A274203 Expansion of x*(1 - x - x^3)/((1 - x)*(1 - 2*x - 3*x^2 - 2*x^3 - x^4)).

0, 1, 2, 7, 21, 67, 212, 673, 2136, 6781, 21527, 68341, 216960, 688777, 2186642, 6941875, 22038189, 69964063, 222113084, 705136609, 2238578784, 7106757625, 22561637903, 71625842857, 227388693456, 721884948913, 2291749301810, 7275556680127, 23097519856965, 73327093306843, 232789608846644

Offset: 0

Author

Ilya Gutkovskiy, Jun 13 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-1,-1)

Crossrefs

Cf. A000045, A003151, A014176, A190179, A272362.

Programs

Mathematica

LinearRecurrence[{3, 1, -1, -1, -1}, {0, 1, 2, 7, 21}, 31]
RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == Floor[(Sqrt[2] + 1) a[n - 1] + (Sqrt[2] + 1) a[n - 2]]}, a, {n, 30}]

PARI

concat(0, Vec(x*(1-x-x^3)/((1-x)*(1-2*x-3*x^2-2*x^3-x^4)) + O(x^99))) \\ Altug Alkan, Jun 26 2016

Formula

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

a(n) = 3*a(n-1) + a(n-2) - a(n-3) - a(n-4) - a(n-5).

a(n) = floor((1 + sqrt(2))*a(n-1) + (1 + sqrt(2))*a(n-2)), a(0)=0, a(1)=1 (empirically).

Lim_{n->infinity} a(n)/a(n+1) = sqrt(sqrt(2) - sqrt(sqrt(2) + sqrt(sqrt(2) - sqrt(sqrt(2) + ...)))) = (sqrt(4*sqrt(2) - 3) - 1)/2 = A190179 - 1.