A002537 a(2n) = a(2n-1) + 3a(2n-2), a(2n+1) = 2a(2n) + 3a(2n-1). (Formerly M3409 N1379)

1, 1, 4, 11, 23, 79, 148, 533, 977, 3553, 6484, 23627, 43079, 157039, 286276, 1043669, 1902497, 6936001, 12643492, 46094987, 84025463, 306335887, 558412276, 2035832213, 3711069041, 13529634721, 24662841844, 89914587851

Offset

0,3

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Links

Table of n, a(n) for n=0..27.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]

Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -9).

Formula

a(n)=8a(n-2)-9a(n-4). - Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

G.f.: (1+x-4x^2+3x^3)/(1-8x^2+9x^4). a(n)/A002536(n) converges to sqrt(7). - Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

a(n+1) = x^n + (-1)^n*(x-2)^n where x = (1+sqrt(7)) and the term is divided by 2 for a(2) and a(3), 4 for a(4) and a(5)... 2^n for a(2n) and a(2n+1). - Ben Paul Thurston, Aug 30 2006

Maple

A002537:=(1+z-4*z**2+3*z**3)/(1-8*z**2+9*z**4); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{0, 8, 0, -9}, {1, 1, 4, 11}, 40] (* Harvey P. Dale, Jul 24 2012 *)

Crossrefs

Sequence in context: A182707 A022495 A238489 * A295001 A230150 A301109

Adjacent sequences: A002534 A002535 A002536 * A002538 A002539 A002540

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Sep 08 2000

Status

approved