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

1,3

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

Links

Table of n, a(n) for n=1..105.

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, ...},

Example

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...

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 *)

Prog

 (PARI) /* Prints initial 1000 terms: */
{A=vector(1000); for(n=1, #A, Bn=binary(n*2^(n-1)); for(k=1, min(#Bn, #A), A[k]=A[k]+Bn[#Bn-k+1]) ); print(A)}

Crossrefs

Cf. A089399, A089400, A089401.

Sequence in context: A016442 A076360 A246702 * A345272 A331183 A284082

Adjacent sequences: A089395 A089396 A089397 * A089399 A089400 A089401

Keyword

base,nonn

Author

Paul D. Hanna, Oct 30 2003

Status

approved