A195818 Generalized 14-gonal numbers: m*(6*m-5), m = 0,+1,-1,+2,-2,+3,-3,...

0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175, 186, 246, 259, 329, 344, 424, 441, 531, 550, 650, 671, 781, 804, 924, 949, 1079, 1106, 1246, 1275, 1425, 1456, 1616, 1649, 1819, 1854, 2034, 2071, 2261, 2300, 2500, 2541, 2751, 2794, 3014, 3059, 3289

Offset: 0

Author

Omar E. Pol, Sep 29 2011

Keywords

• nonn
• easy

Comments

Also generalized tetradecagonal numbers or generalized tetrakaidecagonal numbers.

Also A211014 and positive terms of A051866 interleaved. - Omar E. Pol, Aug 04 2012

Exponents in expansion of Product_{n >= 1} (1 + x^(12*n-11))*(1 + x^(12*n-1))*(1 - x^(12*n)) = 1 + x + x^11 + x^14 + x^34 + .... - Peter Bala, Dec 10 2020

Links

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

Crossrefs

Partial sums of A195817.

Column 10 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), A195162 (k=12), A195313 (k=13), this sequence (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. A051866, A211014.

Programs

Magma

[(3*n*(n+1)+(2*n+1)*(-1)^n-1)/2: n in [0..60]]; // Vincenzo Librandi, Sep 30 2011

Magma

A195818:=func; [0] cat [A195818(n*m): m in [1,-1], n in [1..25]];

Maple

a:= n-> (m-> m*(6*m-5))(ceil(-(n+1)/2)*(-1)^n):
seq(a(n), n=0..46);  # Alois P. Heinz, Jun 08 2021

Mathematica

LinearRecurrence[{1,2,-2,-1,1},{0,1,11,14,34},50] (* Harvey P. Dale, Mar 13 2018 *)

PARI

Vec(-x*(x^2+10*x+1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 15 2013

Formula

a(n) = (3*n*(n+1) + (2*n+1)*(-1)^n - 1)/2. - Vincenzo Librandi, Sep 30 2011

G.f.: -x*(x^2+10*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Sep 15 2013

Sum_{n>=1} 1/a(n) = 6/25 + sqrt(3)*Pi/5. - Vaclav Kotesovec, Oct 05 2016

E.g.f.: (x*(3*x + 4)*cosh(x) + (3*x^2 + 8*x - 2)*sinh(x))/2. - Stefano Spezia, Jun 08 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = (5*log(432)-6)/25. - Amiram Eldar, Feb 28 2022