A193867 Odd central polygonal numbers.

1, 7, 11, 29, 37, 67, 79, 121, 137, 191, 211, 277, 301, 379, 407, 497, 529, 631, 667, 781, 821, 947, 991, 1129, 1177, 1327, 1379, 1541, 1597, 1771, 1831, 2017, 2081, 2279, 2347, 2557, 2629, 2851, 2927, 3161, 3241, 3487, 3571, 3829, 3917, 4187, 4279

Offset: 1

Author

Omar E. Pol, Aug 15 2011

Keywords

• nonn
• easy

Comments

Even triangular numbers plus 1.

Union of A188135 and A185438 without repetitions (A188135 is a bisection of this sequence. Another bisection is A185438 but without its initial term).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000124, A014494, A014601, A185438, A188135, A193868.

Programs

Mathematica

Select[Accumulate[Range[0,100]],EvenQ]+1 (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,7,11,29,37},50] (* Harvey P. Dale, Nov 29 2014 *)

PARI

Vec(-x*(x^2+1)*(x^2+6*x+1) / ((1+x)^2*(x-1)^3) + O(x^100)) \\ Colin Barker, Jan 27 2016

Formula

a(n) = A000124(A014601(n-1)).

a(n) = 1 + A014494(n-1).

G.f.: -x*(x^2+1)*(x^2+6*x+1) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011

From Colin Barker, Jan 27 2016: (Start)

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

a(n) = 2*n^2-n+1 for n even.

a(n) = 2*n^2-3*n+2 for n odd. (End)

Sum_{n>=1} 1/a(n) = 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(2*cosh(sqrt(7)*Pi/4) - sqrt(2))). - Amiram Eldar, May 11 2025