%I #17 Feb 13 2024 03:21:39

%S 1,1,1,5,1,16,1,45,1,121,1,320,1,841,1,2205,1,5776,1,15125,1,39601,1,

%T 103680,1,271441,1,710645,1,1860496,1,4870845,1,12752041,1,33385280,1,

%U 87403801,1,228826125,1

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

%C The function F(x) := product {n = 0..inf} (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 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 Peter Bala, <a href="/A174500/a174500_2.pdf">Some simple continued fraction expansions for an infinite product, Part 1</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-4,0,1).

%F a(2*n-1) = (1/2*(3 + sqrt(5)))^n + (1/2*(3 - sqrt(5)))^n - 2 = A004146(n); a(2*n) = 1.

%F a(4n+1) = A081071(n) = A002878(n)^2;

%F a(4*n-1) = A081070(n) = 5*A049684(n) = 5*(A001906(n))^2.

%F 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 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 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 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 Cf. A001906, A002878, A004146, A049684, A081070, A081071, A174500 (N = 4), A221365 (N = 5), A221366 (N = 7), A221369 (N = 9).

%K nonn,easy,cofr

%O 0,4

%A _Peter Bala_, Jan 15 2013