A177020 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.

1, 3, 1, 12, 5, 1, 53, 25, 7, 1, 247, 126, 42, 9, 1, 1192, 642, 239, 63, 11, 1, 5897, 3306, 1330, 400, 88, 13, 1, 29723, 17187, 7327, 2419, 617, 117, 15, 1, 152020, 90099, 40187, 14233, 4033, 898, 150, 17, 1

Offset: 0

Author

N. J. A. Sloane, Dec 08 2010

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins
1
3    1
12   5   1
53   25  7   1
247  126 42  9  1
1192 642 239 63 11 1
...
		

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.

Crossrefs

Cf. A177011.

Programs

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 *)

Extensions

a(15)-a(44) from Nathaniel Johnston, Apr 15 2011