decompwlj 3D

a(n) = a(floor(n/2)) + n; also denominators in expansion of 1/sqrt(1-x) are 2^a(n); also 2n - number of 1's in binary expansion of 2n

A005187 on the OEIS · family summatory

Weight–level plate of a(n) = a(floor(n/2)) + n; also denominators in expansion of 1/sqrt(1-x) are 2^a(n); also 2n - number of 1's in binary expansion of 2n
Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.
Open in the 3-D viewerA005187 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,997
Level class, k > L12,501 · 12.50 %
Weight class, k ≤ L87,496 · 87.50 %
Ties, k = L46
On the level line L = 18,775
Forced level, l ≤ d²2
Range of a(n)0 … 199,988
Range of the jump d1 … 17
Largest weight k, level L199,961, 99,993

17 different gaps occur, from 1 to 17; the level share is 12.50 %; L = 1 holds 70 % of the level class.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.