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 A062717 Numbers m such that 6*m+1 is a perfect square. 21
 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 X values of solutions to the equation 6*X^3 + X^2 = Y^2. - Mohamed Bouhamida, 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, Nov 06 2007 a(n) = n^2 - n + 2*ceiling((n-1)/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 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 21 2021 From Amiram Eldar, Mar 11 2022: (Start) Sum_{n>=2} 1/a(n) = (9-sqrt(3)*Pi)/6. Sum_{n>=2} (-1)^n/a(n) = 3*(log(3)-1)/2. (End) 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 (Magma) [(6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4: n in [1..70]]; // Wesley Ivan Hurt, Apr 21 2021 CROSSREFS Equals 4 * A001318. Cf. A005563, A046092, A001082, A002378, A036666. Cf. A160757, A000217. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009 Cf. A007310. Diagonal of array A323674. - Sally Myers Moite, Feb 03 2019 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 November 28 22:51 EST 2022. Contains 358421 sequences. (Running on oeis4.)