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 A181880 Expansion of 1/(1-4*x-3*x^2-x^3). 5
 1, 4, 19, 89, 417, 1954, 9156, 42903, 201034, 942001, 4414009, 20683073, 96916320, 454128508, 2127946065, 9971086104, 46722311119, 218930448853, 1025859814873, 4806952917170, 22524321562152, 105544004814991, 494555936863590, 2317380083461485, 10858732149251701, 50881624784254849, 238420075668235984, 1117183909174960184, 5234877488488803537, 24529481757148330684 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS B(n):=a(n-2)*(-1)^n, B(0):=0, B(1):=0, (o.g.f. x^2/(1 + 4*x + 3*x^2 -x^3))appears in the following formula for the nonpositive 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. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with C(n)= A085810(n)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547. LINKS Table of n, a(n) for n=0..29. P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31. Index entries for linear recurrences with constant coefficients, signature (4, 3, 1). FORMULA O.g.f.: 1/(1-4*x-3*x^2-x^3). a(n) = 4*a(n) + 3*a(n-2) +a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=4. MATHEMATICA CoefficientList[Series[1/(1-4*x-3*x^2-x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{4, 3, 1}, {1, 4, 19}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *) CROSSREFS Sequence in context: A084155 A015530 A256959 * A291016 A010907 A229242 Adjacent sequences: A181877 A181878 A181879 * A181881 A181882 A181883 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 27 2010 STATUS approved

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Last modified December 2 19:08 EST 2023. Contains 367525 sequences. (Running on oeis4.)