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%I #24 Sep 30 2019 06:24:51
%S 1,3,1,12,5,1,53,25,7,1,247,126,42,9,1,1192,642,239,63,11,1,5897,3306,
%T 1330,400,88,13,1,29723,17187,7327,2419,617,117,15,1,152020,90099,
%U 40187,14233,4033,898,150,17,1
%N Define two triangular arrays by: B(0,0)=C(0,0)=1, B(0,r)=C(0,r)=0 for r>0, C(t,-1)=C(t,0); and for t,r >= 0, B(t+1,r)=C(t,r-1)+2C(t,r)-B(t,r), C(t+1,r)=B(t+1,r)+2B(t+1,r+1)-C(t,r). Sequence gives array C(t,r) read by rows.
%H Nathaniel Johnston, <a href="/A177020/b177020.txt">Table of n, a(n) for n = 0..5150</a> (first 100 rows of the triangle)
%H P. Fahr, C. M. Ringel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Fahr/ringel44.html">A partition formula for fibonacci numbers</a>, JIS 11 (2008) 08.1.4, section 4.
%H H. Kwong, <a href="https://www.fq.math.ca/Papers1/48-4/Kwong.pdf">On recurrences of Fahr and Ringel: An Alternate Approach</a>, Fib. Quart., 48 (2010), 363-365.
%e Triangle begins
%e 1
%e 3 1
%e 12 5 1
%e 53 25 7 1
%e 247 126 42 9 1
%e 1192 642 239 63 11 1
%e ...
%p B:=proc(t,r)global b:if(not type(b[t,r],integer))then if(t=0 and r=0)then b[t,r]:=1:elif(t=0)then b[t,r]:=0:else b[t,r]:=C(t-1,r-1)+2*C(t-1,r)-B(t-1,r):fi:fi:return b[t,r]:end:
%p C:=proc(t,r)global c:if(not type(c[t,r],integer))then if(r=-1)then return C(t,0):fi:if(t=0 and r=0)then c[t,r]:=1:elif(t=0)then c[t,r]:=0:else c[t,r]:=B(t,r)+2*B(t,r+1)-C(t-1,r):fi:fi:return c[t,r]:end:
%p for t from 0 to 8 do for r from 0 to t do print(C(t,r)):od:od: # _Nathaniel Johnston_, Apr 15 2011
%t bb[t_, r_] := Module[{}, If[Not[IntegerQ[b[t, r]]], Which[t == 0 && r == 0, b[t, r] = 1, t == 0, b[t, r] = 0, True, b[t, r] = cc[t-1, r-1] + 2*cc[t-1, r] - bb[t-1, r]]]; Return[b[t, r]]]; cc[t_, r_] := Module[{}, If[Not[IntegerQ[c[t, r]]], If[r == -1, Return[cc[t, 0]], Which[t == 0 && r == 0, c[t, r] = 1, t == 0, c[t, r] = 0, True, c[t, r] = bb[t, r] + 2*bb[t, r+1] - cc[t-1, r]]]]; Return[c[t, r]]]; Table[cc[t, r], {t, 0, 8}, {r, 0, t}] // Flatten (* _Jean-François Alcover_, Jan 08 2014, translated from Maple *)
%Y Cf. A177011.
%K nonn,tabl,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 08 2010
%E a(15)-a(44) from _Nathaniel Johnston_, Apr 15 2011
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