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%I #22 Jul 07 2019 20:19:44
%S 1,3,4,9,7,12,13,27,16,21,19,36,25,39,40,81,43,48,37,63,40,57,55,108,
%T 61,75,64,117,79,120,121,243,124,129,91,144,85,111,100,189,103,120,97,
%U 171,112,165,163,324,169,183,136,225,139,192,181,351,196,237,199,360,241
%N a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).
%C 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).
%C 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).
%C 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.
%C The Stern polynomial B(n,x) evaluated at x=3. See A125184. - _T. D. Noe_, Feb 28 2011
%H Antti Karttunen, <a href="/A178590/b178590.txt">Table of n, a(n) for n = 1..8191</a>
%F a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).
%F a(n) = A090880(A260443(n)). - _Antti Karttunen_, Jul 29 2015
%F G.f.: x * Product_{k>=0} (1 + 3*x^(2^k) + x^(2^(k+1))). - _Ilya Gutkovskiy_, Jul 07 2019
%e In groups of 2^n terms (n=0,1,2,...):
%e 1;
%e 3, 4;
%e 9, 7, 12, 13;
%e 27, 16, 21, 19, 36, 25, 39, 40;
%e ...
%e a(6) = 12 = 3*a(3) = 3*4
%e a(7) = 13 = a(3) + a(4) = 4 + 9
%t 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 *)
%o (Scheme, with memoization-macro definec)
%o (definec (A178590 n) (cond ((<= n 1) n) ((even? n) (* 3 (A178590 (/ n 2)))) (else (+ (A178590 (/ (- n 1) 2)) (A178590 (/ (+ n 1) 2))))))
%o ;; _Antti Karttunen_, Jul 29 2015
%Y Row 3 of A178568.
%Y Cf. A081625, A090880, A260443.
%K nonn,look
%O 1,2
%A _Gary W. Adamson_, May 29 2010
%E a(19) onwards from _Robert G. Wilson v_, Jun 11 2010
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