%I #11 May 01 2024 09:02:00

%S 1,1,1,1,2,1,1,2,2,1,1,2,2,2,1,1,2,2,2,2,1,1,2,2,6,2,2,1,1,2,2,6,6,2,

%T 2,1,1,2,2,6,4,6,2,2,1,1,2,2,6,4,4,6,2,2,1,1,2,2,6,4,10,4,6,2,2,1,1,2,

%U 2,6,4,10,10,4,6,2,2,1,1,2,2,6,4,10,6,10,4,6,2,2,1

%N Triangle read by rows: T(n, k) = f(k) for 1 <= k <= floor(n/2), T(n, k) = f(n-k) for floor(n/2) < k <= n-1, with T(n, 0) = 1, T(n, n) = 1, and f(k) = (1/2)*(3 - (-1)^k)*k.

%H G. C. Greubel, <a href="/A143187/b143187.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = f(k) for 1 <= k <= floor(n/2), T(n, k) = f(n-k) for floor(n/2) < k <= n-1, with T(n, 0) = 1, T(n, n) = 1, and f(k) = (1/2)*(3 - (-1)^k)*k.

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

%F Sum_{k=0..n} T(n, k) = (1/16)*(33 + 3*(-1)^n - 4*cos(n*Pi/2) - 4*sin(n*Pi/2)*n + 6*n^2) - [n=0] (row sums). - _G. C. Greubel_, Apr 30 2024

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 2, 2, 1;

%e 1, 2, 2, 2, 1;

%e 1, 2, 2, 2, 2, 1;

%e 1, 2, 2, 6, 2, 2, 1;

%e 1, 2, 2, 6, 6, 2, 2, 1;

%e 1, 2, 2, 6, 4, 6, 2, 2, 1;

%e 1, 2, 2, 6, 4, 4, 6, 2, 2, 1;

%e 1, 2, 2, 6, 4, 10, 4, 6, 2, 2, 1;

%t f[n_]= (3-(-1)^n)*n/2;

%t T[n_, k_]:= If[k*(n-k)==0, 1, If[k <= Floor[n/2], f[k], f[n-k]]];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

%o (Magma)

%o f:= func< n | (3-(-1)^n)*n/2 >;

%o A143187:= func< n,k | k eq 0 or k eq n select 1 else k le Floor(n/2) select f(k) else f(n-k) >;

%o [A143187(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Apr 30 2024

%o (SageMath)

%o def f(n): return (3-(-1)^n)*n/2

%o def A143187(n,k):

%o if k==0 or k==n: return 1

%o elif k<=n//2: return f(k)

%o else: return f(n-k)

%o flatten([[A143187(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Apr 30 2024

%Y Cf. A143188.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 17 2008

%E Edited by _G. C. Greubel_, Apr 30 2024