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

0,2

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.

Example

Triangle begins
1
2   1
7   4   1
29  18  6   1
130 85  33  8  1
611 414 177 52 10 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 9 do for r from 0 to t do print(B(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[bb[t, r], {t, 0, 9}, {r, 0, t}] // Flatten (* Jean-François Alcover, Jan 08 2014, translated from Maple *)

Crossrefs

Cf. A177020.

Sequence in context: A144696 A072248 A317360 * A092276 A011274 A122843

Adjacent sequences: A177008 A177009 A177010 * A177012 A177013 A177014

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Dec 08 2010

Extensions

a(15)-a(54) from Nathaniel Johnston, Apr 15 2011

Status

approved