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 A057569 Numbers of the form k*(5*k+1)/2 or k*(5*k-1)/2. 15
 0, 2, 3, 9, 11, 21, 24, 38, 42, 60, 65, 87, 93, 119, 126, 156, 164, 198, 207, 245, 255, 297, 308, 354, 366, 416, 429, 483, 497, 555, 570, 632, 648, 714, 731, 801, 819, 893, 912, 990, 1010, 1092, 1113, 1199, 1221, 1311, 1334, 1428, 1452, 1550 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) is the set of all m such that 40*m+1 is a perfect square. - Gary Detlefs, Feb 22 2010 Integers of the form (n^2 - n) / 10. Numbers of the form n * (5*n - 1) / 2 where n is an integer. - Michael Somos, Jan 13 2012 Also integers of the form sum_{k=1..n} k/5. - Alonso del Arte, Jan 20 2012 These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 356 on p. 284. See the G.f. of A113428. - Wolfdieter Lang, Oct 28 2016 REFERENCES G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284. LINKS G. C. Greubel, Table of n, a(n) for n = 1..5000 Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1). FORMULA A005475 UNION A005476. G.f.: x^2*(2x^2+x+2)/((1-x)^3*(1+x)^2). a(n) = A132356(n+1)/4. - R. J. Mathar, Apr 07 2008 a(n) = (A090771(n)^2 -1)/40. - Gary Detlefs, Feb 22 2010 |A113428(n)| is the characteristic function of the numbers a(n). a(n) = a(1 - n) for all n in Z. - Michael Somos, Jan 13 2012 From Colin Barker, Jun 13 2017: (Start) a(n) = n*(5*n - 2)/8 for n even. a(n) = (5*n - 3)*(n - 1)/8 for n odd. a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5. (End) From Amiram Eldar, Mar 17 2022: (Start) Sum_{n>=2} 1/a(n) = 10 - 2*sqrt(1+2/sqrt(5))*Pi. Sum_{n>=2} (-1)^n/a(n) = 2*sqrt(5)*log(phi) - 5*(2-log(5)), where phi is the golden ratio (A001622). (End) MATHEMATICA Select[Table[Plus@@Range[n]/5, {n, 0, 199}], IntegerQ] (* Alonso del Arte, Jan 20 2012 *) LinearRecurrence[{1, 2, -2, -1, 1}, {0, 2, 3, 9, 11}, 50] (* Harvey P. Dale, Jul 05 2021 *) PROG (PARI) {a(n) = (10 * (n^2 - n) + 12 * (-1)^n * (n\2)) / 16}; \\ Michael Somos, Jan 13 2012 (PARI) Vec(x^2*(2*x^2+x+2) / ((1-x)^3*(1+x)^2) + O(x^60)) \\ Colin Barker, Jun 13 2017 (Magma) [(10*(n^2-n)+12*(-1)^n*(n div 2))/16: n in [1..60]]; // Vincenzo Librandi, Oct 29 2016 CROSSREFS Cf. A074378, A001318, A057570, A154260. - Vladimir Joseph Stephan Orlovsky, Jan 06 2009 Cf. A001622, A113428. Sequence in context: A257027 A271548 A110350 * A177950 A049618 A057292 Adjacent sequences: A057566 A057567 A057568 * A057570 A057571 A057572 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Oct 04 2000 STATUS approved

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Last modified December 3 09:50 EST 2022. Contains 358517 sequences. (Running on oeis4.)