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A084058 a(n) = 2*a(n-1)+7*a(n-2) for n>1, a(0)=1, a(1)=1. 10
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..24.

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

FORMULA

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

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

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

a(n) = Sum_{k, 0<=k<=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, 25, 0] (* Or *) CoefficientList[Series[(1 + 7 x)/(1 - 2 x - 7 x^2), {x, 0, 23}], x] (* Or *) LinearRecurrence[{2, 7}, {1, 1}, 28] (* 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..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 16 2008

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

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Last modified May 23 19:07 EDT 2019. Contains 323528 sequences. (Running on oeis4.)