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A177011 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 B(t,r) read by rows.

%I A177011 #21 Jan 05 2025 19:51:39
%S A177011 1,2,1,7,4,1,29,18,6,1,130,85,33,8,1,611,414,177,52,10,1,2965,2062,
%T A177011 943,313,75,12,1,14726,10447,5023,1817,501,102,14,1,74443,53650,26818,
%U A177011 10348,3152,749,133,16,1,381617,278568,143655,58305,19147,5080,1065,168,18,1
%N A177011 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 B(t,r) read by rows.
%H A177011 Nathaniel Johnston, <a href="/A177011/b177011.txt">Table of n, a(n) for n = 0..5150</a> (first 100 rows of triangle)
%H A177011 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 A177011 Harris Kwong, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-4/Kwong.pdf">On recurrences of Fahr and Ringel: an alternate approach</a>, Fibonacci Quart. 48 (2010), no. 4, 363-365.
%e A177011 Triangle begins
%e A177011 1
%e A177011 2   1
%e A177011 7   4   1
%e A177011 29  18  6   1
%e A177011 130 85  33  8  1
%e A177011 611 414 177 52 10 1
%e A177011 ...
%p A177011 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 A177011 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 A177011 for t from 0 to 9 do for r from 0 to t do print(B(t,r)):od:od: # _Nathaniel Johnston_, Apr 15 2011
%t A177011 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[bb[t, r], {t, 0, 9}, {r, 0, t}] // Flatten (* _Jean-François Alcover_, Jan 08 2014, translated from Maple *)
%Y A177011 Cf. A177020.
%K A177011 nonn,tabl,easy
%O A177011 0,2
%A A177011 _N. J. A. Sloane_, Dec 08 2010
%E A177011 a(15)-a(54) from _Nathaniel Johnston_, Apr 15 2011