A176284 - OEIS

%I #16 Sep 08 2022 08:45:52

%S 1,1,1,1,7,1,1,19,19,1,1,37,49,37,1,1,61,91,91,61,1,1,91,145,163,145,

%T 91,1,1,127,211,253,253,211,127,1,1,169,289,361,385,361,289,169,1,1,

%U 217,379,487,541,541,487,379,217,1,1,271,481,631,721,751,721,631,481,271,1

%N Triangle T(n,k) = 1 + 3*n*k*(n-k) read by rows.

%C This could be written T(n,k) = 1 - (n-k)^3 - k^3 + n^3, where squares (instead of cubes) would define A130154.

%C Row sums are {1, 2, 9, 40, 125, 306, 637, 1184, 2025, 3250, 4961, ...} = (n+1)^2*(n^2 -2*n +2)/2.

%H G. C. Greubel, <a href="/A176284/b176284.txt">Rows n = 0..100 of triangle, flattened</a>

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

%F T(n,k) = 1 + 3*n*k*(n-k).

%e Triangle begins as:

%e   1;

%e   1,   1;

%e   1,   7,   1;

%e   1,  19,  19,   1;

%e   1,  37,  49,  37,   1;

%e   1,  61,  91,  91,  61,   1;

%e   1,  91, 145, 163, 145,  91,   1;

%e   1, 127, 211, 253, 253, 211, 127,   1;

%e   1, 169, 289, 361, 385, 361, 289, 169,   1;

%e   1, 217, 379, 487, 541, 541, 487, 379, 217,   1;

%e   1, 271, 481, 631, 721, 751, 721, 631, 481, 271, 1;

%p seq(seq(1 + 3*n*k*(n-k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 25 2019

%t Flatten[Table[1+3n k(n-k),{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Jul 03 2013 *)

%o (PARI) T(n,k) = 1 + 3*n*k*(n-k); \\ _G. C. Greubel_, Nov 25 2019

%o (Magma) [1 + 3*n*k*(n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 25 2019

%o (Sage) [[1 + 3*n*k*(n-k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 25 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> 1 + 3*n*k*(n-k) ))); # _G. C. Greubel_, Nov 25 2019

%Y Cf. A130154.

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Apr 14 2010

%E Edited by _R. J. Mathar_, May 03 2013