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A048881
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a(n) = A000120(n+1) - 1 = wt(n+1) - 1.
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19
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0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Highest power of 2 dividing n-th Catalan number (A000108).
a(n) = 0 iff n = 2^k - 1, k=0,1,...
Appears to be number of binary left-rotations (iterations of A006257) to reach fixed point of form 2^k-1. Right-rotation analogue is A063250. This would imply that for n >= 0, a(n)=f(n), recursively defined to be 0 for n=0, otherwise as f( ( (1-n)(1-p)(1-s) - (1-n-p-s) ) / 2) + p (s+1) / 2, where p = n mod 2 and s = - signum(n) (f(n<0) is A000120(-n)). - Marc LeBrun (mlb(AT)well.com), Jul 11 2001.
Let f(0) = 01, f(1) = 12, f(2) = 23, f(3) = 34, f(4) = 45, etc. Sequence gives concatenation of 0, f(0), f(f(0)), f(f(f(0))), ... Also f(f(...f(0)...)) converges to A000120. - DELEHAM Philippe(kolotoko(AT)wanadoo.fr), Aug 14, 2003.
Highest power of 2 dividing binomial(n,floor(n/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 20 2003
2^a(n) are numerators in the maclaurin series for (sin x)^2. [From Jacob A. Siehler (siehlerj(AT)wlu.edu), Nov 11 2009]
a(n) = A000120(A129760(n+1)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 30 2010]
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REFERENCES
| R. Alter and K. K. Kubota, Prime and Prime Power Divisibility of Catalan Numbers, J. Com. Th. A 15 (1973) pp. 243-256.
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FORMULA
| Writing n as 2^m+k with -1<=k<2^m-1, then a(n)=A000120(k+1) - Henry Bottomley (se16(AT)btinternet.com), Mar 28 2000
a(n)= k if 2^k divides A000108(n) but 2^(k+1) does not divide A000108(n).
a(2*n) = a(n-1)+1, a(2*n+1) = a(n). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 10 2002
G.f.: 1/(x-x^2) * {x^2/(1-x) - Sum_{k>=1} x^(2^k)/(1-x^(2^k))}. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 13 2002
C(n, k) is the number of occurrence of k in the n-th group of terms in this sequence read by rows: {0}; {0, 1}; {0, 1, 1, 2}; {0, 1, 1, 2, 1, 2, 2, 3}; {0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4 }; ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 01 2004
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EXAMPLE
| Contribution from Omar E. Pol, Mar 08 2011 (Start)
Sequence can be written in the following form (irregular triangle):
0,
0,1,
0,1,1,2,
0,1,1,2,1,2,2,3,
0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,
0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,
Row sums are A001787.
(End)
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PROG
| (PARI) { a(n) = if( n<0, 0, n++; n /= 2^valuation(n, 2); subst( Pol( binary( n ) ), x, 1) - 1 ) } /* Michael Somos Aug 23 2007 */
(PARI) {a(n) = if( n<0, 0, valuation( (2*n)! / n! / (n+1)!, 2 ) ) } /* Michael Somos Aug 23 2007 */
(Haskell)
a048881 n = a048881_list !! n
a048881_list = c [0] where c (x:xs) = x : c (xs ++ [x, x+1])
-- Reinhard Zumkeller, Mar 07 2011
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CROSSREFS
| Cf. A000120, A006257, A007318, A063250.
First differences of A078903.
Sequence in context: A066360 A061358 A025866 * A026931 A127506 A007968
Adjacent sequences: A048878 A048879 A048880 * A048882 A048883 A048884
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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EXTENSIONS
| Entry revised by N. J. A. Sloane, Jun 07 2009
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