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A181879 Expansion of x*(1+x)/(1-3*x-4*x^2-x^3). 5
0, 1, 4, 16, 65, 263, 1065, 4312, 17459, 70690, 286218, 1158873, 4692181, 18998253, 76922356, 311452261, 1261044460, 5105864780, 20673224441, 83704176903, 338911293253, 1372223811812, 5556020785351, 22495868896554, 91083913642878, 368791237300201, 1493205235368669, 6045864568949689, 24479205885623944, 99114281168039257, 401305531615563236 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) appears in the following formula for the nonnegative powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7)= rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field, called there Q(rho). (rho*sigma)^n = C(n) + B(n)*rho + a(n)*sigma,n>=0, with C(n)= A120757(n) with C(0):=1, and B(n)= |A122600(n-1)| with B(0)=1. For the nonpositive powers see A085810(n)*(-1)^n, A181880(n-2)*(-1)^n and A116423(n+1)*(-1)^(n+1), respectively. See also a comment under A052547.

REFERENCES

P. Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, 70,1 (1997) 22-31.

LINKS

Table of n, a(n) for n=0..30.

FORMULA

a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), n>=2, a(-1):=1, a(0)=0, a(1)=1.

PROG

(PARI) Vec((1+x)/(1-3*x-4*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

CROSSREFS

Sequence in context: A013149 A175065 A033140 * A243872 A052927 A012781

Adjacent sequences:  A181876 A181877 A181878 * A181880 A181881 A181882

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 26 2010

STATUS

approved

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Last modified July 22 11:25 EDT 2014. Contains 244815 sequences.