A290075 Number of monomials in c(n) where c(1) = x, c(2) = y, c(n+2) = c(n+1) + c(n)^2.
1, 1, 2, 3, 5, 8, 14, 24, 44, 80, 152, 288, 560, 1088, 2144, 4224, 8384, 16640, 33152, 66048, 131840, 263168, 525824, 1050624, 2100224, 4198400, 8394752, 16785408, 33566720, 67125248, 134242304, 268468224, 536920064, 1073807360, 2147581952, 4295098368
Offset: 1
Examples
G.f. = x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 14*x^7 + 24*x^8 + 44*x^9 + ... c(3) = x^2 + y so a(3) = 2, c(4) = x^2 + (y + y^2) so a(4) = 3, c(5) = x^4 + x^2(2*y) + (y + 2*y^2) so a(5) = 5.
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Kai Williams, Number of monomials in a_n = a_{n-1} + (a_{n-2})^2 with a_1 = a, a_2 = b, Math StackExchange question 2363374, Jul 18 2017.
- Index entries for linear recurrences with constant coefficients, signature (2,2,-4).
Programs
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Mathematica
nn:=36; nn:=10; Rest[CoefficientList[Series[(x - x^2 - 2*x^3 + x^4 - x^5) / ((1 - 2*x) * (1 - 2*x^2)),{x, 0, nn}], x]] (* Georg Fischer, May 10 2020 *) -
PARI
{a(n) = if( n<3, n>0, my(e=n%2, m=2^((n+e)/2-2)); m * (m+2+e) / (1+e))}; -
PARI
Vec(x*(1 - x - 2*x^2 + x^3 - x^4) / ((1 - 2*x)*(1 - 2*x^2)) + O(x^50)) \\ Colin Barker, Jul 22 2017
Formula
G.f.: (x - x^2 - 2*x^3 + x^4 - x^5) / ((1 - 2*x) * (1 - 2*x^2)).
0 = 4*a(n) - 2*a(n+1) - 2*a(n+2) + 1*a(n+3) for n>=3.
From Colin Barker, Jul 22 2017: (Start)
a(n) = 2^(n/2-1) + 2^(n-4) for n>2 and even.
a(n) = 3*2^((n-5)/2) + 2^(n-4) for n>2 and odd.
(End)
Given the sequence c(n, x, y), then the coefficients of: (1) c(n+2, sqrt(t), 0), (2) c(n+1, 0, t), and (3) c(n, t, t), each form the triangular sequence A103484. - Michael Somos, Jul 24 2017