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A163311
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Triangle read by rows in which the diagonals give the infinite set of Toothpick sequences.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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.
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EXAMPLE
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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;
...
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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