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A221364 The simple continued fraction expansion of F(x) := Product_{n >= 0} (1 - x^(4*n+3))/(1 - x^(4*n+1)) when x = 1/2*(3 - sqrt(5)).

%I A221364 #23 Feb 21 2025 06:16:35
%S A221364 1,1,1,5,1,16,1,45,1,121,1,320,1,841,1,2205,1,5776,1,15125,1,39601,1,
%T A221364 103680,1,271441,1,710645,1,1860496,1,4870845,1,12752041,1,33385280,1,
%U A221364 87403801,1,228826125,1,599074576,1,1568397605,1,4106118241,1,10749957120,1,28143753121
%N A221364 The simple continued fraction expansion of F(x) := Product_{n >= 0} (1 - x^(4*n+3))/(1 - x^(4*n+1)) when x = 1/2*(3 - sqrt(5)).
%C A221364 The function F(x) := Product_{n >= 0} (1 - x^(4*n+3))/(1 - x^(4*n+1)) is analytic for |x| < 1. When x is a quadratic irrational of the form x = 1/2*(N - sqrt(N^2 - 4)), N an integer greater than 2, the real number F(x) has a predictable simple continued fraction expansion. The first examples of these expansions, for N = 2, 4, 6 and 8, are due to Hanna. See A174500 through A175503. The present sequence is the case N = 3. See also A221365 (N = 5), A221366 (N = 7), A221369 (N = 9).
%C A221364 If we denote the present sequence by [1, c(1), 1, c(2), 1, c(3), ...] then for k = 1, 2, ..., the simple continued fraction expansion of F((1/2*(3 - sqrt(5)))^k) is given by the sequence [1; c(k), 1, c(2*k), 1, c(3*k), 1, ...]. Examples are given below.
%H A221364 Peter Bala, <a href="/A174500/a174500_2.pdf">Some simple continued fraction expansions for an infinite product, Part 1</a>
%H A221364 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-4,0,1).
%F A221364 a(2*n-1) = (1/2*(3 + sqrt(5)))^n + (1/2*(3 - sqrt(5)))^n - 2 = A004146(n); a(2*n) = 1.
%F A221364 a(4*n+1) = A081071(n) = A002878(n)^2;
%F A221364 a(4*n-1) = A081070(n) = 5*A049684(n) = 5*(A001906(n))^2.
%F A221364 a(n) = 4*a(n-2)-4*a(n-4)+a(n-6). G.f.: -(x^4+x^3-3*x^2+x+1) / ((x-1)*(x+1)*(x^2-x-1)*(x^2+x-1)). - _Colin Barker_, Jan 20 2013
%e A221364 F(1/2*(3 - sqrt(5))) = 1.53879 34992 88095 08323 ... = 1 + 1/(1 + 1/(1 + 1/(5 + 1/(1 + 1/(16 + 1/(1 + 1/(45 + ...))))))).
%e A221364 F((1/2*(3 - sqrt(5)))^2) = 1.16725 98258 10214 95210 ... = 1 + 1/(5 + 1/(1 + 1/(45 + 1/(1 + 1/(320 + 1/(1 + 1/(2205 + ...))))))).
%e A221364 F((1/2*(3 - sqrt(5)))^3) = 1.05883 42773 67371 19975 ... = 1 + 1/(16 + 1/(1 + 1/(320 + 1/(1 + 1/(5776 + 1/(1 + 1/(103680 + ...))))))).
%Y A221364 Cf. A001906, A002878, A004146, A049684, A081070, A081071, A174500 (N = 4), A221365 (N = 5), A221366 (N = 7), A221369 (N = 9).
%K A221364 nonn,easy,cofr
%O A221364 0,4
%A A221364 _Peter Bala_, Jan 15 2013
%E A221364 More terms from _Michel Marcus_, Feb 21 2025