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A143183 Triangle T(n,k) = 1 + (2+n)*abs(n-2*k), read by rows, for 0 <= k <= n.

%I A143183 #12 Apr 24 2024 03:06:34
%S A143183 1,4,4,9,1,9,16,6,6,16,25,13,1,13,25,36,22,8,8,22,36,49,33,17,1,17,33,
%T A143183 49,64,46,28,10,10,28,46,64,81,61,41,21,1,21,41,61,81,100,78,56,34,12,
%U A143183 12,34,56,78,100,121,97,73,49,25,1,25,49,73,97,121
%N A143183 Triangle T(n,k) = 1 + (2+n)*abs(n-2*k), read by rows, for 0 <= k <= n.
%H A143183 G. C. Greubel, <a href="/A143183/b143183.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A143183 T(n, k) = 1 + (2+n)*abs(n-2*k), for 0 <= k <= n.
%F A143183 T(n, k) = T(n, n-k).
%F A143183 Sum_{k=0..n} T(n, k) = (n+2)*A007590(n+1) + n + 1 (row sums).
%F A143183 From _G. C. Greubel_, Apr 23 2024: (Start)
%F A143183 T(n, 0) = A000290(n+1).
%F A143183 T(2*n-1, n) = A005843(n+1), n >= 1.
%F A143183 Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*((n^2 + 3*n + 3) - (-1)^(n/2)*(n + 2)). (End)
%e A143183 Triangle begins as:
%e A143183     1;
%e A143183     4,  4;
%e A143183     9,  1,  9;
%e A143183    16,  6,  6, 16;
%e A143183    25, 13,  1, 13, 25;
%e A143183    36, 22,  8,  8, 22, 36;
%e A143183    49, 33, 17,  1, 17, 33, 49;
%e A143183    64, 46, 28, 10, 10, 28, 46, 64;
%e A143183    81, 61, 41, 21,  1, 21, 41, 61, 81;
%e A143183   100, 78, 56, 34, 12, 12, 34, 56, 78, 100;
%e A143183   121, 97, 73, 49, 25,  1, 25, 49, 73,  97, 121;
%p A143183 A143183 := proc(n,k)
%p A143183         1+(2+n)*abs(n-2*m) ;
%p A143183 end proc: # _R. J. Mathar_, Jul 12 2012
%t A143183 T[n_, m_]:= 1 + Abs[(n-m+1)^2 - (m+1)^2];
%t A143183 Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten
%o A143183 (Magma)
%o A143183 [1+(n+2)*Abs(n-2*k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 23 2024
%o A143183 (SageMath)
%o A143183 flatten([[1+(n+2)*abs(n-2*k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 23 2024
%Y A143183 Cf. A000290, A005843, A007590, A143182.
%K A143183 nonn,tabl,easy
%O A143183 0,2
%A A143183 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 17 2008
%E A143183 Row sums corrected by _R. J. Mathar_, Jul 12 2012