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A062717 Numbers m such that 6*m+1 is a perfect square. 19
0, 4, 8, 20, 28, 48, 60, 88, 104, 140, 160, 204, 228, 280, 308, 368, 400, 468, 504, 580, 620, 704, 748, 840, 888, 988, 1040, 1148, 1204, 1320, 1380, 1504, 1568, 1700, 1768, 1908, 1980, 2128, 2204, 2360, 2440, 2604, 2688, 2860, 2948, 3128, 3220, 3408, 3504 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sequence allows us to find X values of the equation: 6*X^3 + X^2 = Y^2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007

Arithmetic averages of the k triangular numbers 0, 1, 3, 6, ..., (k-1)*k/2 that take integer values. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009 [edited by Jon E. Schoenfield, Jan 10 2015]

Even terms in A186423; union of A033579 and A033580, A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011

a(n) are integers produced by Sum_{i = 1..k-1} i*(k-i)/k, for some k > 0. Values for k are given by A007310 = sqrt(6*a(n)+1), the square roots of those perfect squares. - Richard R. Forberg, Feb 16 2015

Equivalently, numbers of the form 2*h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the sixth comment of A152749). - Bruno Berselli, Feb 02 2017

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000

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

FORMULA

G.f.: 4*x^2*(1 + x + x^2) / ( (1+x)^2*(1-x)^3 ).

a(2*k) = k*(6*k+2), a(2*k+1) = 6*k^2 + 10*k + 4. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007

a(n) = n^2 + n + 2*ceiling(n/2)^2. - Gary Detlefs, Feb 23 2010

From Bruno Berselli, Nov 28 2010: (Start)

a(n) = (6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4.

6*a(n) + 1 = A007310(n)^2. (End)

E.g.f.: (3*x^2*exp(x) - x*exp(-x) + sinh(x))/2. - Ilya Gutkovskiy, Jun 18 2016

MAPLE

seq(n^2+n+2*ceil(n/2)^2, n=0..48); # Gary Detlefs, Feb 23 2010

MATHEMATICA

Select[Range[0, 3999], IntegerQ[Sqrt[6# + 1]] &] (* Harvey P. Dale, Mar 10 2013 *)

PROG

(PARI) je=[]; for(n=0, 7000, if(issquare(6*n+1), je=concat(je, n))); je

(PARI) { n=0; for (m=0, 10^9, if (issquare(6*m + 1), write("b062717.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 09 2009

CROSSREFS

Equals 4 * A001318.

Cf. A005563, A046092, A001082, A002378, A036666.

Cf. A160757, A000217. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009

Cf. A007310.

Sequence in context: A087254 A160726 A191483 * A084922 A180794 A047185

Adjacent sequences:  A062714 A062715 A062716 * A062718 A062719 A062720

KEYWORD

nonn,easy

AUTHOR

Jason Earls, Jul 14 2001

STATUS

approved

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Last modified February 19 10:40 EST 2018. Contains 299330 sequences. (Running on oeis4.)