0,2

Also called Dress's sequence.

All terms are powers of 2. The first occurrence of 2^k is when n = 2^k - 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. Equivalently, number of live cells in generation n of a one-dimensional cellular automaton, Rule 90. [Ben Branman (137ben(AT)comcast.net), Feb 28 2009]

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)gmail.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" -> "1,2", "2" -> "2,4", "4" -> "4,8", ..., "2^k" -> "2^k,2^(k+1)", ... starting with 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

Or the number of solutions of the equation: A000120(x)+A000120(n-x)=A000120(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 19 2009]

Equals left border of triangle A166548 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 16 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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, Pascal knows 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

Index entries for sequences related to cellular automata

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

a(0) = 1; for n>0, write n = 2^i + j where 0 <= j < 2^i; then a(n) = 2*a(j).

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

a(n) = A038573(n) + 1

G.f.: Prod_{k>=0} (1+2*z^(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

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009: G.f.: (1/2)*z^(1/2)*sinh(2*z^(1/2))

Equals infinite convolution product of [1,2,0,0,0,0,0,0,0] aerated A000079 - 1 times, i.e. [1,2,0,0,0,0,0,0,0] * [1,0,2,0,0,0,0,0,0] * [1,0,0,0,2,0,0,0,0]. [From Mats Granvik, Gary W. Adamson (mats.granvik(AT)abo.fi), Oct 02 2009]

a(n) = f(n, 1) with f(x, y) = if x = 0 then y else f(floor(x/2), y*(1 + x mod 2)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2009]

Has a natural structure as a triangle:

.1,

.2,

.2,4,

.2,4,4,8,

.2,4,4,8,4,8,8,16,

.2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,

.2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,

....

The rows converge to A117973.

Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 07 2009: (Start)

Also, triangle begins:

.1;

.2,2;

.4,2,4,4;

.8,2,4,4,8,4,8,8;

16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16;

32,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32;

64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,...

(End)

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

S:=[1]; S:=[op(S), op(2*s)]; # repeat ad infinitum!

a := n -> 2^add(i, i=convert(n, base, 2)); [From Peter Luschny (peter(AT)luschny.de), Mar 11 2009]

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)

Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[90, {{1}, 0}, 100]] (N. J. A. Sloane, Aug 10 2009)

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

(PARI) A001316(n)=1<<norml2(binary(n)) [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 03 2009]

For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.

For partial sums see A006046. Fir first differences see A151930.

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

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

Cf. A008977, A139541, A048883, A102376.

Cf. A156769 = Gould's sequence appears in the numerators. - Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009

Cf. A038573, A159913. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 03 2009]

Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 07 2009]

A166548 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 16 2009]

Sequence in context: A094269 A157227 A054536 this_sequence A161831 A096865 A116466

Adjacent sequences: A001313 A001314 A001315 this_sequence A001317 A001318 A001319

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

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

Additional comments from N. J. A. Sloane, May 30 2009