A178590 a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).

1, 3, 4, 9, 7, 12, 13, 27, 16, 21, 19, 36, 25, 39, 40, 81, 43, 48, 37, 63, 40, 57, 55, 108, 61, 75, 64, 117, 79, 120, 121, 243, 124, 129, 91, 144, 85, 111, 100, 189, 103, 120, 97, 171, 112, 165, 163, 324, 169, 183, 136, 225, 139, 192, 181, 351, 196, 237, 199, 360, 241

Offset: 1

Author

Gary W. Adamson, May 29 2010

Keywords

• nonn
• look

Comments

In groups of 1, 2, 4, 8, ... terms; sums of group terms appears to be A081625: (1, 7, 41, 223,...), for example: 41 = (9 + 7 + 12 + 13).

Equals row 3 in the array shown in A178568, an infinite family of sequences of the form a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).

Let M = an infinite lower triangular matrix with (1, 3, 1, 0, 0, 0,...) in each column, and with successive columns shifted down twice from the previous column. A178590 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence.

The Stern polynomial B(n,x) evaluated at x=3. See A125184. - T. D. Noe, Feb 28 2011

Examples

			In groups of 2^n terms (n=0,1,2,...):
1;
3, 4;
9, 7, 12, 13;
27, 16, 21, 19, 36, 25, 39, 40;
...
a(6) = 12 = 3*a(3) = 3*4
a(7) = 13 = a(3) + a(4) = 4 + 9
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8191

Crossrefs

Row 3 of A178568.

Cf. A081625, A090880, A260443.

Programs

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = If[ OddQ@n, a[(n - 1)/2] + a[(n + 1)/2], 3*a[n/2]]; Array[a, 61] (* Robert G. Wilson v, Jun 11 2010 *)

Formula

a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).

a(n) = A090880(A260443(n)). - Antti Karttunen, Jul 29 2015

G.f.: x * Product_{k>=0} (1 + 3*x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Jul 07 2019

Extensions

a(19) onwards from Robert G. Wilson v, Jun 11 2010