OFFSET
1,2
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.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8191
FORMULA
a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).
G.f.: x * Product_{k>=0} (1 + 3*x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Jul 07 2019
EXAMPLE
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
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 *)
PROG
(Scheme, with memoization-macro definec)
(definec (A178590 n) (cond ((<= n 1) n) ((even? n) (* 3 (A178590 (/ n 2)))) (else (+ (A178590 (/ (- n 1) 2)) (A178590 (/ (+ n 1) 2))))))
;; Antti Karttunen, Jul 29 2015
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Gary W. Adamson, May 29 2010
EXTENSIONS
a(19) onwards from Robert G. Wilson v, Jun 11 2010
STATUS
approved