A082046 - OEIS

%I #15 Dec 24 2022 11:17:17

%S 1,1,1,1,5,1,1,11,11,1,1,19,29,19,1,1,29,55,55,29,1,1,41,89,109,89,41,

%T 1,1,55,131,181,181,131,55,1,1,71,181,271,305,271,181,71,1,1,89,239,

%U 379,461,461,379,239,89,1,1,109,305,505,649,701,649,505,305,109,1

%N Square array, A(n, k) = (k*n)^2 + 3*k*n + 1, read by antidiagonals.

%H G. C. Greubel, <a href="/A082046/b082046.txt">Antidiagonals n = 0..50, flattened</a>

%F A(n, k) = (k*n)^2 + 3*k*n + 1 (square array).

%F A(k, n) = A(n, k).

%F A(n, n) = T(2*n, n) = A057721(n).

%F A(n, n+1) = A072025(n).

%F T(n, k) = (k*(n-k))^2 + 3*k*(n-k) + 1 (antidiagonals).

%F Sum_{k=0..n} T(n, k) = A082047(n) (antidiagonal sums).

%F From _G. C. Greubel_, Dec 22 2022: (Start)

%F Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*(1 - 2*n).

%F T(2*n+1, n-1) = T(2*n-1, n-1) = A072025(n-1). (End)

%e Array, A(n, k), begins as:

%e   1,  1,   1,   1,   1,    1,    1,    1, ... A000012;

%e   1,  5,  11,  19,  29,   41,   55,   71, ... A028387;

%e   1, 11,  29,  55,  89,  131,  181,  239, ... A082108;

%e   1, 19,  55, 109, 181,  271,  379,  505, ... A069131;

%e   1, 29,  89, 181, 305,  461,  649,  869, ... ;

%e   1, 41, 131, 271, 461,  701,  991, 1331, ... ;

%e   1, 55, 181, 379, 649,  991, 1405, 1891, ... ;

%e   1, 71, 239, 505, 869, 1331, 1891, 2549, ... ;

%e Antidiagonals, T(n, k), begin as:

%e   1;

%e   1,  1;

%e   1,  5,   1;

%e   1, 11,  11,   1;

%e   1, 19,  29,  19,   1;

%e   1, 29,  55,  55,  29,   1;

%e   1, 41,  89, 109,  89,  41,   1;

%e   1, 55, 131, 181, 181, 131,  55,  1;

%e   1, 71, 181, 271, 305, 271, 181, 71,  1;

%t T[n_, k_]:= (k*(n-k))^2 + 3*(k*(n-k)) + 1;

%t Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 22 2022 *)

%o (Magma) [(k*(n-k))^2 + 3*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // _G. C. Greubel_, Dec 22 2022

%o (SageMath)

%o def A082046(n,k): return (k*(n-k))^2 + 3*(k*(n-k)) + 1

%o flatten([[A082046(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Dec 22 2022

%Y Cf. A028387, A057721, A069131, A072025, A082039, A082043, A082047, A082105, A082108.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Apr 03 2003