A272362 Expansion of (1 + x - x^2 - x^3 - x^4)/((1 - x)*(1 - x - 2*x^2 - 2*x^3 - x^4)).

1, 3, 6, 14, 32, 74, 171, 396, 917, 2124, 4920, 11397, 26401, 61158, 141673, 328187, 760249, 1761126, 4079670, 9450606, 21892446, 50714123, 117479896, 272143639, 630424122, 1460385314, 3383000731, 7836763241, 18153959452, 42053872709, 97418318825, 225670746387, 522769088906, 1211001092038

Offset: 0

Author

Ilya Gutkovskiy, Apr 27 2016

Keywords

• nonn
• easy

Comments

Partial sums of A272642. - Wolfdieter Lang, May 06 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,0,-1,-1).

Crossrefs

Cf. A000045, A000201, A001622, A189970, A190157, A272642.

Programs

Magma

I:=[1,3,6,14,32]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5): n in [1..30]]; // Vincenzo Librandi, May 08 2016

Mathematica

LinearRecurrence[{2, 1, 0, -1, -1}, {1, 3, 6, 14, 32}, 34]
RecurrenceTable[{a[n] == Floor[GoldenRatio a[n - 1] + GoldenRatio a[n - 2]], a[0] == 1, a[1] == 3}, a, {n, 33}]
CoefficientList[Series[(1 + x - x^2 - x^3 - x^4)/((1 - x) (1 - x - 2 x^2 - 2 x^3 - x^4)), {x, 0, 50}], x] (* Vincenzo Librandi, May 08 2016 *)

PARI

Vec((1+x-x^2-x^3-x^4)/(1-2*x-x^2+x^4+x^5) + O(x^99)) \\ Altug Alkan, Apr 27 2016

Formula

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

a(n) = 2*a(n-1) + a(n-2) - a(n-4) - a(n-5).

a(n) = floor(phi*a(n-1) + phi*a(n-2)), a(0)=1, a(1)=3, where phi is the golden ratio (A001622).

Limit_{n->infinity} a(n)/a(n-1) = 2/(sqrt(2*sqrt(5)-1) - 1) = sqrt(phi + phi*sqrt(phi + phi*sqrt(phi + ...))) = A189970.

Limit_{n->infinity} a(n-1)/a(n) = (sqrt(2*sqrt(5)-1) - 1)/2 = 1 + A190157.