A174114 Even central polygonal numbers (A193868) divided by 2.

1, 2, 8, 11, 23, 28, 46, 53, 77, 86, 116, 127, 163, 176, 218, 233, 281, 298, 352, 371, 431, 452, 518, 541, 613, 638, 716, 743, 827, 856, 946, 977, 1073, 1106, 1208, 1243, 1351, 1388, 1502, 1541, 1661, 1702, 1828, 1871, 2003, 2048, 2186, 2233, 2377, 2426, 2576

Offset: 1

Author

Reinhard Zumkeller, Mar 08 2010

Keywords

• nonn
• easy

Comments

Central terms of A170950, seen as a triangle of rows with an odd number of terms.

Equivalently, numbers of the form m*(4*m+3)+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - Bruno Berselli, Jan 05 2016

Conjecure: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n} 1 - q^k) = q + q^2 + q^8 + q^11 + q^23 + q^28 + .... Cf. A266883. - Peter Bala, May 10 2025

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A002061, A002522, A010696, A170950, A193868, A266883.

Cf. A033951: numbers of the form m*(4*m+3)+1 for nonnegative m.

Programs

Mathematica

Select[Table[(n (n + 1)/2 + 1)/2, {n, 600}], IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)
(Select[PolygonalNumber@ Range@ 100, OddQ] + 1 )/2 (* Version 10.4, or *)
Rest@ CoefficientList[Series[-x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)

PARI

a(n)=(2*n-1)*(2*n-1-(-1)^n)\4+1 \\ Charles R Greathouse IV, Jun 11 2015

Formula

a(n+3) - a(n+2) - a(n+1) + a(n) = A010696(n+1).

a(n) = A170950(A002061(n)).

a(n) = A193868(n)/2. - Omar E. Pol, Aug 16 2011

G.f.: -x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 18 2011

E.g.f.: ((2 + x + 2*x^2)*cosh(x) + (1 - x + 2*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Nov 16 2024

Sum_{n>=1} 1/a(n) = 4*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - Amiram Eldar, May 12 2025

Extensions

New name from Omar E. Pol, Aug 16 2011