A105046 a(n) = 1298*a(n-3) - a(n-6) - 648, for n>6, with a(0)=0, a(1)=1, a(2)=8, a(3)=145, a(4)=505, a(5)=9728, a(6)=187561.

0, 1, 8, 145, 505, 9728, 187561, 654841, 12626288, 243453385, 849982465, 16388911448, 316002305521, 1103276584081, 21272794432568, 410170749112225, 1432052156154025, 27612070784561168, 532401316345361881, 1858802595411339721, 35840446605565962848

Offset: 0

Author

Gerald McGarvey, Apr 03 2005

Keywords

• nonn

Comments

It appears this sequence gives all nonnegative m such that 13*m^2 - 13*m + 1 is a square and that a(n+1) = A104240(n) + 1. (A104240 is nonnegative n such that 13*n^2 + 13*n + 1 is a square.)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..950
• Index entries for linear recurrences with constant coefficients, signature (1,0,1298,-1298,0,-1,1).

Crossrefs

Cf. A104240.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+7*x+137*x^2-938*x^3+137*x^4+7*x^5+x^6) / ((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)) )); // G. C. Greubel, Mar 14 2023

Mathematica

CoefficientList[Series[x (1+7x+137x^2-938x^3+137x^4+7x^5+x^6)/((1-x) (1-11x+x^2)(1+11x+120x^2+11x^3+x^4)),{x,0,30}],x] (* or *)  LinearRecurrence[{1,0,1298,-1298,0,-1,1},{0,1,8,145,505,9728,187561, 654841},30] (* Harvey P. Dale, Jun 12 2012 *)

SageMath

@CachedFunction
def a(n): # a = A105046
    if (n<8): return (0,1,8,145,505,9728,187561,654841)[n]
    else: return a(n-1) +1298*a(n-3) -1298*a(n-4) -a(n-6) +a(n-7)
[a(n) for n in range(41)] # G. C. Greubel, Mar 14 2023

Formula

a(n) = 1298*a(n-3) - a(n-6) - 648 for n > 6.

G.f.: x*(1+7*x+137*x^2-938*x^3+137*x^4+7*x^5+x^6) / ((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)). - R. J. Mathar, Sep 09 2008

Extensions

More terms from Harvey P. Dale, Jun 12 2012