A124731
Triangle, row sums = powers of 3, companion to A124730.
Original entry on oeis.org
1, 2, 1, 4, 3, 2, 8, 7, 10, 2, 16, 15, 34, 12, 4, 32, 31, 98, 46, 32, 4, 64, 63, 258, 144, 156, 36, 8, 128, 127, 642, 402, 600, 192, 88, 8, 256, 255, 1538, 1044, 2004, 792, 560, 96, 16
Offset: 0
Row 2 = (4, 3, 2) since (using the 3 X 3 matrix m = [2,0,0; 1,1,0; 0,2,2]), m^2 * [1,0,0] = [4,3,2].
First few rows of the triangle are:
1;
2, 1;
4, 3, 2;
8, 7, 10, 2;
16, 15, 34, 12, 4;
32, 31, 98, 46, 32, 4;
64, 63, 258, 144, 156, 36, 8;
...
A124572
Triangle read by rows where the n-th row is the first row of M^n, with M the (n+1)-by-(n+1) matrix with (1,3,1,3,1,3,...) on its main diagonal and (3,1,3,1,3,1,...) on its superdiagonal.
Original entry on oeis.org
1, 1, 3, 1, 12, 3, 1, 39, 15, 9, 1, 120, 54, 72, 9, 1, 363, 174, 378, 81, 27, 1, 1092, 537, 1656, 459, 324, 27, 1, 3279, 1629, 6579, 2115, 2349, 351, 81, 1, 9840, 4908, 24624, 8694, 13392, 2700, 1296, 81, 1, 29523, 14748, 88596, 33318, 66258, 16092
Offset: 0
Row 3 = [1, 39, 15, 9] since when n = 3 we have M^3 = [[1, 39, 15, 9], [0, 27, 13, 21], [0, 0, 1, 39], [0, 0, 0, 27]]. The first few rows of the triangle are:
1;
1, 3;
1, 12, 3;
1, 39, 15, 9;
1, 120, 54, 72, 9;
1, 363, 174, 378, 81, 27;
...
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with (LinearAlgebra): for n from 0 to 10 do M := Matrix (n+1, (i, j)->`if`(i=j and i mod 2 = 1, 1, `if`(i=j, 3, `if`(i=j-1 and i mod 2 = 1, 3, `if`(i=j-1, 1, 0))))): X := M^n: for m from 0 to n do printf("%d, ", X[1, m+1]): od: od: # Nathaniel Johnston, Apr 28 2011
A124573
Triangle read by rows where the n-th row is the first row of M^n, with M the (n+1)-by-(n+1) matrix with (3,1,3,1,3,1,...) on its main diagonal and (1,3,1,3,1,3,...) on its superdiagonal.
Original entry on oeis.org
1, 3, 1, 9, 4, 3, 27, 13, 21, 3, 81, 40, 102, 24, 9, 243, 121, 426, 126, 99, 9, 729, 364, 1641, 552, 675, 108, 27, 2187, 1093, 6015, 2193, 3681, 783, 405, 27, 6561, 3280, 21324, 8208, 17622, 4464, 3564, 432, 81, 19683, 9841, 73812, 29532, 77490, 22086
Offset: 0
Row 3 = [27, 13, 21, 3] since when n = 3 we have M^3 = [[27, 13, 21, 3], [0, 1, 39, 15], [0, 0, 27, 13], [0, 0, 0, 1]].
First few rows of the triangle are:
1;
3, 1;
9, 4, 3;
27, 13, 21, 3;
81, 40, 102, 24, 9;
243, 121, 426, 126, 99, 9;
...
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with (LinearAlgebra): for n from 0 to 10 do M := Matrix (n+1, (i, j)->`if`(i=j and i mod 2 = 1, 3, `if`(i=j, 1, `if`(i=j-1 and i mod 2 = 1, 1, `if`(i=j-1, 3, 0))))): X := M^n: for m from 0 to n do printf("%d, ",X[1,m+1]): od: od: # Nathaniel Johnston, Apr 28 2011
A124732
Triangle P*M, where P is the Pascal triangle written as an infinite lower triangular matrix and M is the infinite bidiagonal matrix with (1,2,1,2,...) in the main diagonal and (2,1,2,1,...) in the subdiagonal.
Original entry on oeis.org
1, 3, 2, 5, 5, 1, 7, 9, 5, 2, 9, 14, 14, 9, 1, 11, 20, 30, 25, 7, 2, 13, 27, 55, 55, 27, 13, 1, 15, 35, 91, 105, 77, 49, 9, 2, 17, 44, 140, 182, 182, 140, 44, 17, 1, 19, 54, 204, 294, 378, 336, 156, 81, 11, 2, 21, 65, 285, 450, 714, 714, 450, 285, 65, 21, 1, 23, 77, 385, 660
Offset: 1
First 3 rows of the triangle are (1; 3,2; 5,5,1) since [1,0,0; 1,1,0; 1,2,1] * [1,0,0; 2,2,0; 0,1,1] = [1,0,0; 3,2,0; 5,5,1].
First few rows of the triangle are:
1;
3, 2;
5, 5, 1;
7, 9, 5, 2;
9, 14, 14, 9, 1;
11, 20, 30, 25, 7, 2;
13, 27, 55, 55, 27, 13, 1;
15, 35, 91, 105, 77, 49, 9, 2;
...
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T:=(n,k)->binomial(n,k)*(3*n-(-1)^k*(n-2*k))/2/n: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Showing 1-4 of 4 results.
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