A355809 a(n) is the number at the apex of a triangle whose base contains the distinct powers of 2 summing to n (in ascending order), and each number in a higher row is the sum of the two numbers directly below it; a(0) = 0.

0, 1, 2, 3, 4, 5, 6, 9, 8, 9, 10, 13, 12, 17, 18, 27, 16, 17, 18, 21, 20, 25, 26, 35, 24, 33, 34, 47, 36, 53, 54, 81, 32, 33, 34, 37, 36, 41, 42, 51, 40, 49, 50, 63, 52, 69, 70, 97, 48, 65, 66, 87, 68, 93, 94, 129, 72, 105, 106, 153, 108, 161, 162, 243, 64, 65

Offset: 0

Author

Rémy Sigrist, Jul 18 2022

Keywords

• nonn
• base

Examples

			For n = 27:
- we have the following triangle:
          47
        13  34
       3  10  24
     1   2   8  16
- so a(27) = 47.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Index entries for sequences related to binary expansion of n

Crossrefs

See A355807 for similar sequences.

Cf. A048645, A348296.

Programs

PARI

a(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=b[k]=2^valuation(n, 2)); while (#b>1, b=vector(#b-1, k, b[k+1]+b[k])); if (#b, b[1], 0) }

Formula

a(n) >= n with equality iff n = 0 or n belongs to A048645.

a(2*n) = 2*a(n).