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 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. 3
 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 (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 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 Cf. A177011. Sequence in context: A287985 A117375 A162995 * A226167 A185105 A122844 Adjacent sequences:  A177017 A177018 A177019 * A177021 A177022 A177023 KEYWORD nonn,tabl,easy AUTHOR N. J. A. Sloane, Dec 08 2010 EXTENSIONS a(15)-a(44) from Nathaniel Johnston, Apr 15 2011 STATUS approved

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Last modified February 19 13:23 EST 2020. Contains 332044 sequences. (Running on oeis4.)