A268317 Irregular triangle read by rows: T(n,k) gives the columns sum in the table Fib(n+1) X Fib(n), where k = 1..Fib(n), and 1's are assigned to cells on the longest diagonal path.

0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3

Offset

0,3

Comments

Inspired by sun flower spirals which come in Fib(i) and Fib(i+1) numbers in opposite directions. The present case of the Fib(n+1) X Fib(n) table has the following properties:

(i) Columns sum create the present irregular triangle.

(ii) Rows sum create the irregular triangle A268318.

(iii) The row sum 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.

Links

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

Kival Ngaokrajang, Illustration of initial terms, Sun flower spirals

Example

Irregular triangle begins:
1
2
2  2
2  3  2
2  3  2  3  2
2  3  2  3  3  2  3  2
2  3  2  3  3  2  3  2  3  3  2  3  2
...

Prog

(Small Basic)
TextWindow.Write("0, 1, 2, 2, 2, 2, 3, 2, ")
t[4][1] = 2
t[4][2] = 3
t[4][3] = 2
k[3] = 2
k[4] = 3
For n = 5 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 = i 1+ 1
        endif
      EndIf
    EndIf
    TextWindow.Write(t[n][j]+", ")
  EndFor
EndFor

Crossrefs

Cf. A000071, A191797, A268318.

Sequence in context: A089993 A220431 A351284 * A234092 A047931 A258571

Adjacent sequences: A268314 A268315 A268316 * A268318 A268319 A268320

Keyword

nonn,base,tabf

Author

Kival Ngaokrajang, Feb 01 2016

Status

approved