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 A265288 Decimal expansion of sum{x - c(2n-1), n=1,2,...}, where c = convergents to (x = golden ratio). 21
 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, 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, 9, 0, 7, 3, 6, 6, 1, 7, 8, 3, 5, 7, 6, 6, 9, 7, 9, 5 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Define the lower deviance of x > 0 by dL(x) = sum{x - c(2n-1,x), n=1,2,...}, where c(k,x) = k-th convergent to x. The greatest lower deviance occurs when x = golden ratio, so that the constant in A265288 is the absolute maximal lower deviance. Guide to related constants (as sequences):    x          Sum{x-c(2n-1)}   Sum{c(2n)-x}    Sum{c(2n)-c(2n-1}| (1+sqrt(5))/2   A265288          A265289         A265290 sqrt(2)         A265291          A265292         A265293 sqrt(3)         A265294          A265295         A265296 sqrt(5)         A265297          A265298         A265299 sqrt(6)         A265300          A265301         A265302 sqrt(8)         A265303          A265304         A265305    e            A265306          A265307         A265308 LINKS EXAMPLE sum = 0.75720437504600733864782526067377483... The convergents to x are c(1) = 1, c(2) = 2, c(3) = 3/2, c(4) = 5/3, ..., so that A265288 = (x - 1) + (x - 3/2) + (x - 8/5) + ... ; A265289 = (2 - x) + (5/3 - x) + (13/8 - x ) + ... ; A265290 = (2 - 1) + (5/3 - 3/2) + (13/8 - 8/5) + ... MATHEMATICA x = GoldenRatio; z = 600; c = Convergents[x, z]; s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200] s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200] N[s1 + s2, 200] RealDigits[s1, 10, 120][[1]]  (* A265288 *) RealDigits[s2, 10, 120][[1]]  (* A265289 *) RealDigits[s1 + s2, 10, 120][[1]] (* A265290 *) CROSSREFS Cf. A000045, A265289, A265290. Sequence in context: A248200 A110943 A197379 * A171677 A021573 A080411 Adjacent sequences:  A265285 A265286 A265287 * A265289 A265290 A265291 KEYWORD nonn,cons AUTHOR Clark Kimberling, Dec 06 2015 STATUS approved

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