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A160570 Triangle read by rows, A160552 convolved with (1, 2, 2, 2, ...); row sums = A139250, the Toothpick sequence.

%I A160570 #14 Feb 08 2022 23:07:40
%S A160570 1,1,2,3,2,2,1,6,2,2,3,2,6,2,2,5,6,2,6,2,2,7,10,6,2,6,2,2,1,14,10,6,2,
%T A160570 6,2,2,3,2,14,10,6,2,6,2,2,5,6,2,14,10,6,2,6,2,2,7,10,6,2,14,10,6,2,6,
%U A160570 2,2,5,14,10,6,2,14,10,6,2,6,2,2,11,10,14,10,6,2,14,10,6,2,6
%N A160570 Triangle read by rows, A160552 convolved with (1, 2, 2, 2, ...); row sums = A139250, the Toothpick sequence.
%H A160570 Nathaniel Johnston, <a href="/A160570/b160570.txt">Table of n, a(n) for n = 1..10000</a>
%H A160570 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
%H A160570 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A160570 Construct triangle M = an infinite lower triangular Toeplitz matrix with A160552: (1, 1, 3, 1, 3, 5, 7, ...) in every column. Let Q = an infinite lower triangular matrix with (1, 2, 2, 2, 2, ...) as the main diagonal and the rest zeros. A160570 = M * Q.
%e A160570 First few rows of the triangle:
%e A160570   1;
%e A160570   1,  2;
%e A160570   3,  2,  2;
%e A160570   1,  6,  2,  2;
%e A160570   3,  2,  6,  2,  2;
%e A160570   5,  6,  2,  6,  2,  2;
%e A160570   7, 10,  6,  2,  6,  2,  2;
%e A160570   1, 14, 10,  6,  2,  6,  2,  2;
%e A160570   3,  2, 14, 10,  6,  2,  6,  2,  2;
%e A160570   5,  6,  2, 14, 10,  6,  2,  6,  2,  2;
%e A160570   ...
%e A160570 Example: Row 4 = (1, 6, 2, 2) = (1, 3, 1, 1) dot (1, 2, 2, 2); where (1 + 6 + 2 + 2) = A139250(4), i.e., 4th term in the Toothpick sequence.
%p A160570 T:=proc(n,k)if(k=1)then return A160552(n):else return 2*A160552(n-k+1):fi:end:
%p A160570 for n from 1 to 8 do for k from 1 to n do print(T(n,k));od:od: # _Nathaniel Johnston_, Apr 13 2011
%Y A160570 Cf. A160552, A139250.
%K A160570 nonn,tabl,easy
%O A160570 1,3
%A A160570 _Gary W. Adamson_, May 19 2009