A144152 Triangle read by rows: A128174 * X; X = an infinite lower triangular matrix with a shifted Fibonacci sequence: (1, 1, 1, 2, 3, 5, 8, ...) in the main diagonal and the rest zeros.

1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 5, 1, 0, 1, 0, 3, 0, 8, 0, 1, 0, 2, 0, 5, 0, 13, 1, 0, 1, 0, 3, 0, 8, 0, 21, 0, 1, 0, 2, 0, 0, 5, 0, 13, 0, 34, 1, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55

Offset: 1

Author

Gary W. Adamson, Sep 12 2008

Keywords

• nonn
• tabl

Comments

The original definition was: Eigentriangle, row sums = Fibonacci numbers.

Even n rows are composed of odd-indexed Fibonacci numbers interpolated with zeros.

Odd n rows are composed of even-indexed Fibonacci numbers with alternate zeros.

Sum of n-th row terms = rightmost term of next row, = F(n-1). Row sums = F(n).

Examples

			First few rows of the triangle =
  1;
  0,  1;
  1,  0,  1;
  0,  1,  0,  2;
  1,  0,  1,  0,  3
  0,  1,  0,  2,  0,  5;
  1,  0,  1,  0,  3,  0,  8;
  0,  1,  0,  2,  0,  5,  0, 13;
  1,  0,  1,  0,  3,  0,  8,  0, 21;
  ...
Row 5 = (1, 0, 1, 0, 3) = termwise products of (1, 0, 1, 0, 1) and (1, 1, 1, 2, 3).
		

Crossrefs

Cf. A000045, A128174.

Programs

PARI

MT(n,k) = (1+(-1)^(n-k))/2;
MF(n,k) = n--; k--; if (n==k, if (n==0, 1, fibonacci(n)), 0);
tabl(nn) = {my(T=matrix(nn,nn, n, k, MT(n,k))); my(F=matrix(nn,nn, n, k, MF(n,k))); my(P=T*F); matrix(nn, nn, n, k, if (n>=k, P[n,k], 0));} \\ Michel Marcus, Mar 08 2021

Formula

A128174 = the matrix: (1; 0,1; 1,0,1; 0,1,0,1; ...). These operations are equivalent to termwise products of n terms of A128174 matrix row terms and an equal number of terms in (1, 1, 1, 2, 3, 5, 8, ...).

Extensions

Moved a comment to the Name section. - Omar E. Pol, Mar 08 2021