0,10

Also number of hyperbinary representations of n-1 containing exactly k digits 1. A hyperbinary representation of a nonnegative integer n is a representation of n as a sum of powers of 2, each power being used at most twice. Example: row 9 of the triangle is 1,2,1; indeed the hyperbinary representations of 8 are 200 (2*2^2+0*2^1+0*2^0), 120 (1*2^2+2*2^1+0*2^0), 1000 (1*2^3+0*2^2+0*2^1+0*2^0) and 112 (1*2^2+1*2^1+2*1^0), having 0,1,1,and 2 digits 1, respectively (see S. Klavzar et al. Corollary 3). Number of terms in row n is 1+A057526(n) (n>=2). Row sums yield A002487 (Stern's diatomic series). T(2n+1,1)=A005811(n)= number of 1's in the standard Gray code of n (S. Klavzar et al. Theorem 8). T(4n+1,1)=1, T(4n+3,1)=0 (S. Klavzar et al., Lemma 5).

S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math., (in press).

N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.

D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67.

Triangle starts:

0;

1;

0, 1;

1, 1;

0, 0, 1;

1, 2;

0, 1, 1;

1, 1, 1;

0, 0, 0, 1;

1, 2, 1;

0, 1, 2;

1, 3, 1;

B:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then t*B(n/2) else B((n+1)/2)+B((n-1)/2) fi end: for n from 0 to 36 do B(n):=sort(expand(B(n))) od: dg:=n->degree(B(n)): 0; for n from 0 to 40 do seq(coeff(B(n), t, k), k=0..dg(n)) od; # yields sequence in triangular form

Cf. A057526, A002487, A005811.

Sequence in context: A025913 A123230 A078821 this_sequence A091430 A059282 A056175

Adjacent sequences: A125181 A125182 A125183 this_sequence A125185 A125186 A125187

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 04 2006