A162958 -id:A162958 - cp's OEIS frontend

A162956 a(0) = 0, a(1) = 1; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.

0, 1, 1, 4, 1, 4, 7, 13, 1, 4, 7, 13, 7, 19, 34, 40, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 364, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46

Offset: 0

Gary W. Adamson, Jul 18 2009

nonn

2^n term triangle by rows, analogous to A160552 but multiplier is "3" instead of "2"
Row sums = powers of 5: (1, 5, 25, 125, 625,...).
Rows tend to A162957, obtained by taking (1, 3, 0, 0, 0,...) * A162956.

			The triangle begins:
0;
1;
1, 4;
1, 4, 7, 13;
1, 4, 7, 13, 7, 19, 34, 40;
1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121;
...
Row 4 = (1, 4, 7, 13, 7, 19, 34, 40): brings down (1, 4, 7, 13) then 7 = 3*1 + 4, 19 = 3*4 + 7, 34 = 3*7 + 13, 40 = 3*13 + 1.

Alois P. Heinz, Table of n, a(n) for n = 0..16383
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

Cf. A162957, A162958, A160552.

Cf. A170838-A170852, A170854-A170872.

a:= proc(n) option remember; `if`(n<2, n,
      (j-> 3*a(j)+a(j+1))(n-2^ilog2(n)))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 28 2017

row[0] = {0}; row[1] = {1}; row[n_] := row[n] = Join[row[n-1], 3 row[n-1] + Append[Rest[row[n-1]], 1]]; Table[row[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 13 2017 *)

Follows the same analogous procedure as A160552 but multiplier is 3 instead of 2. (n+1)-th row brings down n-th row and appends to the right and equal number of terms following the rules: from left to right,let a = last term, b = current term, c = next term. Then c = 3*a + b except for the rightmost term = 3*a + 1.

Edited with more terms by N. J. A. Sloane, Jan 02 2010

A163267 Partial sums of A118977.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 10, 13, 14, 16, 19, 22, 25, 30, 36, 40, 41, 43, 46, 49, 52, 57, 63, 67, 70, 75, 81, 87, 95, 106, 116, 121, 122, 124, 127, 130, 133, 138, 144, 148, 151, 156, 162, 168, 176, 187, 197, 202, 205, 210, 216, 222, 230, 241, 251, 258, 266, 277, 289, 303, 322, 343, 358

Offset: 0

Gary W. Adamson, Jul 24 2009

nonn

Can be considered toothpick sequence for N=1.
Based on a consistent set of rules for generating toothpick sequences.
Cf. A139250 where (1, 1, 3, 1, 3, 5, 7, ...) is convolved with (1, 2, 2, 2, ...) and A162958 where A162956 is convolved with (1, 3, 3, 3, ...); the present sequence can be considered toothpick N=1 since A118977 is convolved with (1, 1, 1, ...).
Arranged in array fashion, the first three toothpick sequences would be:
N=1: A163267: (1, 2,  4,  5,  7, 10, 13, 14, ...)
N=2: A139250: (1, 3,  7, 11, 15, 23, 35, 43, ...)
N=3: A162958: (1, 4, 10, 19, 25, 40, 67, 94, ...)
...
Is there an illustration of this sequence using toothpicks? - Omar E. Pol, Dec 13 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
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

Cf. A118977, A139250, A162958.

a[0] = 0; a[1] = 1; a[n_] := a[n] = (j = n - 2^Floor[Log[2, n]]; a[j] + a[j + 1]); Table[Sum[a[n], {n, 0, k}], {k, 0, 20}] (* G. C. Greubel, Dec 12 2016 *)

Edited and extended by N. J. A. Sloane, Jan 07 2010

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

Original entry on oeis.org

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

Gary W. Adamson, Jul 24 2009

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

			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;
...

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

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.

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.

A162957 A162956 convolved with (1, 3, 0, 0, 0, ...).

Original entry on oeis.org

1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 364, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 370, 40, 91, 148, 178, 211, 415, 553, 421, 211, 421, 622, 745, 1048, 1792, 2005, 1093

Offset: 1

Gary W. Adamson, Jul 18 2009

nonn

Rows of A162956 tend to this sequence. Analogous to rows of A160552 tending to A151548.

Cf. A162956, A162958.

Equals (1, 3, 0, 0, 0, ...) * (1, 1, 4, 1, 4, 7, 13, 1, 4, 7, 13, 7, 19, 34, 40, ...).

a(20), a(21) corrected and more terms from Georg Fischer, Jul 17 2025

cp's OEIS Frontend

A162956 a(0) = 0, a(1) = 1; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.

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A163267 Partial sums of A118977.

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

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A162957 A162956 convolved with (1, 3, 0, 0, 0, ...).

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