cp's OEIS Frontend

A265289 Decimal expansion of Sum_{n>=1} (c(2*n) - phi), where phi is the golden ratio (A001622) and c = convergents to phi.

%I A265289 #24 Aug 23 2022 09:52:25
%S A265289 4,3,8,7,5,1,4,1,0,9,7,1,5,0,6,2,5,7,3,5,5,6,4,9,5,3,9,3,4,7,5,2,7,1,
%T A265289 9,0,1,6,9,6,6,4,1,9,3,4,2,5,9,2,0,0,6,7,1,9,4,1,3,7,2,8,5,1,5,0,3,7,
%U A265289 2,1,9,5,3,9,9,5,9,3,2,4,5,5,0,7,4,5
%N A265289 Decimal expansion of Sum_{n>=1} (c(2*n) - phi), where phi is the golden ratio (A001622) and c = convergents to phi.
%C A265289 Define the upper deviance of x > 0 by dU(x) = Sum_{n>=1} (c(2*n,x) - x), where c(k,x) = k-th convergent to x.  The greatest upper deviance occurs when x = golden ratio, so that this constant is the absolute maximal upper deviance.
%F A265289 Equals Sum_{k>=1} 1/(phi^(2*k) * F(2*k)), where F(k) is the k-th Fibonacci number (A000045). - _Amiram Eldar_, Oct 05 2020
%F A265289 Equals sqrt(5)*Sum_{k >= 1} x^(k^2)*(1 + x^k)/(1 - x^k), where x = (7 - 3*sqrt(5))/2. - _Peter Bala_, Aug 21 2022
%e A265289 0.4387514109715062573556495393475271901...
%p A265289 x := (7 - 3*sqrt(5))/2:
%p A265289 evalf(sqrt(5)*add(x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..12), 100); # _Peter Bala_, Aug 21 2022
%t A265289 x = GoldenRatio; z = 600; c = Convergents[x, z];
%t A265289 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265289 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265289 N[s1 + s2, 200]
%t A265289 RealDigits[s1, 10, 120][[1]]  (* A265288 *)
%t A265289 RealDigits[s2, 10, 120][[1]]  (* A265289 *)
%t A265289 RealDigits[s1 + s2, 10, 120][[1]] (* A265290 *)
%Y A265289 Cf. A000005, A000045, A001622, A265288, A265290.
%K A265289 nonn,cons
%O A265289 0,1
%A A265289 _Clark Kimberling_, Dec 06 2015