A332017 a(n) is the sum of the squares of the lengths of the runs of consecutive equal digits in the binary representation of n.

0, 1, 2, 4, 5, 3, 5, 9, 10, 6, 4, 6, 8, 6, 10, 16, 17, 11, 7, 9, 7, 5, 7, 11, 13, 9, 7, 9, 13, 11, 17, 25, 26, 18, 12, 14, 10, 8, 10, 14, 12, 8, 6, 8, 10, 8, 12, 18, 20, 14, 10, 12, 10, 8, 10, 14, 18, 14, 12, 14, 20, 18, 26, 36, 37, 27, 19, 21, 15, 13, 15, 19

Offset

0,3

Comments

a(0) = 0 by convention.

Every nonnegative number k appears A006456(k) times in the sequence, the last occurrence being at index A000975(k).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Index entries for sequences related to binary expansion of n

Formula

a(n) = Sum_{k = 1..A005811(n)} A101211(n, k)^2.

a(A000975(k)) = k for any k >= 0.

a(2^k-1) = k^2 for any k >= 0.

a(2^k) = k^2+1 for any k >= 0.

Example

For n = 49:
- the binary representation of 49 is "110001",
- we have a run of 2 1's followed by a run of 3 0's followed by a run of 1 1's,
- so a(49) = 2^2 + 3^2 + 1^2 = 14.

Prog

(PARI) a(n) = { my (v=0); while (n, my (r=valuation(n+(n%2), 2)); v+=r^2; n\=2^r); v }

Crossrefs

Cf. A000975, A005811, A006456, A101211.

Sequence in context: A021807 A284561 A061728 * A276127 A182115 A362962

Adjacent sequences: A332014 A332015 A332016 * A332018 A332019 A332020

Keyword

nonn,look,base

Author

Rémy Sigrist, Feb 04 2020

Status

approved