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A157716 One-eighth of triangular numbers (integers only). 2
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 (list; graph; refs; listen; history; text; internal format)
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)

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.

Sequence in context: A002155 A290749 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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Last modified December 6 10:03 EST 2022. Contains 358630 sequences. (Running on oeis4.)