A157716 - OEIS

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

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OFFSET

1,2

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

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).

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

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A000217, A101877, A219257.

Sequence in context: A290749 A374005 A091017 * A113968 A093812 A159840

Adjacent sequences: A157713 A157714 A157715 * A157717 A157718 A157719

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Mar 04 2009

EXTENSIONS

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

STATUS

approved

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