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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
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified September 20 18:52 EDT 2019. Contains 327245 sequences. (Running on oeis4.)