A124730
Triangle, row sums = powers of 3.
Original entry on oeis.org
1, 1, 2, 1, 6, 2, 1, 14, 8, 4, 1, 30, 22, 24, 4, 1, 62, 52, 92, 28, 8, 1, 126, 114, 288, 120, 72, 8, 1, 254, 240, 804, 408, 384, 80, 16, 1, 510, 494, 2088, 1212, 1584, 46, 192, 16
Offset: 0
Row 2 = (1, 6, 2) since [1,0,0; 2,2,0; 0,1,1]^2 * [1,0,0] = [1,6,2].
First few rows of the triangle are:
1;
1, 2;
1, 6, 2;
1, 14, 8, 4;
1, 30, 22, 24, 4;
1, 62, 52, 92, 28, 8;
1, 126, 114, 288, 120, 72, 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;
...
-
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;
...
-
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
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