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

Author

N. J. A. Sloane, Oct 04 2000

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A074378, A001318, A057570, A154260. - Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Cf. A001622, A113428.

Programs

Magma

[(10*(n^2-n)+12*(-1)^n*(n div 2))/16: n in [1..60]]; // Vincenzo Librandi, Oct 29 2016

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 *)

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

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)