decompwlj 3D

Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3*a(k), a(2k+1) = 2*a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i)

A006046 on the OEIS · family summatory

Weight–level plate of Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3*a(k), a(2k+1) = 2*a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i)
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 viewerA006046 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L55,209 · 55.21 %
Weight class, k ≤ L44,787 · 44.79 %
Ties, k = L22
On the level line L = 118,284
Forced level, l ≤ d²2,230
Range of a(n)0 … 72,179,939
Range of the jump d1 … 65,536
Largest weight k, level L72,173,987, 10,522,531

17 different gaps occur, from 1 to 65,536; the level share is 55.21 %; 2.2 % of terms are forced level (l <= d^2); L = 1 holds 33 % 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.