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.

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]

Adjacent sequences: A006043 A006044 A006045 this_sequence A006047 A006048 A006049

Sequence in context: A052092 A075991 A002731 this_sequence A161830 A104635 A024903

nonn,nice,easy

Jeffrey Shallit

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