OFFSET
0,2
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (first 100 rows of the triangle)
P. Fahr, C. M. Ringel, A partition formula for fibonacci numbers, JIS 11 (2008) 08.1.4, section 4.
H. Kwong, On recurrences of Fahr and Ringel: An Alternate Approach, Fib. Quart., 48 (2010), 363-365.
EXAMPLE
Triangle begins
1
3 1
12 5 1
53 25 7 1
247 126 42 9 1
1192 642 239 63 11 1
...
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 8 do for r from 0 to t do print(C(t, r)):od:od: # Nathaniel Johnston, Apr 15 2011
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[cc[t, r], {t, 0, 8}, {r, 0, t}] // Flatten (* Jean-François Alcover, Jan 08 2014, translated from Maple *)
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Dec 08 2010
EXTENSIONS
a(15)-a(44) from Nathaniel Johnston, Apr 15 2011
STATUS
approved