A195042 Concentric 9-gonal numbers.

0, 1, 9, 19, 36, 55, 81, 109, 144, 181, 225, 271, 324, 379, 441, 505, 576, 649, 729, 811, 900, 991, 1089, 1189, 1296, 1405, 1521, 1639, 1764, 1891, 2025, 2161, 2304, 2449, 2601, 2755, 2916, 3079, 3249, 3421, 3600, 3781, 3969, 4159, 4356, 4555, 4761, 4969, 5184, 5401, 5625

Offset: 0

Author

Omar E. Pol, Sep 27 2011

Keywords

• nonn
• easy

Comments

Also concentric enneagonal numbers or concentric nonagonal numbers.

A016766 and A069131 interleaved.

Partial sums of A056020. - 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 9 of A195040.

Cf. A016766, A069131, A077221, A195041, A195142, A195043.

Programs

Haskell

a195042 n = a195042_list !! n
a195042_list = scanl (+) 0 a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

[(9*n^2+5/2*((-1)^n-1))/4: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

Mathematica

LinearRecurrence[{2,0,-2,1},{0,1,9,19},60] (* Harvey P. Dale, Nov 24 2019 *)

PARI

a(n)=(9*n^2+5/2*((-1)^n-1))/4 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (9*n^2 + 5/2*((-1)^n - 1))/4.

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

G.f.: -x*(1+7*x+x^2) / ( (1+x)*(x-1)^3 ).

a(n) + a(n+1) = A060544(n+1). (End)

Sum_{n>=1} 1/a(n) = Pi^2/54 + tan(sqrt(5)*Pi/6)*Pi/(3*sqrt(5)). - Amiram Eldar, Jan 16 2023