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

%I A094392 #20 Jan 05 2025 19:51:37
%S A094392 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,
%T A094392 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,
%U A094392 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
%N A094392 Antidiagonals of the tables formed from b(m,2,n,n), which is defined in Du 1989.
%H A094392 Bau-Sen Du, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-2/du.pdf">A Simple Method Which Generates Infinitely Many Congruence Identities</a>, Fib. Quart. 27 (1989), 116-124.
%H A094392 Bau-Sen Du, <a href="http://arXiv.org/abs/0706.2421">A Simple Method Which Generates Infinitely Many Congruence Identities</a>, arXiv:0706.2421 [math.NT], 2007.
%F A094392 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.
%e A094392 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.
%e A094392     1   1  1  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392     1   1  1  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392     2   1  1  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392     3   1  1  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392     5   2  1  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392     8   3  1  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392    13   5  2  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392    21   7  3  1  1 1 1 1 1 1 1 1 1 1 1
%e A094392    34  11  5  2  1 1 1 1 1 1 1 1 1 1 1
%e A094392    55  16  7  3  1 1 1 1 1 1 1 1 1 1 1
%e A094392    89  25 11  5  2 1 1 1 1 1 1 1 1 1 1
%e A094392   144  37 15  7  3 1 1 1 1 1 1 1 1 1 1
%e A094392   233  57 23 11  5 2 1 1 1 1 1 1 1 1 1
%e A094392   377  85 32 15  7 3 1 1 1 1 1 1 1 1 1
%e A094392   610 130 49 23 11 5 2 1 1 1 1 1 1 1 1
%p A094392 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
%t A094392 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]];
%t A094392 a[m_, n_] := b[m, 2, n, n];
%t A094392 Table[a[m - n + 1, n], {m, 1, 14}, {n, m, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 21 2017, adapted from Maple *)
%Y A094392 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.
%K A094392 nonn,tabl
%O A094392 1,6
%A A094392 Amy Robinson (amylou(AT)mchsi.com), Apr 28 2004
%E A094392 Corrected and extended by Chris Deugau (deugaucj(AT)uvic.ca), Dec 19 2005
%E A094392 Typo 891 -> 89,1 corrected by _Jean-François Alcover_, Nov 21 2017