A057569 Numbers of the form k*(5*k+1)/2 or k*(5*k-1)/2.

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

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: A271548 A369338 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