A144152 - OEIS

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

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OFFSET

1,10

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).

LINKS

Table of n, a(n) for n=1..67.

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, ...).

EXAMPLE

First few rows of the triangle =

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).

PROG

(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

CROSSREFS

Cf. A000045, A128174.

Sequence in context: A158461 A281498 A118269 * A265674 A229297 A116675

Adjacent sequences: A144149 A144150 A144151 * A144153 A144154 A144155

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Sep 12 2008

EXTENSIONS

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

STATUS

approved