A170903 - OEIS

%I #23 Feb 24 2021 02:48:19

%S 1,1,5,1,5,9,13,1,5,9,13,9,21,33,29,1,5,9,13,9,21,33,29,9,21,33,37,41,

%T 77,97,61,1,5,9,13,9,21,33,29,9,21,33,37,41,77,97,61,9,21,33,37,41,77,

%U 97,69,41,77,105,117,161,253,257,125,1,5,9,13,9,21,33,29,9,21,33,37,41,77

%N a(n) = 2*A160552(n)-1.

%H 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 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 It appears that a(n) = A160164(n) - A169707(n). - _Omar E. Pol_, Feb 17 2015

%e When written as a triangle:

%e 1

%e 1, 5;

%e 1, 5, 9, 13;

%e 1, 5, 9, 13, 9, 21, 33, 29;

%e ...

%e Rows sums are A006516 (this is immediate from the definition).

%e From _Omar E. Pol_, Feb 17 2015: (Start)

%e Also, written as an irregular triangle in which the row lengths are the terms of A011782:

%e 1;

%e 1;

%e 5,1;

%e 5,9,13,1;

%e 5,9,13,9,21,33,29,1;

%e 5,9,13,9,21,33,29,9,21,33,37,41,77,97,61,1;

%e 5,9,13,9,21,33,29,9,21,33,37,41,77,97,61,9,21,33,37,41,77,97,69,41,77,105,117,161,253,257,125,1;

%e Row sums give 1 together with the positive terms of A006516.

%e It appears that the right border (A000012) gives the smallest difference between A160164 and A169707 in every period.

%e (End)

%Y Cf. A006516, A139250, A139251, A160164, A160552, A169707.

%K nonn

%O 1,3

%A _Gary W. Adamson_, Jan 21 2010