Wesley Ivan Hurt, Nov 22 2015

Sum_{n>=1} 1/a(n) = Pi^2/78 + tan(3*Pi/(2*sqrt(13)))*Pi/(3*sqrt(13)). - Amiram Eldar, Jan 16 2023