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%I A006046 M2445
%S A006046 0,1,3,5,9,11,15,19,27,29,33,37,45,49,57,65,81,83,87,91,99,103,111,119,
%T A006046 135,139,147,155,171,179,195,211,243,245,249,253,261,265,273,281,297,
%U A006046 301,309,317,333,341,357,373,405,409,417,425,441,449,465,481,513,521
%N A006046 Total number of odd entries in first n rows of Pascal's triangle.
%C A006046 The following alternative construction of this sequence is due to Thomas 
               Nordhaus (tnordh(AT)t-online.de), Oct 31 2000: For each n >= 0 let 
               f_n be the piecewise linear function given by the points (k /(2^n), 
               a(k) / 3^n), k =0, 1, ..., 2^n. f_n is a monotonic map from the interval 
               [0,1] into itself, f_n(0) = 0, f_n(1) = 1. This sequence of functions 
               converges uniformly. But the limiting function is not differentiable 
               on a dense subset of this interval.
%C A006046 Comment from D. E. Knuth, Jun 18 2007: I submitted a problem to the Amer. 
               Math. Monthly about an infinite family of non-convex sequences that 
               solve a recurrence that involves minimization: a(1) = 1; a(n) = max 
               { ua(k)+a(n-k) | 1 <= k <= n/2 }, for n>1; here u is any real-valued 
               constant >= 1. The case u=2 gives the present sequence. Cf. A130665 
               - A130667.
%C A006046 a(n) = sum of (n-1)-th row terms of triangle A166556 [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Oct 17 2009]
%D A006046 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006046 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.
%D A006046 H. Harborth, Number of odd binomial coefficients, Proc. Amer. Math. Soc., 
               62 (1977), 19-22.
%D A006046 A. Lakhtakia and R. Messier, Self-similar sequences and chaos from Gauss 
               sums, Computers and Graphics, 13 (1989), 59-62.
%D A006046 A. Lakhtakia et al., Fractal sequences derived from the self-similar 
               extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928.
%D A006046 K. B. Stolarsky, Power and exponential sums of digital sums related to 
               binomial coefficient parity, SIAM J. Appl. Math., 32 (1977), 717-730.
%H A006046 T. D. Noe, <a href="b006046.txt">Table of n, a(n) for n=0..1000</a>
%H A006046 S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">
               Odd Entries in Pascal's Trinomial Triangle</a> (arXiv:0802.2654)
%H A006046 P. Flajolet et al., <a href="http://algo.inria.fr/flajolet/Publications/
               FlGrKiPrTi94.pdf">Mellin Transforms And Asymptotics: Digital Sums</
               a>, Theoret. Computer Sci. 23 (1994), 291-314.
%H A006046 P. J. Grabner and H.-K. Hwang, <a href="http://algo.stat.sinica.edu.tw/
               ">Digital sums and divide-and-conquer recurrences: Fourier expansions 
               and absolute convergence</a>
%H A006046 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer 
               generating functions. I. Elementary sequences</a>
%H A006046 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Stolarsky-HarborthConstant.html">Link to a section of The World of 
               Mathematics.</a>
%H A006046 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PascalsTriangle.html">Pascal's Triangle</a>
%F A006046 a(0) = 0, a(1) = 1, a(2k) = 3*a(k), a(2k+1) = 2*a(k) + a(k+1).
%F A006046 a(n) = a(n-1)+A001316(n-1). a(2^n) = 3^n. - Henry Bottomley (se16(AT)btinternet.com), 
               Apr 05 2001
%F A006046 a(n) = n^(log2(3))*G(log2(n)) where G(x) is a function of period 1 defined 
               by its Fourier series. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Aug 16 2002; formula modified by S. R. Finch, Dec 31 2007
%F A006046 G.f.: (x/(1-x))prod(k>=0, 1+2x^2^k) - Ralf Stephan, Jun 01 2003; corrected 
               by Herbert S. Wilf (wilf(AT)math.upenn.edu), Jun 16 2005
%F A006046 a(1) = 1, a(n) = 2a([n/2]) + a(ceil(n/2)).
%F A006046 a(n)=sum{k=0..n-1, 2^A000120(n-k-1)} - Paul Barry (pbarry(AT)wit.ie), 
               Jan 05 2005
%F A006046 a(n) = 3 a([n/2]) + (n%2)*2^A000120(n-1), where n%2 = parity of n (= 
               1 if odd, 0 else). [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 
               03 2009]
%t A006046 f[n_] := Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ]; Table[ Sum[ f[k], 
               {k, 0, n} ], {n, 0, 100} ]
%o A006046 (PARI) A006046(n)={ n<2 & return(n); A006046(n\2)*3+if(n%2,1<<norml2(binary(n\2))) 
               } [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 03 2009]
%Y A006046 Partial sums of A001316.
%Y A006046 a(n) = A074330(n-1)+1 for n>=2. A080978(n) = 2*a(n)+1. Cf. A080263.
%Y A006046 Cf. A159912. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 03 2009]
%Y A006046 A166556 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 17 2009]
%Y A006046 Sequence in context: A052092 A075991 A002731 this_sequence A161830 A151922 
               A104635
%Y A006046 Adjacent sequences: A006043 A006044 A006045 this_sequence A006047 A006048 
               A006049
%K A006046 nonn,nice,easy
%O A006046 0,3
%A A006046 Jeffrey Shallit
%E A006046 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 21 2000

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