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A268318 Irregular triangle read by rows: T(n,k) gives the row sums in the table Fib(n+1) X Fib(n), where k = 1..Fib(n+1), and 1's are assigned to cells on the longest diagonal path.

%I A268318 #9 Mar 03 2016 16:10:06
%S A268318 0,1,1,1,1,2,1,1,2,1,2,1,1,2,1,2,2,1,2,1,1,2,1,2,2,1,2,1,2,2,1,2,1,1,
%T A268318 2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,1,2,1,2,2,1,2,1,2,2,1,2,2,1,
%U A268318 2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,1,2,1,2,2,1,2,1,2,2,1,2,2
%N A268318 Irregular triangle read by rows: T(n,k) gives the row sums in the table Fib(n+1) X Fib(n), where k = 1..Fib(n+1), and 1's are assigned to cells on the longest diagonal path.
%C A268318 Inspired by sun flower spirals which come in  Fib(i) and Fib(i+1) numbers in opposite directions. The present Fib(n+1) X Fib(n) table has the following properties:
%C A268318 (i) Columns sum create the irregular triangle A268317.
%C A268318 (ii) Rows sum create the present irregular triangle.
%C A268318 (iii) The row sums of each of these irregular triangles is conjectured to be A000071.
%C A268318 (iv) The first differences of the sequence of half of the  voids (0's) are conjectured to give A191797.
%C A268318 See illustrations in the links of A268317.
%e A268318 Irregular triangle begins:
%e A268318 0
%e A268318 1
%e A268318 1 1
%e A268318 1 2 1
%e A268318 1 2 1 2 1
%e A268318 1 2 1 2 2 1 2 1
%e A268318 1 2 1 2 2 1 2 1 2 2 1 2 1
%e A268318 1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 1
%e A268318 ...
%o A268318 (Small Basic)
%o A268318 TextWindow.Write("0, 1, 1, 1, 1, 2, 1, ")
%o A268318 t[3][1] = 1
%o A268318 t[3][2] = 2
%o A268318 t[3][3] = 1
%o A268318 k[2] = 2
%o A268318 k[3] = 3
%o A268318 For n = 4 To 12
%o A268318   k[n] = k[n-1] + k[n-2]
%o A268318   c = math.Ceiling(k[n]/2)
%o A268318   i1 = 1
%o A268318   For j = 1 To k[n]
%o A268318     If Math.Remainder(k[n],2)<>0 Then
%o A268318       If j > c then
%o A268318         t[n][j] = t[n][j-2*i1]
%o A268318         i1 = i1 + 1
%o A268318       Else
%o A268318         t[n][j] = t[n-1][j]
%o A268318       EndIf
%o A268318     Else
%o A268318       If j <= c then
%o A268318         t[n][j] = t[n-1][j]
%o A268318       Else
%o A268318         if j = c+1 Then
%o A268318           t[n][j] = t[n][j-1]
%o A268318         else
%o A268318         t[n][j] = t[n][j-(2*i1+1)]
%o A268318         i1 = i1 + 1
%o A268318         endif
%o A268318       EndIf
%o A268318     EndIf
%o A268318     TextWindow.Write(t[n][j]+", ")
%o A268318   EndFor
%o A268318 EndFor
%Y A268318 Cf. A000071, A191797, A268317.
%K A268318 nonn,base,tabf
%O A268318 0,6
%A A268318 _Kival Ngaokrajang_, Feb 01 2016