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).

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

Author

Floor van Lamoen, Jul 11 2001

Keywords

• easy
• nonn

Examples

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

Links

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

Crossrefs

Cf. A000667, A059216, A063179.

Programs

Maple

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

Mathematica

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 *)

Extensions

More terms from Alois P. Heinz, Feb 08 2011