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A299639

a(0) = 0, and for n > 0, (a(n)) gives the indices n for which d(n) > d(k) for k < n, where d is the difference sequence of (cos k + sin k).

0, 4, 5, 52168, 52878, 53588, 54298, 55008, 55718, 56428, 57138, 57848, 58558, 59268, 59978, 60688, 61398, 62108, 62818, 63528, 64238, 64948, 65658, 66368, 67078, 67788, 68498, 69208, 69918, 70628, 71338, 72048, 72758, 73468, 74178, 74888, 75598, 76308

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OFFSET

0,2

COMMENTS

Conjecture: d(n) -> 1.356...

This is true and the value of the limit is 2*sqrt(1 - cos(1)). Proof: d(n) simplifies to this number times a phase-shifted sin(n). Because multiples of 1 are dense in R mod 2*Pi, we can get arbitrarily close to the maximum. - Jonas Karlsson, Mar 10 2026

LINKS

Table of n, a(n) for n=0..37.

EXAMPLE

Records for n - 0,1,2, read from the first 6 values of d(n) approximated by

0.381, -0.088, -1.34, -0.56, 0.73, 1.35; viz.,

d(0) = cos(1) + sin(1) - cos(0) - sin(0) = 0.38177...

d(4) = cos(5) + sin(5) - cos(4) - sin(4) = 0.73518...

d(5) = 1.356016878...

d(52168) = 1.356016794...

MATHEMATICA

z = 100000; d[n_] := N[Cos[n + 1] + Sin[n + 1] - Cos[n] - Sin[n], 10];

max = d[0]; k = 0; s = {0};