A053430 a(n) = (6^(n+1) - (-5)^(n+1))/11.

1, 1, 31, 61, 991, 2821, 32551, 117181, 1093711, 4609141, 37420471, 175694701, 1298308831, 6569149861, 45518414791, 242592910621, 1608145354351, 8885932672981, 57130293303511, 323708273492941, 2037617072598271

Offset: 0

Author

Barry E. Williams, Jan 10 2000

Keywords

• easy
• nonn

Comments

Hankel transform is := 1,30,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
• Index entries for linear recurrences with constant coefficients, signature (1,30).

Crossrefs

Cf. A001045, A015441, A053404.

Programs

Magma

[(6^(n+1)-(-5)^(n+1))/11: n in [0..30]]; // Vincenzo Librandi, Mar 11 2014

Maple

A053430:=n->( 6^(n+1)-(-5)^(n+1) )/11; seq(A053430(n), n=0..20); # Wesley Ivan Hurt, Mar 11 2014

Mathematica

Join[{a=1,b=1},Table[c=b+30*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
Table[(6^(n+1)-(-5)^(n+1))/11,{n,0,20}] (* Harvey P. Dale, May 09 2012 *)
LinearRecurrence[{1,30},{1,1},21] (* Harvey P. Dale, May 09 2012 *)
CoefficientList[Series[-1/(5 x + 1)/(6 x - 1), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 11 2014 *)

PARI

a(n) = ( 6^(n+1)-(-5)^(n+1) )/11; \\ Joerg Arndt, Mar 10 2014

Formula

G.f.: -1/(5*x+1)/(6*x-1). - R. J. Mathar, Nov 16 2007

a(0)=1, a(1)=1, a(n) = a(n-1) + 30*a(n-2). - Harvey P. Dale, May 09 2012

Extensions

More terms from James Sellers, Feb 02 2000