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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. 5

%I #19 Oct 12 2016 09:57:06

%S 1,2,1,7,4,1,29,18,6,1,130,85,33,8,1,611,414,177,52,10,1,2965,2062,

%T 943,313,75,12,1,14726,10447,5023,1817,501,102,14,1,74443,53650,26818,

%U 10348,3152,749,133,16,1,381617,278568,143655,58305,19147,5080,1065,168,18,1

%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 B(t,r) read by rows.

%H Nathaniel Johnston, <a href="/A177011/b177011.txt">Table of n, a(n) for n = 0..5150</a> (first 100 rows of 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 Harris Kwong, <a href="http://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 Triangle begins

%e 1

%e 2 1

%e 7 4 1

%e 29 18 6 1

%e 130 85 33 8 1

%e 611 414 177 52 10 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 9 do for r from 0 to t do print(B(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[bb[t, r], {t, 0, 9}, {r, 0, t}] // Flatten (* _Jean-François Alcover_, Jan 08 2014, translated from Maple *)

%Y Cf. A177020.

%K nonn,tabl,easy

%O 0,2

%A _N. J. A. Sloane_, Dec 08 2010

%E a(15)-a(54) from _Nathaniel Johnston_, Apr 15 2011

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)