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A290075 Number of monomials in c(n) where c(1) = x, c(2) = y, c(n+2) = c(n+1) + c(n)^2.

%I A290075 #23 May 10 2020 09:18:06
%S A290075 1,1,2,3,5,8,14,24,44,80,152,288,560,1088,2144,4224,8384,16640,33152,
%T A290075 66048,131840,263168,525824,1050624,2100224,4198400,8394752,16785408,
%U A290075 33566720,67125248,134242304,268468224,536920064,1073807360,2147581952,4295098368
%N A290075 Number of monomials in c(n) where c(1) = x, c(2) = y, c(n+2) = c(n+1) + c(n)^2.
%H A290075 Colin Barker, <a href="/A290075/b290075.txt">Table of n, a(n) for n = 1..1000</a>
%H A290075 Kai Williams, <a href="http://math.stackexchange.com/questions/2363374">Number of monomials in a_n = a_{n-1} + (a_{n-2})^2 with a_1 = a, a_2 = b</a>, Math StackExchange question 2363374, Jul 18 2017.
%H A290075 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4).
%F A290075 G.f.: (x - x^2 - 2*x^3 + x^4 - x^5) / ((1 - 2*x) * (1 - 2*x^2)).
%F A290075 0 = 4*a(n) - 2*a(n+1) - 2*a(n+2) + 1*a(n+3) for n>=3.
%F A290075 A088966(n) = a(2*n+2). A257273(n) = a(2*n+3). A211525(n) = a(n+8).
%F A290075 From _Colin Barker_, Jul 22 2017: (Start)
%F A290075 a(n) = 2^(n/2-1) + 2^(n-4) for n>2 and even.
%F A290075 a(n) = 3*2^((n-5)/2) + 2^(n-4) for n>2 and odd.
%F A290075 (End)
%F A290075 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
%e A290075 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 + ...
%e A290075 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.
%t A290075 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 *)
%o A290075 (PARI) {a(n) = if( n<3, n>0, my(e=n%2, m=2^((n+e)/2-2)); m * (m+2+e) / (1+e))};
%o A290075 (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
%Y A290075 Cf. A088966, A103484, A211525, A257273.
%K A290075 nonn,easy
%O A290075 1,3
%A A290075 _Michael Somos_, Jul 19 2017