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A163311 Triangle read by rows in which the diagonals give the infinite set of Toothpick sequences. 3
1, 1, 2, 1, 3, 4, 1, 4, 7, 5, 1, 5, 10, 11, 7, 1, 6, 13, 19, 15, 10, 1, 7, 16, 29, 25, 23, 13, 1, 8, 19, 41, 37, 40, 35, 14, 1, 9, 22, 55, 51, 61, 67, 43, 16, 1, 10, 25, 71, 67, 86, 109, 94, 47, 19, 1, 11, 28, 89, 85, 115, 161, 173, 100, 55, 22, 1, 12, 31, 109, 105, 148, 223, 286, 181 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Apart from the second diagonal (which gives the toothpick sequence A139250), the rest of the diagonals cannot be represented with toothpick structures. - Omar E. Pol, Dec 14 2016

LINKS

Table of n, a(n) for n=1..75.

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

FORMULA

See A162958 for rules governing the generation of N-th Toothpick sequences. By way of example, (N+2), A139250. The generator is A160552, which uses the multiplier "2". Then A160552 convolved with (1, 2, 2, 2,...) = A139250 the Toothpick sequence for N=2. Similarly, we create an array for Toothpick sequences N=1, 2, 3,...etc = A163267, A139250, A162958,...; then take the antidiagonals, creating triangle A163311.

EXAMPLE

Triangle begins:

1;

1, 2;

1, 3,  4;

1, 4,  7,  5;

1, 5,  10, 11,  7;

1, 6,  13, 19,  15,  10;

1, 7,  16, 29,  25,  23,  13;

1, 8,  19, 41,  37,  40,  35,  14;

1, 9,  22, 55   51,  61,  67,  43,  16;

1, 10, 25, 71,  67,  86,  109, 94,  47,  19;

1, 11, 28, 89,  85,  115, 161, 173, 100, 55,  22;

1, 12, 31, 109, 105, 148, 223, 286, 181, 115, 67,  25;

1, 13, 34, 131, 127, 185, 295, 439, 296, 205, 142, 79,  30;

1, 14, 37, 155, 151, 226, 377, 638, 451, 331, 253, 175, 95, 36;

...

CROSSREFS

Row sums = A163312: (1, 3, 8, 17, 34, 64,...).

Right border = A163267, toothpick sequence for N=1.

Next diagonal going to the left = A139250, toothpick sequence for N=2.

Then 1, 4, 10, 19,... = A162958, toothpick sequence for N=3.

Cf. A160552, A162958, A163311, A163312.

Sequence in context: A174829 A132110 A039912 * A210555 A008949 A076832

Adjacent sequences:  A163308 A163309 A163310 * A163312 A163313 A163314

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jul 24 2009

STATUS

approved

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Last modified October 21 06:01 EDT 2019. Contains 328291 sequences. (Running on oeis4.)