A098111 Inverse binomial transform of A098149.

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

Offset

0,3

Comments

A030191(n) + 2*a(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 A049310).

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 *)

Crossrefs

Cf. A030191, A049310, A098149, A187360.

Sequence in context: A316778 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