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 A001316 Gould's sequence: a(n) = Sum_{k=0..n} (C(n,k) mod 2); number of odd entries in row n of Pascal's triangle (A007318); 2^A000120(n). (Formerly M0297 N0109) 173

%I M0297 N0109

%S 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,

%T 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,

%U 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

%N Gould's sequence: a(n) = Sum_{k=0..n} (C(n,k) mod 2); number of odd entries in row n of Pascal's triangle (A007318); 2^A000120(n).

%C Also called Dress's sequence.

%C This sequence might be better called Glaisher's sequence, since James Glaisher showed that odd binomial coefficients are counted by 2^A000120(n) in 1899. - _Eric Rowland_, Mar 17 2017 [However, the name "Gould's sequence" is deeply entrenched in the literature. - _N. J. A. Sloane_, Mar 17 2017]

%C All terms are powers of 2. The first occurrence of 2^k is at n = 2^k - 1; e.g., the first occurrence of 16 is at n = 15. - _Robert G. Wilson v_, Dec 06 2000

%C a(n) is the highest power of 2 dividing C(2n,n)=A000984(n). - _Benoit Cloitre_, Jan 23 2002

%C Also number of 1's in n-th row of triangle in A070886. - _Hans Havermann_, May 26 2002. Equivalently, number of live cells in generation n of a one-dimensional cellular automaton, Rule 90, starting with a single live cell. - _Ben Branman_, Feb 28 2009. Ditto for Rule 18. - _N. J. A. Sloane_, Aug 09 2014. This is also the odd-rule cellular automaton defined by OddRule 003 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - _N. J. A. Sloane_, Feb 25 2015

%C 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_, Jan 28 2003

%C 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_, Jan 30 2003

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

%C The odd entries in Pascal's triangle form the Sierpiński Gasket (a fractal). - _Amarnath Murthy_, Nov 20 2004

%C Row sums of Sierpiński's Gasket A047999. - _Johannes W. Meijer_, Jun 05 2011

%C 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 Deléham_, Jun 18 2005

%C 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

%C a[33..63]=A117973[1..31]. - _Stephen Crowley_, Mar 21 2007

%C Or the number of solutions of the equation: A000120(x)+A000120(n-x)=A000120(n). - _Vladimir Shevelev_, Jul 19 2009

%C For positive n, a(n) equals the denominator of the permanent of the n X n matrix consisting entirely of (1/2)'s. - _John M. Campbell_, May 26 2011

%C Companions to A001316 are A048896, A105321, A117973, A151930 and A191488. They all have the same structure. We observe that for all these sequences a((2*n+1)*2^p-1) = C(p)*A001316(n), p>=0. If C(p) = 2^p then a(n) = A001316(n), if C(p) = 1 then a(n) = A048896(n), if C(p) = 2^p+2 then a(n) = A105321(n+1), if C(p) = 2^(p+1) then a(n) = A117973(n), if C(p) = 2^p-2 then a(n) = (-1)*A151930(n) and if C(p) = 2^(p+1)+2 then a(n) = A191488(n). Furthermore for all a(2^p - 1) = C(p). - _Johannes W. Meijer_, Jun 05 2011

%C a(n) = number of zeros in n-th row of A219463 = number of ones in n-th row of A047999. - _Reinhard Zumkeller_, Nov 30 2012

%C a(n) = A226078(n,1). - _Reinhard Zumkeller_, May 25 2013

%C This is the Run Length Transform of S(n) = {1,2,4,8,16,...} (cf. A000079). The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - _N. J. A. Sloane_, Sep 05 2014

%C A105321(n+1) = a(n+1) + a(n). - _Reinhard Zumkeller_, Nov 14 2014

%C a(n) = A261363(n,n) = number of distinct terms in row n of A261363 = number of odd terms in row n+1 of A261363. - _Reinhard Zumkeller_, Aug 16 2015

%C From _Gary W. Adamson_, Aug 26 2016: (Start)

%C A production matrix for the sequence is Lim_{k=1..inf} M^k, the left-shifted vector of M:

%C 1, 0, 0, 0, 0, ...

%C 2, 0, 0, 0, 0, ...

%C 0, 1, 0, 0, 0, ...

%C 0, 2, 0, 0, 0, ...

%C 0, 0, 1, 0, 0, ...

%C 0, 0, 2, 0, 0, ...

%C 0, 0, 0, 1, 0, ...

%C ...

%C The result is equivalent to the g.f. of Apr 06 2003: Product_{k>=0} (1 + 2*z^(2^k)). (End)

%C Number of binary palindromes of length n for which the first floor(n/2) symbols are themselves a palindrome (Ji and Wilf 2008). - _Jeffrey Shallit_, Jun 15 2017

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

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

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

%D O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm Math. Physics, 93 (1984), pp. 219-258. Reprinted in Theory and Applications of Cellular Automata, S Wolfram, Ed., World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113

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

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

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

%D Andrew Wuensche, Exploring Discrete Dynamics, Luniver Press, 2011. See Fig. 2.3.

%H Indranil Ghosh, <a href="/A001316/b001316.txt">Table of n, a(n) for n = 0..50000</a> (terms 0..1000 from T. D. Noe)

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://neilsloane.com/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer Sci., 98 (1992), 163-197.

%H E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/abs/math.CO/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, arXiv:math/0407326 [math.CO], 2004.

%H E. Deutsch and B. E. Sagan, <a href="http://dx.doi.org/10.1016/j.jnt.2005.06.005">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Num. Theory 117 (2006), 191-215.

%H Philippe Dumas, <a href="http://algo.inria.fr/dumas/DC/asympt.html">Diviser pour regner Comportement asymptotique</a> (has many references)

%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796 [math.CO], 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.

%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/stlrsky/stlrsky.html">Stolarsky-Harborth Constant</a>

%H James W. L. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quarterly Journal of Pure and Applied Mathematics 30 (1899), 150-156.

%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>

%H Kathy Q. Ji and Herbert S. Wilf, <a href="http://www.jstor.org/stable/27642506">Extreme Palindromes</a>, Amer. Math. Monthly 115 (2008), 447-451.

%H A. J. Macfarlane, <a href="http://www.damtp.cam.ac.uk/user/ajm/Papers2016/GFsForCAsOfEvenRuleNo.ps">Generating functions for integer sequences defined by the evolution of cellular automata...</a>

%H Sam Northshield, <a href="http://www.jstor.org/stable/10.4169/000298910X496714">Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...</a>, Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.

%H T. Pisanski and T. W. Tucker, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.5375">Growth in Repeated Truncations of Maps</a>, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), suppl., 167-176.

%H D. G. Poole, <a href="http://www.jstor.org/stable/2690991">The towers and triangles of Professor Claus (or, Pascal knows Hanoi)</a>, Math. Mag., 67 (1994), 323-344.

%H N. J. A. Sloane, <a href="/A001316/a001316.png">Illustration of first 20 generations of Rule 90</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.

%H L. Spiegelhofer, M. Wallner, <a href="https://arxiv.org/abs/1710.10884">Divisibility of binomial coefficients by powers of two</a>, arXiv:1710.10884

%H R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>, arXiv:math/0307027 [math.CO], 2003.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://dx.doi.org/10.1103/RevModPhys.55.601">Statistical mechanics of cellular automata</a>, Rev. Mod. Phys., 55 (1983), 601-644.

%H S. Wolfram, <a href="http://www.jstor.org/stable/2323743">Geometry of Binomial Coefficients</a>, Amer. Math. Monthly, Volume 91, Number 9, November 1984, pages 566-571.

%H Chai Wah Wu, <a href="https://arxiv.org/abs/1610.06166">Sums of products of binomial coefficients mod 2 and run length transforms of sequences</a>, arXiv preprint arXiv:1610.06166 [math.CO], 2016.

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%H <a href="/index/Ru#rlt">Index entries for sequences computed with run length transform</a>

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

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

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

%F a(n) = A038573(n) + 1.

%F G.f.: Product_{k>=0} (1+2*z^(2^k)). - _Ralf Stephan_, Apr 06 2003

%F a(n) = Sum_{i=0..2*n} (binomial(2*n, i) mod 2)*(-1)^i. - _Benoit Cloitre_, Nov 16 2003

%F a(n) mod 3 = A001285(n). - _Benoit Cloitre_, May 09 2004

%F a(n) = 2^n - 2*Sum_{k=0..n} floor(C(n, k)/2). - _Paul Barry_, Dec 24 2004

%F a(n) = Product_{k=0..log_2(n)} 2^b(n, k), b(n, k) = coefficient of 2^k in binary expansion of n. - _Paul D. Hanna_

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

%F a(n) = n/2 + 1/2 + (1/2)*Sum_{k=0..n}(-(-1)^binomial(n,k))/2. - _Stephen Crowley_, Mar 21 2007

%F G.f.: (1/2)*z^(1/2)*sinh(2*z^(1/2)). - _Johannes W. Meijer_, Feb 20 2009

%F 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]. - _Mats Granvik_, _Gary W. Adamson_, Oct 02 2009

%F a(n) = f(n, 1) with f(x, y) = if x = 0 then y else f(floor(x/2), y*(1 + x mod 2)). - _Reinhard Zumkeller_, Nov 21 2009

%F a(n) = 2 ^ (number of 1's in binary form of (n-1)). - _Gabriel C. Benamy_, Dec 08 2009

%F a((2*n+1)*2^p-1) = (2^p)*a(n), p>=0. - _Johannes W. Meijer_, Jun 05 2011

%F a(n) = A000120(A001317(n)). - _Reinhard Zumkeller_, Nov 24 2012

%F a(n) = lcm(n!, 2^n) / n!. - _Daniel Suteu_, Apr 28 2017

%F a(n) = A061142(A005940(1+n)). - _Antti Karttunen_, May 29 2017

%e Has a natural structure as a triangle:

%e .1,

%e .2,

%e .2,4,

%e .2,4,4,8,

%e .2,4,4,8,4,8,8,16,

%e .2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,

%e .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,

%e ....

%e The rows converge to A117973.

%e From _Omar E. Pol_, Jun 07 2009: (Start)

%e Also, triangle begins:

%e .1;

%e .2,2;

%e .4,2,4,4;

%e .8,2,4,4,8,4,8,8;

%e 16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16;

%e 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;

%e 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,...

%e (End)

%e G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 8*x^7 + 2*x^8 + ...

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

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

%p a := n -> 2^add(i,i=convert(n,base,2)); # _Peter Luschny_, Mar 11 2009

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

%t Nest[ Join[#, 2#] &, {1}, 7] (* _Robert G. Wilson v_, Jan 24 2006 and modified Jul 27 2014 *)

%t Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[90,{{1},0},100]] (* Produces counts of ON cells. _N. J. A. Sloane_, Aug 10 2009 *)

%t ArrayPlot[CellularAutomaton[90, {{1}, 0}, 20]] (* Illustration of first 20 generations. - _N. J. A. Sloane_, Aug 14 2014 *)

%t Table[2^(RealDigits[n - 1, 2][[1]] // Total), {n, 1, 100}] (* _Gabriel C. Benamy_, Dec 08 2009 *)

%t CoefficientList[Series[Exp[2*x], {x, 0, 100}], x] // Numerator (* _Jean-François Alcover_, Oct 25 2013 *)

%t Count[#,_?OddQ]&/@Table[Binomial[n,k],{n,0,90},{k,0,n}] (* _Harvey P. Dale_, Sep 22 2015 *)

%o (PARI) {a(n) = if( n<0, 0, numerator(2^n / n!))};

%o (PARI) A001316(n)=1<<norml2(binary(n)) \\ _M. F. Hasler_, May 03 2009

%o (PARI) a(n)=2^hammingweight(n) \\ _Charles R Greathouse IV_, Jan 04 2013

%o import Data.List (transpose)

%o a001316 = sum . a047999_row -- _Reinhard Zumkeller_, Nov 24 2012

%o a001316_list = 1 : zs where

%o zs = 2 : (concat \$ transpose [zs, map (* 2) zs])

%o -- _Reinhard Zumkeller_, Aug 27 2014, Sep 16 2011

%o (Sage)

%o def A001316(n):

%o if n <= 1: return Integer(n+1)

%o return A001316(n//2) << n%2

%o [A001316(n) for n in range(88)] # _Peter Luschny_, Nov 19 2012

%o (Python)

%o def A001316(n):

%o ....return 2**bin(n)[2:].count("1") # _Indranil Ghosh_, Feb 06 2017

%o (Scheme) (define (A001316 n) (let loop ((n n) (z 1)) (cond ((zero? n) z) ((even? n) (loop (/ n 2) z)) (else (loop (/ (- n 1) 2) (* z 2)))))) ;; _Antti Karttunen_, May 29 2017

%Y Equals left border of triangle A166548. - _Gary W. Adamson_, Oct 16 2009

%Y 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.

%Y For partial sums see A006046. For first differences see A151930.

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

%Y Cf. A051638, A048967, A007318, A094959, A048896, A117973, A008977, A139541, A048883, A102376, A038573, A159913, A000079, A166548, A006047, A089898, A105321, A061142.

%Y Cf. A156769 = Gould's sequence appears in the numerators. - _Johannes W. Meijer_, Feb 20 2009

%Y Cf. A047999, A261363, A261366.

%Y If we subtract 1 from the terms we get a pair of essentially identical sequences, A038573 and A159913.

%Y A163000 and A163577 count binomial coefficients with 2-adic valuation 1 and 2. A275012 gives a measure of complexity of these sequences. - _Eric Rowland_, Mar 15 2017

%Y Cf. A286575 (run-length transform of this sequence), also A037445.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_