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Antidiagonals of the tables formed from b(m,2,n,n), which is defined in Du 1989.
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%I #18 May 17 2019 02:35:47

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

%H Bau-Sen Du, <a href="http://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 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 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 5 2 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 8 3 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 13 5 2 1 1 1 1 1 1 1 1 1 1 1 1

%e 21 7 3 1 1 1 1 1 1 1 1 1 1 1 1

%e 34 11 5 2 1 1 1 1 1 1 1 1 1 1 1

%e 55 16 7 3 1 1 1 1 1 1 1 1 1 1 1

%e 89 25 11 5 2 1 1 1 1 1 1 1 1 1 1

%e 144 37 15 7 3 1 1 1 1 1 1 1 1 1 1

%e 233 57 23 11 5 2 1 1 1 1 1 1 1 1 1

%e 377 85 32 15 7 3 1 1 1 1 1 1 1 1 1

%e 610 130 49 23 11 5 2 1 1 1 1 1 1 1 1

%p 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 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 a[m_, n_] := b[m, 2, n, n];

%t Table[a[m - n + 1, n], {m, 1, 14}, {n, m, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 21 2017, adapted from Maple *)

%Y 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 nonn,tabl

%O 1,6

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

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

%E Typo 891 -> 89,1 corrected by _Jean-François Alcover_, Nov 21 2017