A094392 Antidiagonals of the tables formed from b(m,2,n,n), which is defined in Du 1989.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 8, 1, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 1, 1, 5, 21, 1, 1, 1, 1, 1, 1, 2, 7, 34, 1, 1, 1, 1, 1, 1, 1, 3, 11, 55, 1, 1, 1, 1, 1, 1, 1, 1, 5, 16, 89, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 25, 144, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 11, 37, 233, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amy Robinson (amylou(AT)mchsi.com), Apr 28 2004

Keywords

• nonn
• tabl

Examples

			E.g., for m = 5 and n = 2, b(5,2,2,2)= b(3,2,1,2) + b(4,2,2,2)= 2 because of the definition in the reference.
    1   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    1   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    2   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    3   1  1  1  1 1 1 1 1 1 1 1 1 1 1
    5   2  1  1  1 1 1 1 1 1 1 1 1 1 1
    8   3  1  1  1 1 1 1 1 1 1 1 1 1 1
   13   5  2  1  1 1 1 1 1 1 1 1 1 1 1
   21   7  3  1  1 1 1 1 1 1 1 1 1 1 1
   34  11  5  2  1 1 1 1 1 1 1 1 1 1 1
   55  16  7  3  1 1 1 1 1 1 1 1 1 1 1
   89  25 11  5  2 1 1 1 1 1 1 1 1 1 1
  144  37 15  7  3 1 1 1 1 1 1 1 1 1 1
  233  57 23 11  5 2 1 1 1 1 1 1 1 1 1
  377  85 32 15  7 3 1 1 1 1 1 1 1 1 1
  610 130 49 23 11 5 2 1 1 1 1 1 1 1 1
		

Links

• Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, Fib. Quart. 27 (1989), 116-124.
• Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, arXiv:0706.2421 [math.NT], 2007.

Crossrefs

Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006208 (A_{n,3}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

Programs

Maple

b := proc(k,i,j,n) option remember; if k = 1 then if i = 1 then return 0; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if k = 2 then if i = 1 then return 1; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if j = n then return b(k-2, i, 1, n) + b(k-1, i, n, n); end if; return b(k-2, i, 1, n) + b(k-2, i, j+1, n); end proc; # Chris Deugau (deugaucj(AT)uvic.ca), Dec 19 2005

Mathematica

b[k_, i_, j_, n_] := b[k, i, j, n] = Which[k == 1, Which[i == 1, 0, i == 2 , If[j == n, 1, 0], True, 0], k == 2, Which[i == 1, 1, i == 2, If[j == n, 1, 0], True, 0], j == n, b[k - 2, i, 1, n] + b[k - 1, i, n, n], True, b[k - 2, i, 1, n] + b[k - 2, i, j + 1, n]];
a[m_, n_] := b[m, 2, n, n];
Table[a[m - n + 1, n], {m, 1, 14}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Nov 21 2017, adapted from Maple *)

Formula

For i=2 and k >= 1 b(k+2, 2, n, n)=b(k, 2, 1, n) + b(k+1, 2, n, n). The remaining portion for the recurrence is defined in Du 1989.

Extensions

Corrected and extended by Chris Deugau (deugaucj(AT)uvic.ca), Dec 19 2005

Typo 891 -> 89,1 corrected by Jean-François Alcover, Nov 21 2017