A384947 Positive integers m for which A183136(m) != f(m), where f(m) = floor( (m*(m+1)/2)/phi - m/2 + 1/(2*phi) ) and phi = (1+sqrt(5))/2 is the golden ratio.

15, 18, 36, 39, 41, 47, 49, 52, 91, 94, 96, 102, 103, 104, 107, 109, 123, 125, 128, 130, 136, 138, 141, 235, 238, 240, 246, 247, 248, 251, 252, 253, 267, 268, 269, 272, 273, 274, 277, 280, 281, 282, 285, 287, 303, 306, 322, 324, 327, 328

Offset

1,1

Comments

f(m) is an approximation to A183136(m) = Sum_{k=1..m} floor(k/phi) based on assuming the floor in each term decreases it by 1/2 from what is otherwise a triangular sum; and further offset + 1/(2*phi) in f(m) chosen to improve the accuracy of this approximation.

The actual values of frac(k/phi) can differ from 1/2 each by a net amount which is enough to make m a term of this sequence.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Hoang Xuan Thanh)

Example

41 is term, because A183136(41) = 512 != 511 = floor(((41^2+1)*phi - 41) / (2*phi^2)).

Mathematica

PositionIndex[MapIndexed[# != Floor[PolygonalNumber[#2[[1]]]/GoldenRatio - #2[[1]]/2 + 1/(2*GoldenRatio)] &, Accumulate[Floor[Range[500]/GoldenRatio]]]][True] (* Paolo Xausa, Jun 20 2025 *)

Crossrefs

Cf. A183136, A001622, A060143.

Sequence in context: A390601 A390320 A389981 * A274979 A083823 A222682

Adjacent sequences: A384944 A384945 A384946 * A384948 A384949 A384950

Keyword

nonn

Author

Hoang Xuan Thanh, Jun 13 2025

Status

approved