A084058 a(n) = 2*a(n-1) + 7*a(n-2) for n>1, a(0)=1, a(1)=1.

1, 1, 9, 25, 113, 401, 1593, 5993, 23137, 88225, 338409, 1294393, 4957649, 18976049, 72655641, 278143625, 1064876737, 4076758849, 15607654857, 59752621657, 228758827313, 875786006225, 3352883803641, 12836269650857, 49142725927201

Offset

0,3

Comments

Binomial transform of expansion of cosh(sqrt(8)x) (A001018 with interpolated zeros : 1, 0, 8, 0, 64, 0, 512, 0, ...); inverse binomial transform of A084128.

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators 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

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,7).

Formula

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

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

E.g.f.: exp(x) * cosh(sqrt(8)*x).

a(n) = Sum_{k=0..n} A098158(n,k)*8^(n-k). - Philippe Deléham, Dec 26 2007

G.f.: G(0)/2, 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

Satisfies recurrence relation system a(n) = 2*b(n-1) - a(n-1), b(n) = 3*b(n-1) + 2*a(n-1), a(0)=1, b(0)=1. - Ilya Gutkovskiy, Apr 11 2017

Mathematica

a[n_]:= Simplify[((1 + Sqrt[8])^n + (1 - Sqrt[8])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-7x^2), {x, 0, 30}], x] (* Or *) LinearRecurrence[{2, 7}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r8>:=NumberField(x^2-8); S:=[ ((1+r8)^n+(1-r8)^n)/2: n in [0..30] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 16 2008
(Magma) I:=[1, 1]; [n le 2 select I[n] else 2*Self(n-1) +7*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019
(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-2*x-7*x^2)) \\ G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-2*x-7*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 1];; for n in [3..30] do a[n]:=2*a[n-1]+7*a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Essentially a duplicate of A083100.

Sequence in context: A083672 A193644 A083100 * A108570 A092769 A263951

Adjacent sequences: A084055 A084056 A084057 * A084059 A084060 A084061

Keyword

easy,nonn

Author

Paul Barry, May 10 2003

Status

approved