A090171 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 1, 0.
0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0
Offset: 0
Examples
Triangle begins: 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 ...
Links
- Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121; see Fig. 2.
Crossrefs
Formula
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.
Extensions
Edited and extended by Christian G. Bower, Jan 20 2004
A091562 Triangle read by rows, related to Pascal's triangle, starting with 1, 0, 0.
1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 5, 7, 5, 2, 3, 10, 17, 17, 10, 3, 5, 20, 41, 51, 41, 20, 5, 8, 38, 91, 136, 136, 91, 38, 8, 13, 71, 195, 339, 405, 339, 195, 71, 13, 21, 130, 403, 799, 1107, 1107, 799, 403, 130, 21, 34, 235, 812, 1807, 2845, 3297, 2845, 1807, 812, 235, 34
Offset: 0
Examples
Triangle begins: 1; 0,0; 1,1,1; 1,2,2,1; 2,5,7,5,2; ...
Crossrefs
Formula
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.
G.f.: A(x, y) = (1-x-x*y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
A090173 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 0, 1.
0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1
Offset: 0
Examples
Triangle begins 0; 0,1; 0,1,1; 0,0,1,0; 0,1,1,1,1; ...
Links
- Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121; see Fig. 2.
Crossrefs
Formula
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.
Extensions
Edited and extended by Christian G. Bower, Jan 20 2004
A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.
0, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 10, 14, 10, 3, 5, 20, 36, 36, 20, 5, 8, 38, 83, 106, 83, 38, 8, 13, 71, 182, 281, 281, 182, 71, 13, 21, 130, 382, 690, 834, 690, 382, 130, 21, 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34, 55, 420, 1546, 3586, 5780, 6750
Offset: 0
Examples
Triangle begins : 0 1, 1 1, 2, 1 2, 5, 5, 2 3, 10, 14, 10, 3 5, 20, 36, 36, 20, 5 8, 38, 83, 106, 83, 38, 8 13, 71, 182, 281, 281, 182, 71, 13 21, 130, 382, 690, 834, 690, 382, 130, 21 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34 55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55
Formula
G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).
Comments