A137693 Numbers n such that 3n^2-n = 6k^2-2k for some integer k>0.

7, 7887, 9101399, 10503006367, 12120460245927, 13987000620793199, 16140986595935105527, 18626684544708490984767, 21495177823607002661315399, 24805416581757936362666985487, 28625429240170834955515039936407, 33033720537740561780727993419627999

Offset: 1

Author

M. F. Hasler, Feb 08 2008

Keywords

• easy
• nonn

Comments

Also indices of pentagonal numbers which are twice some other pentagonal number.

Note that A000326(n) = 2 A000326(k) <=> n(3n-1)=2k(3k-1), which is easily solved by standard Pell-type techniques (cf. link to D. Alpern's quadratic solver). Here we consider only positive solutions.

Inspired by a recent comment on A000326 by R. J. Mathar.

Links

• Colin Barker, Table of n, a(n) for n = 1..250
• Dario Alpern, Quadratic two integer variable equation solver
• Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

Crossrefs

Cf. A000326, A136112-A136118, A135768-A135769, A135771, A137694.

Programs

Mathematica

CoefficientList[Series[x (-7+198x+x^2)/((x-1)(x^2-1154x+1)),{x,0,20}],x] (* or *) Join[{0},LinearRecurrence[{1155,-1155,1},{7,7887,9101399}, 20]] (* Harvey P. Dale, Jun 21 2011 *)

PARI

vector(20,i, (v=if(i>1,[577,408; 816,577]*v-[164;232], [5;7]))[2,1])

Formula

a(n) = f^{2n-2}(5,7)[2], where f(x,y) = (577x + 408y - 164, 816x + 577y - 232)

a(n) = (7,7,9,7,7,9,...) mod 10

G.f. x*(-7+198*x+x^2) / ( (x-1)*(x^2-1154*x+1) ). - R. J. Mathar, Apr 17 2011

a(0)=0, a(1)=7, a(2)=7887, a(3)=9101399, a(n)=1155*a(n-1)-1155*a(n-2)+ a(n-3). - Harvey P. Dale, Jun 21 2011

Extensions

More terms from Harvey P. Dale, Jun 21 2011