A163311 Triangle read by rows in which the diagonals give the infinite set of Toothpick sequences.

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

Offset: 1

Author

Gary W. Adamson, Jul 24 2009

Keywords

• nonn
• tabl

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

Examples

			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;
...
		

Links

• 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

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.

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.