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A178243 a(2n) = a(n), a(2n+1) = 10*a(n) + a(n+1).

%I A178243 #17 Apr 28 2025 09:03:37
%S A178243 1,1,11,1,21,11,111,1,31,21,221,11,221,111,1111,1,41,31,331,21,431,
%T A178243 221,2221,11,331,221,2321,111,2221,1111,11111,1,51,41,441,31,641,331,
%U A178243 3331,21,641,431,4531,221,4431,2221,22221,11,441,331,3531,221,4531,2321,23321
%N A178243 a(2n) = a(n), a(2n+1) = 10*a(n) + a(n+1).
%C A178243 Parsed into blocks of 1, 2, 4, 8,...term sums = powers of 12: (1, 12, 144,...).
%C A178243 A178569 is generated from a(2n) = 10*a(n); a(2n+1) = a(n) + a(n+1).
%C A178243 Polcoeff (1 + 10x + 11x^2 + ...) satisfies f(x)/f(x^2) = (1 + x + 10x^2).
%C A178243 Let q(x) = (1 + x + 10*x^2). Then (1 + 10x + 11x^2 + 100x^3 + ...) = q(x) * q(x^2) * q(x^4) * q(x^8) * ...
%H A178243 Michael De Vlieger, <a href="/A178243/b178243.txt">Table of n, a(n) for n = 1..10000</a>
%F A178243 a(2n) = a(n), a(2n+1) = 10*a(n) + a(n+1) = row 10 in the array of A178239.
%F A178243 Let M = an infinite lower triangular matrix with (1, 1, 10, 0, 0, 0,...) in each column, shifted down twice from the previous column. This sequence is Lim_{n->inf} M^n, the left-shifted vector considered as a sequence.
%e A178243 a(6) = a(3) = 10 since a(2n) = a(n);
%e A178243 a(7) = 111 = 10*a(n) + a(n+1) = 10*11 + 1.
%t A178243 a[1] = 1; a[n_] := a[n] = If[EvenQ@ n, a[n/2], 10 a[(n - 1)/2] + a[(n - 1)/2 + 1]]; Array[a, 55] (* _Michael De Vlieger_, May 20 2017 *)
%Y A178243 Cf. A178239, A178569.
%K A178243 nonn,look
%O A178243 1,3
%A A178243 _Gary W. Adamson_, May 29 2010
%E A178243 More terms from _Ilya Gutkovskiy_, May 18 2017