A151548 -id:A151548 - cp's OEIS frontend

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 127, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137

Offset: 1

Omar E. Pol, Apr 04 2015

Row sums give A002001.
The sum of all terms of first n rows gives A000302(n-1).
The rows of triangle A256263 converge to this sequence.
Rows converge to A016969.
First 11 terms agree with A151548.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
5,7;
5,11,17,15;
5,11,17,23,29,35,41,31;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,131,137,143,149,155,161,167,173,179,185,127;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1    1      |_| | | |_ _  | |_ _ _ _ _ _  | |
2    3      |_ _| | |_  | | |_ _ _ _ _  | | |
3    5      |_ _ _| | | | | |_ _ _ _  | | | |
4    7      |_ _ _ _| | | | |_ _ _  | | | | |
5    5      | | |_ _ _| | | |_ _  | | | | | |
6   11      | |_ _ _ _ _| | |_  | | | | | | |
7   17      |_ _ _ _ _ _ _| | | | | | | | | |
8   15      |_ _ _ _ _ _ _ _| | | | | | | | |
9    5      | | | | | | |_ _ _| | | | | | | |
10  11      | | | | | |_ _ _ _ _| | | | | | |
11  17      | | | | |_ _ _ _ _ _ _| | | | | |
12  23      | | | |_ _ _ _ _ _ _ _ _| | | | |
13  29      | | |_ _ _ _ _ _ _ _ _ _ _| | | |
14  35      | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
15  41      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
16  31      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
It appears that A241717 can be represented by a similar diagram.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A000225, A000302, A002001, A011782, A016969, A141548, A241717, A256260, A256261, A256263, A256264.

Nest[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 7] (* Ivan Neretin, Feb 14 2017 *)

A162958 Equals A162956 convolved with (1, 3, 3, 3, ...).

Original entry on oeis.org

1, 4, 10, 19, 25, 40, 67, 94, 100, 115, 142, 175, 208, 280, 388, 469, 475, 490, 517, 550, 583, 655, 763, 850, 883, 955, 1069, 1201, 1372, 1696, 2101, 2344, 2350, 2365, 2392, 2425, 2458, 2530, 2638, 2725, 2758, 2830, 2944, 3076, 3247, 3571, 3976, 4225, 4258

Offset: 1

Gary W. Adamson, Jul 18 2009

nonn

Can be considered a toothpick sequence for N=3, following rules analogous to those in A160552 (= special case of "A"), A151548 = special case "B", and the toothpick sequence A139250 (N=2) = special case "C".
To obtain the infinite set of toothpick sequences, (N = 2, 3, 4, ...), replace the multiplier "2" in A160552 with any N, getting a triangle with 2^n terms. Convolve this A sequence with (1, N, 0, 0, 0, ...) = B such that row terms of A triangles converge to B.
Then generalized toothpick sequences (C) = A convolved with (1, N, N, N, ...).
Examples: A160552 * (1, 2, 0, 0, 0,...) = a B-type sequence A151548.
A160552 * (1, 2, 2, 2, 2,...) = the toothpick sequence A139250 for N=2.
A162956 is analogous to A160552 but replaces "2" with the multiplier "3".
Row terms of A162956 tend to A162957 = (1, 3, 0, 0, 0, ...) * A162956.
Toothpick sequence for N = 3 = A162958 = A162956 * (1, 3, 3, 3, ...).
Row sums of "A"-type triangles = powers of (N+2); since row sums of A160552 = (1, 4, 16, 64, ...), while row sums of A162956 = (1, 5, 25, 125, ...).
Is there an illustration of this sequence using toothpicks? - Omar E. Pol, Dec 13 2016

Alois P. Heinz, Table of n, a(n) for n = 1..16384
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. A139250, A152548, A160552, A162956, A163267.

Third diagonal of A163311.

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

b[n_] := b[n] = If[n<2, n, Function[j, 3*b[j]+b[j+1]][n-2^Floor[Log[2, n]] ]];
a[n_] := a[n] = If[n == 0, 0, a[n-1] + 2*b[n-1] + b[n]];
Array[a, 100] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Clarified definition by Omar E. Pol, Feb 06 2017

A113679 Expansion of (1-x-2x^2)/(1-x).

Original entry on oeis.org

1, 0, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2

Offset: 0

Paul Barry, Nov 04 2005

From Gary W. Adamson, Mar 06 2012: (Start)
Signed (1, 0, -2, 2, -2, 2, ...) and convolved with the Toothpick sequence A139250 = A151548: (1, 3, 5, 7, 5, 11, ...).  The inverse of (1, 0, -2, 2, -2, ...) = A151575: (1, 0, 2, -2, 6, -10, 22, ...).
The unsigned sequence convolved with:
(1, 2, 3, ...) = A002523, (n^2 + 1). Convolved with:
(A001045) = .... A097064: (1, 1, 5, 9, 21, 41, ...).
(End)

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A113680.

Cf. A139250, A151548, A151575, A002523. - Gary W. Adamson, Mar 06 2012

CoefficientList[Series[(1-x-2x^2)/(1-x),{x,0,80}],x] (* or *) Join[{1,0}, PadRight[{},80,-2]] (* Harvey P. Dale, Mar 05 2012 *)

a(n) = C(0, n) + 2*C(1, n) - 2.

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

A255045 a(n) = (1 + A160552(n))/2.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 3, 6, 9, 8, 1, 2, 3, 4, 3, 6, 9, 8, 3, 6, 9, 10, 11, 20, 25, 16, 1, 2, 3, 4, 3, 6, 9, 8, 3, 6, 9, 10, 11, 20, 25, 16, 3, 6, 9, 10, 11, 20, 25, 18, 11, 20, 27, 30, 41, 64, 65, 32, 1, 2, 3, 4, 3, 6, 9, 8, 3, 6, 9, 10, 11, 20, 25, 16, 3, 6, 9, 10, 11, 20, 25, 18, 11, 20, 27, 30, 41, 64, 65, 32, 3, 6, 9, 10, 11, 20, 25, 18

Offset: 1

Omar E. Pol, Feb 13 2015

Triangle of numbers related to cellular automata.

It appears that this is also a triangle read by rows in which row n lists the first 2^(n-1) terms of A255046, with n >= 1.

			Written as an irregular triangle in which the length of row j is 2^j, j >= 0, the sequence begins:
1;
1,2;
1,2,3,4;
1,2,3,4,3,6,9,8;
1,2,3,4,3,6,9,8,3,6,9,10,11,20,25,16;
1,2,3,4,3,6,9,8,3,6,9,10,11,20,25,16,3,6,9,10,11,20,25,18,11,20,27,30,41,64,65,32;
...
It appears that the right border gives A000079.
It appears that the row sums give A007582.
It appears that rows converge to A255046.

Index entries for sequences related to cellular automata

Cf. A000079, A007582, A139251, A151548, A160552, A169708, A255046, A255049.

cp's OEIS Frontend

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

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A162958 Equals A162956 convolved with (1, 3, 3, 3, ...).

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A113679 Expansion of (1-x-2x^2)/(1-x).

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

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A255045 a(n) = (1 + A160552(n))/2.

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