0,2

Also called Dress's sequence.

All terms are powers of 2. The first occurrence of 2^k is when n = 2^n - 1: e.g. the first occurrence of 16 is at n = 15 - Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 06 2000

a(n) is the highest power of 2 dividing C(2n,n)=A000984(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 23 2002

Also number of 1's in n-th row of triangle in A070886. - Hans Havermann (pxp(AT)rogers.com), May 26 2002

Also number of numbers k, 0<=k<=n, such that (k OR n) = n (bitwise logical OR): a(n) = #{k : T(n,k)=n, 0<=k<=n}, where T is defined as in A080098. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 28 2003

To construct the sequence, start with 1 and use the rule: If k>=0 and a(0),a(1),...,a(2^k-1) are the 2^k first terms, then the next 2^k terms are 2*a(0),2*a(1),...,2*a(2^k-1). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 30 2003

Also, numerator((2^k)/k!). - Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 03 2004

The odd entries in Pascal's triangle form the Seirpinski Gasket (a fractal). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 20 2004

Fixed point of the morphism 1 -> 12, 2 -> 24, 4 -> 48, ... = 2^k -> 2^k 2^(k+1), ... starting from a(0) = 1; 1 -> 12 -> 1224 -> = 12242448 -> 122424482448488(16) -> . . . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 18 2005

a(n) = number of 1's of stage n of the one-dimensional cellular automaton with rule 90. - Andras Erszegi (erszegi.andras(AT)chello.hu), Apr 01 2006

a[33..63]=A117973[1..31] - Stephen Crowley (crow(AT)crowlogic.net), Mar 21 2007

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 75ff.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.

H. W. Gould, Exponential Binomial Coefficient Series. Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sep 1961.

D. G. Poole, The towers and triangles of Professor Claus (or, Pacal known Hanoi), Math. Mag., 67 (1994), 323-344.

M. R. Schroeder, "Fractals, Chaos, Power Laws," W.H. Freeman, NY, 1991, page 383.

T. D. Noe, Table of n, a(n) for n=0..1000

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Philippe Dumas, Diviser pour regner Comportement asymptotique (has many references)

S. R. Finch, Stolarsky-Harborth Constant

Michael Gilleland, Some Self-Similar Integer Sequences

T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), suppl., 167-176.

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

a(n) = 2^A000120(n).

a(n) = 2a(n-1)/A006519(n) = A000079(n)*A049606(n)/A000142(n)

a(n) = A038573(n) + 1

G.f.: prod(k>=0, 1+2z^(2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 06 2003

a(n)=sum(i=0, 2*n, (binomial(2*n, i) (mod 2))*(-1)^i) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 16 2003

a(n) {mod 3}=A001285(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2004

2^n-2*sum{k=0..n, floor(C(n, k)/2)} - Paul Barry (pbarry(AT)wit.ie), Dec 24 2004

a(n)=product{k=0..log_2(n), 2^b(n, k)}, b(n, k)=coefficient of 2^k in binary expansion of n. Formula from Paul D. Hanna.

Sum_{k<n} a(k) = A006046(n).

a(n)=(n/2)+(1/2)+(sum(-(-1)^binomial(n,k),k=0..n)/2) - Stephen Crowley (crow(AT)crowlogic.net), Mar 21 2007

A001316 := proc(n) local k; add(binomial(n, k) mod 2, k=0..n); end;

Table[ Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ], {n, 0, 100} ]

Flatten[ Nest[ Flatten[ # /. a_Integer -> {a, 2a}] &, {1}, 7]] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 24 2006)

(PARI) a(n)=if(n<0, 0, numerator(2^n/n!))

This is the numerator of 2^n/n!, while A049606 gives the denominator.

Cf. A051638, A048967, A007318, A094959, A048896, A117973.

Adjacent sequences: A001313 A001314 A001315 this_sequence A001317 A001318 A001319

Sequence in context: A036845 A094269 A054536 this_sequence A096865 A116466 A116467

nonn,easy,nice

njas

Additional comments from Henry Bottomley (se16(AT)btinternet.com), Mar 12 2001