A178569 a(2*n) = 10*a(n), a(2*n+1) = a(n) + a(n+1).

1, 10, 11, 100, 21, 110, 111, 1000, 121, 210, 131, 1100, 221, 1110, 1111, 10000, 1121, 1210, 331, 2100, 341, 1310, 1231, 11000, 1321, 2210, 1331, 11100, 2221, 11110, 11111, 100000, 11121, 11210, 2331, 12100, 1541, 3310, 2431, 21000, 2441, 3410, 1651, 13100, 2541, 12310, 12231, 110000, 12321, 13210

Offset: 1

Author

Gary W. Adamson, May 29 2010

Keywords

• nonn

Comments

Equals row 10 in the array of A178568.

Polcoeff f(x)= (1 + 10x + 11x^2 + ...) satisfies f(x)/f(x^2) = (1 +10*x + x^2).

Let q(x) = (1 + 10*x + x^2). Then polcoeff f(x) = q(x) * q(x^2) * q(x^4) * q(x^8) * ...

The sequence mod 10 = (1, 0, 1, 0, 1, 0,...)

Examples

			a(6) = 110 = 10*a(5) = 10*11.
a(7) = 111 = a(3) + a(4) = 111 + 100.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A178568.

Programs

Maple

A178569 := proc(n)
    option remember;
    if n <= 2 then
        10^(n-1);
    elif type(n,'even') then
        10*procname(n/2);
    else
        procname((n-1)/2)+procname((n+1)/2) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2015

Mathematica

a[1]=1; a[2]=10; a[n_]:= a[n]= If[OddQ@n, a[(n-1)/2] + a[(n+1)/2], 10*a[n/2]]; Array[a, 50] (* G. C. Greubel, Jan 30 2019 *)

Formula

a(2*n) = 10*a(n), a(2*n+1) = a(n) + a(n+1). Let M = an infinite lower triangular matrix with (1, 10, 1, 0, 0, 0,...) in each column; with each column >1 shifted down twice from the previous column. Then A178570 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence.