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A265288 Decimal expansion of Sum_{n >= 1} (phi - c(2*n-1)), where phi is the golden ratio (A001622), and c(n) is the n-th convergent to the continued fraction expansion of phi.

%I A265288 #38 Aug 23 2022 09:40:07
%S A265288 7,5,7,2,0,4,3,7,5,0,4,6,0,0,7,3,3,8,6,4,7,8,2,5,2,6,0,6,7,3,7,7,4,8,
%T A265288 3,0,1,0,5,8,5,2,0,1,6,1,5,6,6,7,8,4,1,9,2,9,3,2,0,1,5,5,1,1,3,4,7,1,
%U A265288 9,0,7,3,6,6,1,7,8,3,5,7,6,6,9,7,9,5
%N A265288 Decimal expansion of Sum_{n >= 1} (phi - c(2*n-1)), where phi is the golden ratio (A001622), and c(n) is the n-th convergent to the continued fraction expansion of phi.
%C A265288 Define the lower deviance of x > 0 by dL(x) = Sum_{n>=1} (x - c(2*n-1,x)), where c(k,x) = k-th convergent to x. The greatest lower deviance occurs when x = golden ratio, so that this constant is the absolute maximal lower deviance.
%C A265288 Guide to related constants (as sequences):
%C A265288    x          Sum{x-c(2*n-1)}  Sum{c(2*n)-x}  Sum|c(2*n)-c(2*n-1)|
%C A265288 (1+sqrt(5))/2   A265288          A265289         A265290
%C A265288 sqrt(2)         A265291          A265292         A265293
%C A265288 sqrt(3)         A265294          A265295         A265296
%C A265288 sqrt(5)         A265297          A265298         A265299
%C A265288 sqrt(6)         A265300          A265301         A265302
%C A265288 sqrt(8)         A265303          A265304         A265305
%C A265288    e            A265306          A265307         A265308
%F A265288 Equals Sum_{k >= 1} 1/(phi^(2*k-1) * F(2*k-1)), where F(k) is the k-th Fibonacci number (A000045). - _Amiram Eldar_, Oct 05 2020
%F A265288 From _Peter Bala_, Aug 19 2022: (Start)
%F A265288 The constant equals Sum_{k >= 1} (-1)^(k+1)/F(2*k). The constant also equals (3/5)*Sum_{k >= 1} (-1)^(k+1)/(F(2*k)*F(2*k+2)*F(2*k+4)) + 11/15.
%F A265288 A rapidly converging series for the constant is sqrt(5) * Sum_{k >= 1} x^(k*(k+1)/2)/ (x^k - 1) at x = phi - 2 = -(3 - sqrt(5))/2. (End)
%e A265288 0.75720437504600733864782526067377483...
%e A265288 The convergents to x are c(1) = 1, c(2) = 2, c(3) = 3/2, c(4) = 5/3, ..., so that
%e A265288 A265288 = (x - 1) + (x - 3/2) + (x - 8/5) + ... ;
%e A265288 A265289 = (2 - x) + (5/3 - x) + (13/8 - x ) + ... ;
%e A265288 A265290 = (2 - 1) + (5/3 - 3/2) + (13/8 - 8/5) + ...
%p A265288 x := -(3 - sqrt(5))/2:
%p A265288 evalf(sqrt(5)*add(x^(n*(n+1)/2)/(x^n - 1), n = 1..24), 100); # _Peter Bala_, Aug 21 2022
%t A265288 x = GoldenRatio; z = 600; c = Convergents[x, z];
%t A265288 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265288 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265288 N[s1 + s2, 200]
%t A265288 RealDigits[s1, 10, 120][[1]]  (* A265288 *)
%t A265288 RealDigits[s2, 10, 120][[1]]  (* A265289 *)
%t A265288 RealDigits[s1 + s2, 10, 120][[1]] (* A265290 *)
%Y A265288 Cf. A000045, A001227, A001622, A153386, A265289, A265290.
%K A265288 nonn,cons
%O A265288 0,1
%A A265288 _Clark Kimberling_, Dec 06 2015