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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

Table of n, a(n) for n=0..85.

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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Last modified August 19 13:20 EDT 2017. Contains 290807 sequences.