A231181 Expansion of 1/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).

1, 1, 5, 6, 20, 27, 75, 110, 275, 429, 1001, 1637, 3639, 6172, 13243, 23104, 48280, 86090, 176341, 319792, 645150, 1185305, 2363596, 4386331, 8669142, 16212913, 31825005, 59873834, 116914020, 220964744, 429737220, 815057639, 1580244061

Offset: 0

Author

Wolfdieter Lang, Nov 05 2013

Keywords

• nonn
• easy

Comments

This sequence is fundamental for the coefficient sequences for the nonnegative powers of rho(11) = 2*cos(Pi/n) (length ration (smallest diagonal)/side in the regular 11-gon (Hendecagon)) when written in the power basis of the degree 5 number field Q(rho(11)). See A187360 for the minimal polynomial of rho(11) which is C(11, x) = x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1. See A231182-5 for these coefficient sequences.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3532
• Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
• Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

Crossrefs

Cf. A231182, A231183, A231184, A231185.

Programs

Mathematica

CoefficientList[Series[1/(1-x-4x^2+3x^3+3x^4-x^5),{x,0,50}],x] (* or *) LinearRecurrence[{1,4,-3,-3,1},{1,1,5,6,20},50] (* Harvey P. Dale, Nov 13 2013 *)

Formula

G.f.: 1/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).

a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n>=0, with a(-5)=1, a(-4)=a(-3)=a(-2)=a(-1)=0.