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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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 * A331183 A284082 A047040

Adjacent sequences:  A089395 A089396 A089397 * A089399 A089400 A089401

KEYWORD

base,nonn

AUTHOR

Paul D. Hanna, Oct 30 2003

STATUS

approved

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Last modified August 10 19:52 EDT 2020. Contains 336381 sequences. (Running on oeis4.)