A307116 -id:A307116 - cp's OEIS frontend

A307357 Row sums of triangle A307116.

1, 2, 4, 8, 5, 10, 11, 10, 17, 22, 12, 24, 25, 18, 30, 36, 20, 40, 47, 24, 42, 42, 32, 46, 58, 34, 50, 58, 52, 60, 63, 50, 73, 64, 56, 84, 82, 56, 83, 88, 58, 86, 88, 74, 99, 96, 76, 106, 96, 90, 136, 124, 75, 108, 131, 82, 106, 142, 130, 132, 127, 104, 129

Offset: 0

Rémy Sigrist, Apr 04 2019

nonn

The scatterplot of the sequence shows 3 beams of points; this could be related to the regular structure visible on both sides of the triangle A307116.

			The first terms, alongside the corresponding rows in A307116, are:
  n  a(n)  Row n of A307116
  -  ----  ----------------
  0     1           1
  1     2          1 1
  2     4         1 2 1
  3     8        1 3 3 1
  4     5       1 1 1 1 1
  5    10      1 2 2 2 2 1
  6    11     1 3 1 1 1 3 1
  7    10    1 1 1 2 2 1 1 1
  8    17   1 2 2 3 1 3 2 2 1
  9    22  1 3 1 5 1 1 5 1 3 1

Rémy Sigrist, Table of n, a(n) for n = 0..25000
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..100000 (where the color is function of n mod 3)

Cf. A307116.

fibs = Set(vector(100,k,fibonacci(k)))
f(s) = if (setsearch(fibs, s), s, 1)
{ for (r=0, 62, row = vector(r+1, k, if (k==1||k==r+1, 1, f(row[k-1]+row[k]))); print1 (vecsum(row) ", ")) }

a(n) = Sum_{k = 0..n} A307116(n, k).

A307069 Given a special version of Pascal's triangle where only Fibonacci numbers are permitted, a(n) is the row number in which the n-th Fibonacci number first appears.

Original entry on oeis.org

0, 0, 2, 3, 9, 50, 51, 70, 71, 133, 134, 135, 136, 2543, 2544

Offset: 1

Elliott Line, Mar 22 2019

Consider a version of Pascal's Triangle: a triangular array with a single 1 on row 0, with numbers below equal to the sum of the two numbers above it if and only if that sum appears in the Fibonacci sequence A000045. If the sum does not appear in A000045, a 1 is put in its place.
So the first few rows would be as follows:
  row 0:            1
  row 1:           1 1
  row 2:          1 2 1
  row 3:         1 3 3 1
  row 4:        1 1 1 1 1
  row 5:       1 2 2 2 2 1
  row 6:      1 3 1 1 1 3 1
  row 7:     1 1 1 2 2 1 1 1
  row 8:    1 2 2 3 1 3 2 2 1
  row 9:   1 3 1 5 1 1 5 1 3 1
  ...
a(n) is the row number in which the n-th Fibonacci number first appears in this triangular array.
a(16) > 2.2*10^5. - David A. Corneth, Mar 25 2019
a(16) > 3.2*10^6. - Daniel Suteu, Mar 26 2019
a(16) > 1.5*10^7. - Bert Dobbelaere, Apr 02 2019

David A. Corneth, PARI program

Cf. A000045, A307116 (the special Pascal's triangle).

Block[{s = Array[Fibonacci, 20], t}, t = Nest[Append[#1, (PadLeft[#1[[-1]], #2] + PadRight[#1[[-1]], #2]) /. k_Integer /; FreeQ[s, k] -> 1] & @@ {#, Length@ # + 1} &, {{1}}, 10^4]; -1 + TakeWhile[Map[FirstPosition[t, #][[1]] &, s], IntegerQ]] (* Michael De Vlieger, Mar 24 2019 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
lista(nn) = {print1(0, ", ", 0, ", "); v = [1,1]; nextf = 3; for (n=2, nn, w = vector(n+1); w[1] = v[1]; for (j=2, n, w[j] = v[j-1]+ v[j]; if (!isfib(w[j]), w[j] = 1)); w[n+1] = v[n]; sw = vecsort(w,,8); if (vecsearch(sw, fibonacci(nextf)), print1(n, ", "); nextf++); v = w;);} \\ Michel Marcus, Mar 22 2019

See Corneth link \\ David A. Corneth, Mar 25 2019

a(14)-a(15) from Michel Marcus, Mar 22 2019

A307433 A special version of Pascal's triangle where only powers of 2 are permitted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 4, 4, 1, 1, 1, 2, 1, 8, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 4, 4, 4, 4, 4, 4, 1, 1, 1, 2, 1, 8, 8, 8, 8, 8, 1, 2, 1, 1, 1, 1, 1, 16, 16, 16, 16, 1, 1, 1, 1, 1, 2, 2, 2, 1, 32, 32, 32, 1, 2, 2, 2, 1

Offset: 0

Rémy Sigrist, May 05 2019

If the sum of the two numbers above in the triangular array is not a power of 2 (A000079), then a 1 is put in its place.
The ones in the table form a Sierpinski gasket (A047999).
Apparently, for any k > 0, the value 2^k first occurs on row A206332(k).
From Bernard Schott, May 05 2019: (Start)
For any m, at row 2^m - 1, there is only a string of 2^m times the number 1, then at row 2^(m+1) - 2, comes out for the first time and only once, the power of 2 equals to 2^(2^m-1). At row 2^(m+1) - 1, there are again 2^(m+1) times the number 1. This cycle can go on. Under, a part of this triangle between row 2^3 -1 and 2^4 - 2 that visualizes the explanations.
                 1   1   1   1   1   1   1   1
                   2   2   2   2   2   2   2
                     4   4   4   4   4   4
                       8   8   8   8   8
                         16  16  16  16
                           32  32  32
                             64  64
                               128
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1 (End)

			The triangle begins:
                                1
                              1   1
                            1   2   1
                          1   1   1   1
                        1   2   2   2   1
                      1   1   4   4   1   1
                    1   2   1   8   1   2   1
                  1   1   1   1   1   1   1   1
                1   2   2   2   2   2   2   2   1
              1   1   4   4   4   4   4   4   1   1
            1   2   1   8   8   8   8   8   1   2   1
          1   1   1   1  16  16  16  16   1   1   1   1
        1   2   2   2   1  32  32  32   1   2   2   2   1
      1   1   4   4   1   1  64  64   1   1   4   4   1   1
    1   2   1   8   1   2   1  128  1   2   1   8   1   2   1
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1

Rémy Sigrist, Table of n, a(n) for n = 0..8255 (rows n = 0..127)
Rémy Sigrist, Colored representation of the first 1024 rows (where the hue is function of log(T(n,k)))
Rémy Sigrist, Colored representation of the first 1024 rows (where black pixels correspond to ones)

Cf. A000079, A007318, A047999, A206332, A307116 (analog with Fibonacci numbers).

for (r=1, 13, apply(v -> print1 (v", "), row=vector(r, k, if (k==1 || k==r, 1, hammingweight(s=row[k-1]+row[k])==1, s, 1))))

cp's OEIS Frontend

A307357 Row sums of triangle A307116.

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A307069 Given a special version of Pascal's triangle where only Fibonacci numbers are permitted, a(n) is the row number in which the n-th Fibonacci number first appears.

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A307433 A special version of Pascal's triangle where only powers of 2 are permitted.

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PARI