A157716 One-eighth of triangular numbers (integers only).

0, 15, 17, 62, 66, 141, 147, 252, 260, 395, 405, 570, 582, 777, 791, 1016, 1032, 1287, 1305, 1590, 1610, 1925, 1947, 2292, 2316, 2691, 2717, 3122, 3150, 3585, 3615, 4080, 4112, 4607, 4641, 5166, 5202, 5757, 5795, 6380, 6420, 7035, 7077, 7722, 7766, 8441

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Mar 04 2009

Keywords

• nonn
• easy

Comments

From Lamine Ngom, Oct 27 2020: (Start)

Numbers of the form (4*k)^2-k (A157446) or (4*k)^2+k (A157474).

Also numbers k such that 1+64*k is a square. (End)

The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(32*n))*(1 - q^(32*n-15))*(1 - q^(32*n-17)) = 1 - q^15 - q^17 + q^62 + q^66 - q^141 - q^147 + + - - .... - Peter Bala, Dec 24 2024

Examples

			The first three members of A000217 that are divisible by 8 are A000217(0), A000217(15) and A000217(16), so a(1) = A000217(0)/8 = 0, a(2) = A000217(15)/8 = 15, a(3) = A000217(16)/8 = 17.
		

Links

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

Crossrefs

Cf. A000217, A101877, A219257.

Programs

Maple

seq((2*n-1 + 7/8*(-1)^n)^2 - 1/64, n = 1 .. 1000); # Robert Israel, Apr 20 2014

Mathematica

Array[(2 # - 1 + 7/8*(-1)^#)^2 - 1/64 &, 46] (* or *)
Rest@ CoefficientList[Series[x^2*(15 + 2 x + 15 x^2)/((1 + x)^2*(1 - x)^3), {x, 0, 46}], x] (* Michael De Vlieger, Nov 05 2020 *)

Formula

G.f.: x^2*(15+2*x+15*x^2)/((1+x)^2*(1-x)^3 ). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009]

a(n) = (2*n-1 + 7/8*(-1)^n)^2 -1/64. - Robert Israel, Apr 20 2014

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Nov 10 2020

Sum_{n>=2} 1/a(n) = 16 - (sqrt(2*(2+sqrt(2))) + sqrt(2) + 1)*Pi. - Amiram Eldar, Mar 17 2022

Extensions

Definition edited by N. J. A. Sloane, Mar 08 2009