A121338 Pentagonal numbers P(k) that are one-third of other pentagonal numbers: P(k) such that 3*P(k)=P(m) for some m>k.

70, 511258935, 3732600255368600, 27250975409595074561065, 198953975772318806945317308330, 1452523584226469439408576900215922395, 10604587088767577582197244731443261336155260, 77421990626847055423676582260371016672624778798925

Offset: 1

Author

Franz Vrabec, Aug 28 2006

Keywords

• nonn
• easy

Comments

The k values are (A001835(6n-2)+1)/6, the m values are (A001834(6n-3)+1)/6.

Examples

			a(1) = ((A001835(4))^2-1)/24 = (41^2-1)/24 = 70; this number and 3*70=210 are pentagonal numbers (in A000326).
		

Links

• Colin Barker, Table of n, a(n) for n = 1..146
• Index entries for linear recurrences with constant coefficients, signature (7300803,-7300803,1).

Crossrefs

Cf. A001834, A001835.

Programs

Mathematica

CoefficientList[Series[5 (x^2 + 40545 x + 14)/((1 - x) (x^2 - 7300802 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 21 2015 *)

PARI

Vec(-5*x*(x^2+40545*x+14)/((x-1)*(x^2-7300802*x+1)) + O(x^20)) \\ Colin Barker, Jun 20 2015

Formula

a(n) = ((A001835(6n-2))^2-1)/24.

a(n) = 7300803*a(n-1)-7300803*a(n-2)+a(n-3). - Colin Barker, Jun 20 2015

G.f.: -5*x*(x^2+40545*x+14) / ((x-1)*(x^2-7300802*x+1)). - Colin Barker, Jun 20 2015

Extensions

Added more terms, Colin Barker, Jun 20 2015