A351996 -id:A351996 - cp's OEIS frontend

A352000 A chain reaction sequence: a digit d1 from a(n) is expelled towards a(n+1) where it hits a digit d2 [from a(n+1)] and replaces it; d2 in turn is expelled towards a(n+2), hits a digit d3 there and replaces it; d3 in turn is expelled towards a(n+3), hits a digit there, and replaces it; d4 is expelled... etc. At the end of the chain reaction, only square numbers will be left. This is the lexicographically earliest sequence of distinct positive integers with this property.

1, 2, 15, 3, 16, 4, 5, 20, 100, 101, 6, 10, 102, 25, 35, 26, 45, 7, 129, 8, 11, 9, 40, 103, 36, 46, 19, 56, 21, 66, 12, 55, 22, 65, 13, 76, 184, 80, 104, 29, 75, 229, 85, 31, 86, 41, 39, 96, 42, 95, 43, 124, 81, 82, 111, 83, 161, 84, 49, 59, 23, 224, 121, 125, 87, 284, 131, 261, 141, 60, 105, 24, 61, 88

Offset: 1

Eric Angelini and Carole Dubois, Feb 27 2022

The sequence is a permutation of the integers > 0.

			1 is expelled from a(1) = 1 and hits the 2 of a(2) = 2, turning this integer into 1, a square number;
2 is expelled from a(2) = 2 and hits the 1 of a(3) = 15, turning this integer into 25, a square number;
1 is expelled from a(3) = 15 and hits the 3 of a(4) = 3, turning this integer into 1, a square number;
3 is expelled from a(4) = 3 and hits the 1 of a(5) = 16, turning this integer into 36, a square number;
1 is expelled from a(5) = 16 and hits the 4 of a(6) = 4, turning this integer into 1, a square number;
4 is expelled from a(6) = 4 and hits the 5 of a(7) = 5, turning this integer into 4, a square number;
5 is expelled from a(7) = 5 and hits the 0 of a(8) = 20, turning this integer into 25, a square number;
0 is expelled from a(8) = 20 and hits a 0 of a(9) = 100, "turning" this integer into 100, a square number;
0 is expelled from a(9) = 100 and hits the rightmost 1 of a(10) = 101, turning this integer into 100, a square number; etc.

Eric Angelini, A chain reaction producing primes, personal blog of the author, Feb. 2022.

Cf. A351996 (prime numbers left), A351997 (odd numbers left), A351998 (even numbers left), A351999 (Fibonacci numbers left), A000290 (the squares).

A351997 A chain reaction sequence: a digit d1 from a(n) is expelled towards a(n+1) where it hits a digit d2 [from a(n+1)] and replaces it; d2 in turn is expelled towards a(n+2), hits a digit d3 there and replaces it; d3 in turn is expelled towards a(n+3), hits a digit there, and replaces it; d4 is expelled... etc. At the end of the chain reaction, only odd numbers will be left. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 2, 11, 3, 4, 13, 5, 6, 15, 7, 8, 17, 9, 10, 101, 103, 105, 107, 109, 111, 12, 19, 14, 21, 23, 25, 27, 29, 31, 16, 33, 18, 35, 20, 113, 22, 37, 24, 39, 26, 41, 43, 45, 47, 49, 51, 28, 53, 30, 115, 32, 55, 34, 57, 36, 59, 38, 61, 63, 65, 67, 69, 71, 40, 117, 42, 73, 44, 75, 46, 77, 48, 79, 50, 119, 52, 81

Offset: 1

Eric Angelini and Carole Dubois, Feb 27 2022

The sequence is a permutation of the integers > 0.

			1 is expelled from a(1) = 1 and hits the 2 of a(2) = 2, turning this integer into 1, an odd number;
2 is expelled from a(2) = 2 and hits the leftmost 1 of a(3) = 11, "turning" this integer into 11, an odd number;
1 is expelled from a(3) = 11 and hits the 3 of a(4) = 3, turning this integer into 1, an odd number;
3 is expelled from a(4) = 3 and hits the 4 of a(5) = 4, turning this integer into 3, an odd number;
4 is expelled from a(5) = 4 and hits the 1 of a(6) = 13, turning this integer into 43, an odd number;
1 is expelled from a(6) = 13 and hits the 5 of a(7) = 5, turning this integer into 1, an odd number; etc.

Eric Angelini, A chain reaction producing primes, personal blog of the author, Feb. 2022.

Cf. A351996 (prime numbers left), A351998 (even numbers left), A351999 (Fibonacci numbers left), A352000 (square numbers left).

A351998 A chain reaction sequence: a digit d1 from a(n) is expelled towards a(n+1) where it hits a digit d2 [from a(n+1)] and replaces it; d2 in turn is expelled towards a(n+2), hits a digit d3 there and replaces it; d3 in turn is expelled towards a(n+3), hits a digit there, and replaces it; d4 is expelled... etc. At the end of the chain reaction, only even numbers will be left. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 10, 12, 14, 16, 18, 20, 2, 3, 22, 4, 5, 24, 6, 7, 26, 8, 9, 28, 11, 30, 32, 34, 36, 38, 40, 13, 42, 15, 44, 17, 46, 19, 48, 21, 50, 52, 54, 56, 58, 60, 23, 62, 25, 64, 27, 66, 29, 68, 31, 70, 72, 74, 76, 78, 80, 33, 82, 35, 84, 37, 86, 39, 88, 41, 90, 92, 94, 96, 98, 100, 43, 102, 45, 104, 47, 106, 49

Offset: 1

Eric Angelini ansd Carole Dubois, Feb 27 2022

The sequence is a permutation of the integers > 0.

			1 is expelled from a(1) = 1 and hits the 1 of a(2) = 10, "turning" this integer into 10, an even number;
1 is expelled from a(2) = 10 and hits the 1 of a(3) = 12, "turning" this integer into 12, an even number;
1 is expelled from a(3) = 12 and hits the 1 of a(4) = 14, "turning" this integer into 14, an even number;
1 is expelled from a(4) = 14 and hits the 1 of a(5) = 16, "turning" this integer into 16, an even number;
1 is expelled from a(5) = 16 and hits the 1 of a(6) = 18, "turning" this integer into 18, an even number;
1 is expelled from a(6) = 18 and hits the 2 of a(7) = 20, turning this integer into 10, an even number;
2 is expelled from a(7) = 20 and hits the 2 of a(8) = 2, "turning" this integer into 2, an even number;
2 is expelled from a(8) = 2 and hits the 3 of a(9) = 3, turning this integer into 2, an even number;
3 is expelled from a(9) = 3 and hits the leftmost 2 of a(10) = 22, turning this integer into 32, an even number; etc.

Eric Angelini, A chain reaction producing primes, personal blog of the author, Feb. 2022.

Cf. A351996 (prime numbers left), A351997 (odd numbers left), A351999 (Fibonacci numbers left), A352000 (square numbers left).

A351999 A chain reaction sequence: a digit d1 from a(n) is expelled towards a(n+1) where it hits a digit d2 [from a(n+1)] and replaces it; d2 in turn is expelled towards a(n+2), hits a digit d3 there and replaces it; d3 in turn is expelled towards a(n+3), hits a digit there, and replaces it; d4 is expelled... etc. At the end of the chain reaction, only Fibonacci numbers will be left. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 2, 3, 4, 30, 610, 611, 5, 6, 110, 7, 307, 612, 8, 9, 80, 613, 10, 614, 31, 13, 20, 615, 15, 21, 22, 11, 23, 12, 41, 32, 51, 25, 61, 210, 71, 317, 24, 33, 14, 26, 310, 16, 410, 34, 35, 45, 36, 510, 50, 616, 710, 327, 81, 19, 27, 337, 17, 347, 37, 357, 52, 91, 82, 133, 28, 29, 233, 333, 18, 39, 44

Offset: 1

Eric Angelini and Carole Dubois, Feb 27 2022

The sequence is a permutation of the integers > 0.

			1 is expelled from a(1) = 1 and hits the 2 of a(2) = 2, "turning" this integer into 2, a Fibonacci number;
2 is expelled from a(2) = 2 and hits the 3 of a(3) = 3, "turning" this integer into 3, a Fibonacci number;
3 is expelled from a(3) = 3 and hits the 4 of a(4) = 4, turning this integer into 3, a Fibonacci number;
4 is expelled from a(4) = 4 and hits the 0 of a(5) = 30, turning this integer into 34, a Fibonacci number;
0 is expelled from a(5) = 30 and hits the 0 of a(6) = 610, "turning" this integer into 610, a Fibonacci number;
0 is expelled from a(6) = 610 and hits the rightmost 1 of a(7) = 611, turning this integer into 610, a Fibonacci number;
1 is expelled from a(7) = 611 and hits the 5 of a(8) = 5, turning this integer into 1, a Fibonacci number;
5 is expelled from a(8) = 5 and hits the 6 of a(9) = 6, turning this integer into 5, a Fibonacci number;
6 is expelled from a(9) = 110 and hits the leftmost 1 of a(7) = 110, turning this integer into 610, a Fibonacci number; etc.

Eric Angelini, A chain reaction producing primes, personal blog of the author, Feb. 2022.

Cf. A351996 (prime numbers left), A351997 (odd numbers left), A351998 (even numbers left), A352000 (square numbers left), A000045 (Fibonacci numbers).

A361996 Order array of A361994, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 7, 4, 15, 17, 11, 5, 39, 43, 28, 14, 8, 102, 112, 73, 38, 20, 9, 268, 292, 191, 100, 51, 23, 10, 568, 592, 491, 263, 132, 61, 27, 12, 868, 892, 791, 563, 345, 159, 72, 32, 13, 1168, 1192, 1091, 863, 645, 416, 189, 83, 35, 16, 1468, 1492, 1391

Offset: 1

Clark Kimberling, Apr 05 2023

This array is an interspersion (hence a dispersion, as in A114537 and A163255), so every positive integer occurs exactly once. See A333029 for the definition of order array.

			Corner:
  1    2    6   15   39  102  268 ...
  3    7   17   43  112  292  592 ...
  4   11   28   73  191  491  791 ...
  5   14   38  100  263  563  863 ...
  8   20   51  132  345  645  945 ...
  9   23   61  159  416  716 1016 ...
  ...

Cf. A114537, A163255, A333029, A361993, A361996.

zz = 300; z = 30;
w[n_, k_] := w[n, k] = Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
b[h_, k_] := b[h, k] = w[2 h - 1, 2 k - 1] + w[2 h - 1, 2 k] + w[2 h, 2 k - 1] + w[2 h, 2 k];
s = Flatten[Table[b[h, k], {h, 1, zz}, {k, 1, z}]];
r[h_, k_] := Length[Select[s, # <= b[h, k] &]]
TableForm[Table[r[h, k], {h, 1, 50}, {k, 1, 12}]](*A351996, array*)
v = Table[r[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A351996, sequence *)

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A361996 Order array of A361994, read by descending antidiagonals.

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