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.
1, 2, 1, 7, 4, 1, 29, 18, 6, 1, 130, 85, 33, 8, 1, 611, 414, 177, 52, 10, 1, 2965, 2062, 943, 313, 75, 12, 1, 14726, 10447, 5023, 1817, 501, 102, 14, 1, 74443, 53650, 26818, 10348, 3152, 749, 133, 16, 1, 381617, 278568, 143655, 58305, 19147, 5080, 1065, 168, 18, 1
Offset: 0
Examples
Triangle begins 1 2 1 7 4 1 29 18 6 1 130 85 33 8 1 611 414 177 52 10 1 ...
Links
- Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (first 100 rows of triangle)
- P. Fahr, C. M. Ringel, A partition formula for fibonacci numbers, JIS 11 (2008) 08.1.4, section 4.
- Harris Kwong, On recurrences of Fahr and Ringel: an alternate approach, Fibonacci Quart. 48 (2010), no. 4, 363-365.
Crossrefs
Cf. A177020.
Programs
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Maple
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: 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: 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
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Mathematica
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 *)
Extensions
a(15)-a(54) from Nathaniel Johnston, Apr 15 2011