A089398 a(n) = n-th column sum of binary digits of k*2^(k-1), where summation is over k>=1, without carrying between columns.

1, 0, 2, 1, 1, 1, 3, 2, 2, 0, 3, 2, 2, 2, 4, 3, 3, 1, 2, 2, 2, 2, 4, 3, 3, 1, 4, 3, 3, 3, 5, 4, 4, 2, 3, 1, 2, 2, 4, 3, 3, 1, 4, 3, 3, 3, 5, 4, 4, 2, 3, 3, 3, 3, 5, 4, 4, 2, 5, 4, 4, 4, 6, 5, 5, 3, 4, 2, 1, 2, 4, 3, 3, 1, 4, 3, 3, 3, 5, 4, 4, 2, 3, 3, 3, 3, 5, 4, 4, 2, 5, 4, 4, 4, 6, 5, 5, 3, 4, 2, 3, 3, 5, 4, 4

Offset: 1

Author

Paul D. Hanna, Oct 30 2003

Keywords

• base
• nonn

Comments

sum(k=1,n, a(k)*2^(k-1)) = 2^A089399(n)+1 for n>2, with a(1)=a(2)=1.

Row sums of triangular arrays in A103588 and in A103589. - Philippe Deléham, Apr 04 2005

a(k) = 0 for k = 2, 10, 2058, 2058 + 2^2059, ..., that is, for k = A034797(n) - 1, n>=2. - Philippe Deléham, Nov 16 2007

Examples

			Binary expansions of k*2^(k-1), with bits in ascending order by powers of 2, are:
1
001
0011
000001
0000101
00000011
000000111
00000000001
000000001001
0000000000101
00000000001101
000000000000011
0000000000001011
.................
Giving column sums:
10211132203222433...
		

Crossrefs

Cf. A089399, A089400, A089401.

Programs

Mathematica

f[n_] := Block[{lg = Floor[Log[2, n]] + 1}, Sum[ Join[ Reverse[ IntegerDigits[n - i + 1, 2]], {0}][[i]], {i, lg}]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Mar 26 2005 *)

Formula

a(2^n)=n-1 (for n>0), a(2^n-1)=n (for n>0), a(2^n+1)=n-1 (for n>1), a(2^n-k)=n-A089400(k) (for n>k>0), a(2^n+k)=n-A089401(k) (for n>k>0), where sequences have limits: A089400={0, 2, 2, 2, 1, 4, 2, 2, 1, 3, 3, ...} and A089401={1, 1, 3, 2, 4, 5, 6, 5, 7, 8, 11, 9, ...},