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A375723 Square array read by antidiagonals, where the top row is the powers of 2 (A000079) and the other numbers are the sum of the neighbors in the preceding row.

1, 2, 3, 4, 7, 10, 8, 14, 24, 34, 16, 28, 49, 83, 117, 32, 56, 98, 171, 288, 405, 64, 112, 196, 343, 597, 1002, 1407, 128, 224, 392, 686, 1200, 2085, 3492, 4899, 256, 448, 784, 1372, 2401, 4198, 7285, 12184, 17083, 512, 896, 1568, 2744, 4802, 8403, 14686, 25463, 42546, 59629

Offset: 0

WG Zeist, Aug 25 2024

Each number in the top row of the array is determined by the pre-defined sequence (in this case, the powers of 2, A000079). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three). This is the same method as for constructing A217536, which has the top row be the nonnegative integers instead; other similar arrays are described in the comments of that sequence.
The main diagonal is the powers of 7 (A000420), and all numbers above or to the right of the main diagonal are multiples of powers of 2 and powers of 7. Specifically, the number in row m and column n, for n >= m, is 2^(n-m) * 7^m. Above the main diagonal, all numbers in the same column have the same final digit in base 10, and all numbers are 7/2 times the number immediately above.
More broadly, for any similarly constructed array with the powers of x as the top row, then the main diagonal will be the powers of (x^2 + x + 1) and the numbers above the main diagonal will be x^(n-m) * (x^2 + x + 1)^m (see also A062105, which can be interpreted as a similar array with the powers of 1 in the top row, and A020474 with the powers of 0).

			The array starts:
  1  2  4   8
  3  7  14  28
  10 24 49  98
  34 83 171 343

The main diagonal gives A000420 (powers of 7). The first column gives A059738.

Cf. A217536, A062105, A020474.

T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=2^n), m, n >= 0.

A323382 a(n) is the period of the oscillating pattern formed by a diagonal line of 2*n cells in the Life-like cellular automaton B2e3ijkn4cz5/S236.

Original entry on oeis.org

2, 3, 2, 5, 8, 3, 20, 14, 26, 3, 36, 5, 106, 3, 14, 29, 48, 3, 80, 67, 156, 3, 74, 14, 594, 3, 26, 93, 440, 3, 380, 115, 1062, 3, 1826, 82, 1864, 3, 1488, 2603, 328, 3, 1714, 10229, 2330, 3, 1372, 23, 15202, 3, 43186, 7524, 49534, 3, 69612, 9508, 5112, 3, 1260, 54687

Offset: 1

WG Zeist, Jan 12 2019

nonn

An explanation of the Hensel notation used to define the cellular automaton rule can be found on the LifeWiki (see links).
Lines of odd lengths are excluded because they break up into patterns not consisting of diagonal lines.
These diagonal line oscillators are effectively emulating a four-state one-dimensional cellular automaton.
From Charlie Neder, Feb 12 2019: (Start)
Specifically, such an oscillator with 2*n cells is isomorphic to a row of 2*n state-1 cells that evolve according to the following rules:
1) A state-1 cell becomes state-3 if it has a state-1 neighbor, and state-2 otherwise,
2) A state-2 cell becomes state-1 unconditionally,
3) A state-3 cell becomes state-1 if both its neighbors are state-3, and state-2 otherwise. (End)

			a(4) = 5 because a diagonal line of 8 cells oscillates with period 5 in this cellular automaton.

LifeWiki, Isotropic non-totalistic Life-like cellular automaton

If n == 2 (mod 4), a(n) = 3.

A298819 a(n) is the period of the oscillating pattern formed by a 1 X n line of cells in the Life-like cellular automaton B2c3-cekq4ikt5i8/S2-in3-acky4aijry5eiky6i, or 0 if the pattern vanishes.

Original entry on oeis.org

0, 0, 2, 2, 2, 0, 6, 6, 12, 4, 28, 14, 14, 0, 14, 14, 84, 12, 372, 62, 248, 8, 504, 126, 252, 28, 420, 30, 30, 0, 30, 30, 420, 28, 2044, 1022, 12264, 24, 504, 126, 7812, 124, 253828, 4094, 32752, 16, 16368, 2046, 85932, 252, 18396, 1022, 4088, 56, 917448, 32766, 327660, 60, 1860, 62, 62, 0, 62, 62

Offset: 1

WG Zeist, Jan 26 2018

nonn

An explanation of the Hensel notation used to define the cellular automaton rule can be found on the LifeWiki (see links).

			a(4) = 2 because a 1 X 4 line oscillates with period 2 in this cellular automaton.

LifeWiki, Isotropic non-totalistic Life-like cellular automaton

Cf. A000918, A003558, A160657, A268754.

For n divisible by 4, a(n) = A160657(n/4) = (a divisor of) 2^(A003558(n/4) + 1) - 2.
With the exception of a(1), all values of n for which a(n) = 0 are of the form 2^m - 2 for some integer m (A000918).
a(2k) = A268754(k); a(2k+1) = lcm(A268754(k), A268754(k+1)) (conjectured). Note that, by an arbitrary/cosmetic convention, A268754 uses 1 for empty patterns and A298819 uses 0. - WG Zeist and Charlie Neder, Jan 11 2019

A217536 Square array read by antidiagonals, where the top row is the nonnegative integers and the other numbers are the sum of the neighbors in the preceding row.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 3, 6, 10, 14, 4, 9, 18, 32, 46, 5, 12, 27, 55, 101, 147, 6, 15, 36, 81, 168, 315, 462, 7, 18, 45, 108, 244, 513, 975, 1437, 8, 21, 54, 135, 324, 736, 1564, 3001, 4438, 9, 24, 63, 162, 405, 973, 2222, 4761, 9199, 13637, 10, 27, 72, 189, 486, 1215, 2924, 6710, 14472, 28109, 41746

Offset: 0

WG Zeist, Oct 06 2012

Each number in the top row of the array is determined by the pre-defined sequence (in this case, the nonnegative integers). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three).
Replacing the top row with A000012 (the all 1's sequence) and constructing the rest of the array the same way produces A062105. Similarly, replacing the top row with A000007 (a(n) = 0^n) produces A020474. - WG Zeist, Aug 24 2024
For any array constructed with this method, regardless of the sequence chosen for the top row, the sequence in the first column of the array can be computed from the sequence in the top row as follows: let a(0), a(1), a(2), ... be the terms in the top row, and b(0), b(1), b(2), ... the terms in the first column. Then b(n) = Sum_{k=0..n} A064189(n,k) * a(k). The inverse operation, to compute the top row from the first column, is given by a(n) = Sum_{k=0..n} A104562(n,k) * b(k). - WG Zeist, Aug 26 2024

			The array starts:
  0  1  2  3
  1  3  6  9
  4  10 18 27
  14 32 55 81

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Main diagonal gives A036290. First column gives A330796.

Cf. A020474, A062105, A064189, A104562.

A:= proc(n, k) option remember; `if`(k<0, 0,
     `if`(n=0, k, add(A(n-1, k+i), i=-1..1)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 24 2024

T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=n), m, n >= 0. - M. F. Hasler, Oct 09 2012

Offset 0 from Alois P. Heinz, Aug 24 2024

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User: WG Zeist

A375723 Square array read by antidiagonals, where the top row is the powers of 2 (A000079) and the other numbers are the sum of the neighbors in the preceding row.

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A323382 a(n) is the period of the oscillating pattern formed by a diagonal line of 2*n cells in the Life-like cellular automaton B2e3ijkn4cz5/S236.

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A298819 a(n) is the period of the oscillating pattern formed by a 1 X n line of cells in the Life-like cellular automaton B2c3-cekq4ikt5i8/S2-in3-acky4aijry5eiky6i, or 0 if the pattern vanishes.

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A217536 Square array read by antidiagonals, where the top row is the nonnegative integers and the other numbers are the sum of the neighbors in the preceding row.

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