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A098111 Inverse binomial transform of A098149. 1
1, 0, -5, -25, -100, -375, -1375, -5000, -18125, -65625, -237500, -859375, -3109375, -11250000, -40703125, -147265625, -532812500, -1927734375, -6974609375, -25234375000, -91298828125, -330322265625, -1195117187500, -4323974609375, -15644287109375, -56601562500000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A030191(n) + 2a(n) + A093129(n+2) = 4 A093129(n+1). - Creighton Dement, Oct 18 2004

From Wolfdieter Lang, Oct 02 2013: (Start)

These numbers a(n) and those of A030191(n) =: b(n), both interspersed with zeros, appear in the formula for nonnegative powers of the algebraic number rho(10) := 2*cos(pi/10) = phi*sqrt(3-phi), with the golden section phi, in terms of the power basis of the number field Q(rho(10)) of degree 4 (see A187360, n=10). In a (regular) decagon rho(10) is the length ratio of a smallest diagonal to the side. rho(10)^n = sum(A(n,k)*rho(10)^k, k=0..3), with A(2*k+1,0) = 0, A(2*k,0) = a(k), k >= 0; A(2*k,1) = 0, A(2*k+1,1) = a(k), k >= 0; A(2*k+1,2) = 0, k >= 0, A(0,2) = 0, A(2*k,2) = b(k-1), k >= 1; and A(2*k,3) = 0, k >= 0, A(1,3) = 0, A(2*k+1,3) = b(k-1), k >= 1. (End)

LINKS

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (5,-5).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

G.f.: (1-5x)/(1-5x+5x^2).

From - Wolfdieter Lang, Oct 02 2013 (Start)

a(n) = b(n) - 5*b(n-1), n >= 0,  with b(n) = A030191(n) = (sqrt(5))^n*S(n, sqrt(5)), with Chebyshev S-polynomials (see A949130).

a(n) = 5*(a(n-1) - a(n-2)), n >= 1, a(-1) = 1 = a(0). (End)

EXAMPLE

Powers of rho(10) in the Q(rho(10)) power basis for n = 5: rho(10)^5 = 0*1 + a(2)*rho(10) + 0*rho(10)^2 + b(1)*rho(10)^3 = -5*rho(10) + 5*rho(10)^3.  - Wolfdieter Lang, Oct 02 2013

MATHEMATICA

LinearRecurrence[{5, -5}, {1, 0}, 40] (* Harvey P. Dale, Dec 08 2015 *)

PROG

Floretion Algebra Multiplication Program, FAMP

CROSSREFS

Sequence in context: A146830 A255612 A022729 * A224415 A255459 A083877

Adjacent sequences:  A098108 A098109 A098110 * A098112 A098113 A098114

KEYWORD

easy,sign

AUTHOR

Creighton Dement, Sep 23 2004

EXTENSIONS

More terms from David Wasserman, Jan 16 2008

STATUS

approved

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Last modified March 24 05:55 EDT 2017. Contains 283984 sequences.