A195162 Generalized 12-gonal numbers: k*(5*k-4) for k = 0, +-1, +-2, ...

0, 1, 9, 12, 28, 33, 57, 64, 96, 105, 145, 156, 204, 217, 273, 288, 352, 369, 441, 460, 540, 561, 649, 672, 768, 793, 897, 924, 1036, 1065, 1185, 1216, 1344, 1377, 1513, 1548, 1692, 1729, 1881, 1920, 2080, 2121, 2289, 2332, 2508, 2553, 2737, 2784, 2976, 3025

Offset: 0

Author

Omar E. Pol, Sep 10 2011

Keywords

• nonn
• easy

Comments

Also generalized dodecagonal numbers.

Second 12-gonal numbers (A135705) and positive terms of A051624 interleaved. - Omar E. Pol, Aug 04 2012

The characteristic function of this sequence is A205988. - Jason Kimberley, Nov 15 2012

Also, integer values of m*(m+4)/5. - Bruno Berselli, Dec 05 2012

Also, numbers h such that 5*h + 4 is a square. - Bruno Berselli, Oct 10 2013

Exponents in expansion of Product_{n >= 1} (1 + x^(10*n-9))*(1 + x^(10*n-1))*(1 - x^(10*n)) = 1 + x + x^9 + x^12 + x^28 + .... - Peter Bala, Dec 10 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math. , Vol. 274, No. 1-3 (2004), pp. 9-24. See E(q).
• John Elias, Generalized 12-gonal & 20-gonal Cross Configurations
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Partial sums of A195161.

Column 8 of A195152.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), this sequence (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Cf. A001622, A051624, A135705.

Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

Programs

GAP

List([0..50], n-> (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8); # G. C. Greubel, Jul 04 2019

Magma

[0] cat &cat[[5*n^2-4*n, 5*n^2+4*n]: n in [1..25]]; // Vincenzo Librandi, Sep 26 2011

Mathematica

nn = 25; Sort[Table[n*(5*n - 4), {n, -nn, nn}]] (* T. D. Noe, Sep 23 2011 *)

PARI

vector(50, n, n--; (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8) \\ G. C. Greubel, Jul 04 2019

Sage

[(10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8 for n in (0..50)] # G. C. Greubel, Jul 04 2019

Formula

From R. J. Mathar, Sep 24 2011: (Start)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

a(n) = A008805(n-1) + A008805(n-3) + 8*A008805(n-2). (End)

From Bruno Berselli, Sep 26 2011: (Start)

G.f.: x*(1+8*x+x^2)/((1+x)^2*(1-x)^3).

a(n) = (10*n*(n+1) + 3*(2*n+1)*(-1)^n - 3)/8.

a(n) = a(-n-1). (End)

Sum_{n>=1} 1/a(n) = (5 + 4*sqrt(1 + 2/sqrt(5))*Pi)/16. - Vaclav Kotesovec, Oct 05 2016

E.g.f.: (3*(1 - 2*x)*exp(-x) + (-3 +20*x +10*x^2)*exp(x))/8. - G. C. Greubel, Jul 04 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5)/8 + sqrt(5)*log(phi)/4 - 5/16, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 28 2022