0,3

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.

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.

a(n) = sum of (n-1)-th row terms of triangle A166556 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 17 2009]

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

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.

H. Harborth, Number of odd binomial coefficients, Proc. Amer. Math. Soc., 62 (1977), 19-22.

A. Lakhtakia and R. Messier, Self-similar sequences and chaos from Gauss sums, Computers and Graphics, 13 (1989), 59-62.

A. Lakhtakia et al., Fractal sequences derived from the self-similar extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928.

K. B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., 32 (1977), 717-730.

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

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

P. Flajolet et al., Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci. 23 (1994), 291-314.

P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Pascal's Triangle

a(0) = 0, a(1) = 1, a(2k) = 3*a(k), a(2k+1) = 2*a(k) + a(k+1).

a(n) = a(n-1)+A001316(n-1). a(2^n) = 3^n. - Henry Bottomley (se16(AT)btinternet.com), Apr 05 2001

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

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

a(1) = 1, a(n) = 2a([n/2]) + a(ceil(n/2)).

a(n)=sum{k=0..n-1, 2^A000120(n-k-1)} - Paul Barry (pbarry(AT)wit.ie), Jan 05 2005

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]

f[n_] := Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ]; Table[ Sum[ f[k], {k, 0, n} ], {n, 0, 100} ]

(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]

Partial sums of A001316.

a(n) = A074330(n-1)+1 for n>=2. A080978(n) = 2*a(n)+1. Cf. A080263.

Cf. A159912. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 03 2009]

A166556 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 17 2009]

Sequence in context: A052092 A075991 A002731 this_sequence A161830 A151922 A104635

Adjacent sequences: A006043 A006044 A006045 this_sequence A006047 A006048 A006049

nonn,nice,easy

Jeffrey Shallit

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 21 2000