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.

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, 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, 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

Offset

0,6

Comments

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:

(i) Columns sum create the irregular triangle A268317.

(ii) Rows sum create the present irregular triangle.

(iii) The row sums of each of these irregular triangles is conjectured to be A000071.

(iv) The first differences of the sequence of half of the voids (0's) are conjectured to give A191797.

See illustrations in the links of A268317.

Links

Table of n, a(n) for n=0..100.

Example

Irregular triangle begins:
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 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 1
...

Prog

(Small Basic)
TextWindow.Write("0, 1, 1, 1, 1, 2, 1, ")
t[3][1] = 1
t[3][2] = 2
t[3][3] = 1
k[2] = 2
k[3] = 3
For n = 4 To 12
  k[n] = k[n-1] + k[n-2]
  c = math.Ceiling(k[n]/2)
  i1 = 1
  For j = 1 To k[n]
    If Math.Remainder(k[n], 2)<>0 Then
      If j > c then
        t[n][j] = t[n][j-2*i1]
        i1 = i1 + 1
      Else
        t[n][j] = t[n-1][j]
      EndIf
    Else
      If j <= c then
        t[n][j] = t[n-1][j]
      Else
        if j = c+1 Then
          t[n][j] = t[n][j-1]
        else
        t[n][j] = t[n][j-(2*i1+1)]
        i1 = i1 + 1
        endif
      EndIf
    EndIf
    TextWindow.Write(t[n][j]+", ")
  EndFor
EndFor

Crossrefs

Cf. A000071, A191797, A268317.

Sequence in context: A003842 A095771 A245933 * A309414 A007421 A239228

Adjacent sequences: A268315 A268316 A268317 * A268319 A268320 A268321

Keyword

nonn,base,tabf

Author

Kival Ngaokrajang, Feb 01 2016

Status

approved