A063179 -id:A063179 - cp's OEIS frontend

A063180 The array of A063179 read by diagonals in direction of creation.

1, 0, 1, 0, 1, 2, 0, 2, 3, 4, 0, 4, 7, 10, 12, 0, 12, 22, 31, 38, 42, 0, 42, 80, 115, 144, 166, 178, 0, 178, 344, 500, 637, 748, 828, 870, 0, 870, 1698, 2488, 3205, 3820, 4308, 4652, 4830, 0, 4830, 9482, 13968, 18132

Offset: 1

Floor van Lamoen, Jul 09 2001

Cf. A063179, A063181.

A063181 The array of A063179 read by diagonals in the 'up' direction.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 0, 2, 3, 4, 12, 10, 7, 4, 0, 0, 12, 22, 31, 38, 42, 178, 166, 144, 115, 80, 42, 0, 0, 178, 344, 500, 637, 748, 828, 870, 4830, 4652, 4308, 3820, 3205, 2488, 1698, 870, 0, 0, 4830, 9482, 13968, 18132

Offset: 1

Floor van Lamoen, Jul 09 2001

Cf. A063179, A063180.

A063415 Triangle of coefficients of di-Boustrophedon transform (see A063179) read by rows: Let the original sequence be (U0,U1,...) and the transformed sequence (V0,V2,...), then Vn is a linear combination of U0,...,Un. T(n,m) is the coefficient that goes with Um to get Vn.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 12, 14, 9, 4, 1, 42, 51, 32, 14, 5, 1, 178, 214, 137, 60, 20, 6, 1, 870, 1049, 668, 295, 100, 27, 7, 1, 4830, 5820, 3713, 1636, 555, 154, 35, 8, 1, 29976, 36125, 23036, 10160, 3446, 952, 224, 44, 9, 1, 205572, 247734, 157993, 69664

Offset: 0

Floor van Lamoen, Jul 19 2001

			The triangle begins:
......1
....1...1
..2...2...1
4...5...3...1

T(n, 0) is A063179. Row sums form A062704. T(n, n-2) is A000096. Cf. A059718.

A062704 Di-Boustrophedon transform of all 1's sequence: Fill in an array by diagonals alternating in the 'up' and 'down' directions. Each diagonal starts with a 1. When going in the 'up' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the row the new element is in. When going in the 'down' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the column the new element is in. The final element of the n-th diagonal is a(n).

Original entry on oeis.org

1, 2, 5, 13, 40, 145, 616, 3017, 16752, 103973, 713040, 5352729, 43645848, 384059537, 3626960272, 36585357429, 392545057280, 4463791225145, 53622168102640, 678508544425721, 9020035443775264, 125684948107190045, 1831698736650660952, 27866044704218390113

Offset: 1

Floor van Lamoen, Jul 11 2001

			The array begins:
   1   2   1  13   1
   1   3  10  14
   5   6  25
   1  34
  40

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000667, A059216, A063179.

T:= proc(n, k) option remember;
      if n<1 or k<1 then 0
    elif n=1 and irem(k, 2)=1 or k=1 and irem(n, 2)=0 then 1
    elif irem(n+k, 2)=0 then T(n-1, k+1)+T(n-1, k)+T(n-2, k)
                        else T(n+1, k-1)+T(n, k-1)+T(n, k-2)
      fi
    end:
a:= n-> `if`(irem (n, 2)=0, T(1, n), T(n, 1)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 08 2011

T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0
     , n == 1 && Mod[k, 2] == 1 || k == 1 && Mod[n, 2] == 0, 1
     , Mod[n + k, 2] == 0, T[n - 1, k + 1] + T[n - 1, k] + T[n - 2, k]
     , True,               T[n + 1, k - 1] + T[n, k - 1] + T[n, k - 2]];
a[n_] := If[Mod [n, 2] == 0, T[1, n], T[n, 1]];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

More terms from Alois P. Heinz, Feb 08 2011

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A063180 The array of A063179 read by diagonals in direction of creation.

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A063181 The array of A063179 read by diagonals in the 'up' direction.

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A063415 Triangle of coefficients of di-Boustrophedon transform (see A063179) read by rows: Let the original sequence be (U0,U1,...) and the transformed sequence (V0,V2,...), then Vn is a linear combination of U0,...,Un. T(n,m) is the coefficient that goes with Um to get Vn.

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