A177020 -id:A177020 - cp's OEIS frontend

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

N. J. A. Sloane, Dec 08 2010

			Triangle begins
1
2   1
7   4   1
29  18  6   1
130 85  33  8  1
611 414 177 52 10 1
...

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.

Cf. A177020.

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

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 *)

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

A197956 Even-index Fibonacci partition triangle read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 7, 1, 5, 12, 1, 6, 18, 29, 1, 7, 25, 53, 1, 8, 33, 85, 130, 1, 9, 42, 126, 247, 1, 10, 52, 177, 414, 611, 1, 11, 63, 239, 642, 1192, 1, 12, 75, 313, 943, 2062, 2965, 1, 13, 88, 400, 1330, 3306, 5897, 1, 14, 102, 501, 1817, 5023, 10447, 14726

Offset: 1

Claus Michael Ringel, Oct 24 2011

These are the entries of a triangle which starts
           1,
           1,
       1,  2,
       1,  3,
   1,  4,  7,
   1,  5, 12,
1, 6, 18, 29,...
The numbers d(i,n) in the row with index n are recursively defined for 0 <= n and 0 <= i <= n/2, by d(0,n) = 1 for all n, and d(i,n) = 2d(i-1,n-1) + d(i,n-1) - d(i-1,n-2) for 0 < i < n/2, and d(i,2i) = 3d(i-1,n-1) - d(i-1,n-2).
The numbers d(i,n-1) and d(i,n) form the dimension vector of the Fibonacci modules P(n), these are indecomposable quiver representations of the 3-regular tree with bipartite orientation.
A linear combination of the row n (with coefficients either 1 or of the form 3*2^t) gives a partition of the Fibonacci number f_{2n+2}, see A000045 and A001906.
The sequence A177011 is obtained by reading the rows with even index from the right.
The sequence A177020 is obtained by reading the rows with odd index from the right.
The sequence of the entries in the last column of the triangle, with even row index is recorded as A132262, the sequence of the entries in the last column of the triangle, with odd row index is A110122 (Number of Delannoy paths of length n with no EE's crossing the line y=x)
The sequence A197957 is obtained by taking differences of pairs of numbers in neighboring rows of the triangle.

Philipp Fahr and Claus Michael Ringel, The Fibonacci partition triangles, http://arxiv.org/abs/1109.2849
P. Fahr, C. M. Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, J. Int. Seq. 15 (2012) # 12.2.1

cp's OEIS Frontend

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.

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A197956 Even-index Fibonacci partition triangle read by rows.

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