A015519 a(n) = 2*a(n-1) + 7*a(n-2), with a(0) = 0, a(1) = 1.

0, 1, 2, 11, 36, 149, 550, 2143, 8136, 31273, 119498, 457907, 1752300, 6709949, 25685998, 98341639, 376485264, 1441362001, 5518120850, 21125775707, 80878397364, 309637224677, 1185423230902, 4538307034543, 17374576685400

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard, Sep 25 2005

Pisano period lengths: 1, 2, 8, 4, 24, 8, 3, 8, 24, 24, 15, 8, 168, 6, 24, 16, 16, 24, 120, 24, ... . - R. J. Mathar, Aug 10 2012

References

• John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,7).

Crossrefs

The following sequences (and others) belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A063727, A083098, A083099, A083100, A084057.

Programs

Magma

[ n eq 1 select 0 else n eq 2 select 1 else 2*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

Mathematica

LinearRecurrence[{2,7},{0,1},30] (* Harvey P. Dale, Oct 09 2017 *)

PARI

a(n)=([0,1; 7,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

Sage

[lucas_number1(n,2,-7) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

Formula

From Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003: (Start)

a(n) = a(n-1) + A083100(n-2), n>1.

A083100(n)/a(n+1) converges to sqrt(8). (End)

From Paul Barry, Jul 17 2003: (Start)

G.f.: x/ ( 1-2*x-7*x^2 ).

a(n) = ((1+2*sqrt(2))^n-(1-2*sqrt(2))^n)*sqrt(2)/8. (End)

E.g.f.: exp(x)*sinh(2*sqrt(2)*x)/(2*sqrt(2)). - Paul Barry, Nov 20 2003

Second binomial transform is A000129(2n)/2 (A001109). - Paul Barry, Apr 21 2004

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*(7/2)^k*2^(n-k-1). - Paul Barry, Jul 17 2004

a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*8^k. - Paul Barry, Sep 29 2004

G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013