A111006 - OEIS

A111006

Another version of Fibonacci-Pascal triangle A037027.

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144

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OFFSET

0,6

COMMENTS

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.

Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.

T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).

Triangle read by rows: T(n,n-k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)). - Philippe Deléham, Feb 17 2014

Diagonal sums are A013979(n). - Philippe Deléham, Feb 17 2014

T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014

LINKS

Reinhard Zumkeller, Rows n = 0..120 of table, flattened

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).

T(n, k) = A037027(k, n-k). T(n, n) = A000045(n+1). T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).

G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005

Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006

Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014

EXAMPLE

Triangle begins:

0, 1;

0, 1, 2;

0, 0, 2, 3;

0, 0, 1, 5, 5;

0, 0, 0, 3, 10, 8;

0, 0, 0, 1, 9, 20, 13;

0, 0, 0, 0, 4, 22, 38, 21;

0, 0, 0, 0, 1, 14, 51, 71, 34;

0, 0, 0, 0, 0, 5, 40, 111, 130, 55;

0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89;