A236538 - OEIS

%I #43 Mar 28 2024 21:56:50

%S 1,3,5,8,12,16,20,28,36,44,48,64,80,96,112,112,144,176,208,240,272,

%T 256,320,384,448,512,576,640,576,704,832,960,1088,1216,1344,1472,1280,

%U 1536,1792,2048,2304,2560,2816,3072,3328,2816,3328,3840,4352,4864,5376

%N Triangle read by rows: T(n,k) = (n+1)*2^(n-2)+(k-1)*2^(n-1) for 1 <= k <= n.

%C 1, 9, 45, 161, 497, 1409, ... is the sequence of perimeters (sum of border elements) of the triangle.

%C 1, 5, 80, 3520, 394240, 107233280, 68629299200, ... is the sequence of determinants of the triangle.

%C Only the first three terms are odd.

%H Fedor Igumnov, <a href="/A236538/a236538.txt">T(n,k) for n = 1..26</a>

%F T(n,k) = T(n-1,k) + T(n-1,k+1).

%F Sum_{k=1..n} T(n,k) = n^2*2^(n-1) = A014477(n-1).

%e Triangle begins:

%e ================================================

%e \k |    1     2     3     4     5     6     7

%e n\ |

%e ================================================

%e 1  |    1;

%e 2  |    3,    5;

%e 3  |    8,   12,   16;

%e 4  |   20,   28,   36,   44;

%e 5  |   48,   64,   80,   96,  112;

%e 6  |  112,  144,  176,  208,  240,  272;

%e 7  |  256,  320,  384,  448,  512,  576,  640;

%e ...

%t t[n_, k_] := (n + 1)*2^(n - 2) + (k - 1)*2^(n - 1); Table[t[n, k], {n, 10}, {k, n}] // Flatten (* _Bruno Berselli_, Jan 28 2014 *)

%o (C) int a(int n, int k) {return (n+1)*pow(2,n-2)+(k-1)*pow(2,n-1);}

%o (Magma) /* As triangle: */ [[(n+1)*2^(n-2)+(k-1)*2^(n-1): k in [1..n]]: n in [1..10]]; // _Bruno Berselli_, Jan 28 2014

%Y Cf. A001792 (column 1), A053220 (right border). Also:

%Y A014477, row sums;

%Y A036826, partial sums;

%Y A058962, central elements in odd rows;

%Y A045623, second column;

%Y A045891, third column;

%Y A034007, fourth column;

%Y A167667, subdiagonal;

%Y A130129, second subdiagonal.

%K nonn,tabl,easy

%O 1,2

%A _Fedor Igumnov_, Jan 28 2014

%E More terms from _Bruno Berselli_, Jan 28 2014