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A375492
Number of compositions of n into powers of two that each divide the sum of previous powers.
0
1, 1, 2, 2, 5, 5, 10, 10, 26, 26, 52, 52, 130, 130, 260, 260, 677, 677, 1354, 1354, 3385, 3385, 6770, 6770, 17602, 17602, 35204, 35204, 88010, 88010, 176020, 176020, 458330, 458330, 916660, 916660, 2291650, 2291650, 4583300, 4583300, 11916580, 11916580
OFFSET
0,3
COMMENTS
If n = 2^k, a(n) = A003095(k). Otherwise, a(n) is the product of terms from A003095 corresponding to the powers of two in the binary representation of n. If n is odd, the final term of the composition must be 1, so a(n) = a(n-1).
Pieter Mostert points out that, after the first two values, this is a subsequence of A000404 (sums of two nonzero squares), because each term is either a square + 1 or a product of two earlier terms.
FORMULA
Let p be the largest power of two less than n; then a(n) = a(p)a(n-p) if n is not a power of two, or a(n) = a(p)^2 + 1 if n is a power of two.
EXAMPLE
For n = 4 the a(4) = 5 compositions are 1+1+1+1, 1+1+2, 2+1+1, 2+2, and 4. The composition 1+2+1 is not allowed, because 2 does not divide the sum of previous terms.
CROSSREFS
Sequence in context: A214795 A287744 A214253 * A306320 A347449 A325347
KEYWORD
nonn
AUTHOR
David Eppstein, Aug 17 2024
STATUS
approved