A143199 - OEIS

%I #23 Jun 10 2024 08:50:33

%S -1,-1,-1,-1,2,-1,-1,3,3,-1,-1,4,14,4,-1,-1,5,17,17,5,-1,-1,6,20,41,

%T 20,6,-1,-1,7,23,47,47,23,7,-1,-1,8,26,53,89,53,26,8,-1,-1,9,29,59,99,

%U 99,59,29,9,-1,-1,10,32,65,109,164,109,65,32,10,-1

%N Triangle read by rows: T(n, k) = (n+1)*A000096(k-1) + n if k <= floor(n/2), otherwise T(n, k) = (n+1)*A000096(n-k-1) + n.

%H G. C. Greubel, <a href="/A143199/b143199.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, m) = (n + 1)*(if m <= floor(n/2) then (m - 1)*(m + 2) / 2 else (n - m + 2)*(n - (m + 1)) / 2 fi) + n. - _Georg Fischer_, Oct 28 2023

%F From _G. C. Greubel_, Jun 10 2024: (Start)

%F T(n, k) = n + (n+1)*(k-1)*(k+2)/2 if 0 <= k <= floor(n/2), otherwise T(n, k) = T(n, n-k).

%F Sum_{k=0..n} T(n, k) = (1/48)*(n+1)*(-53 - 5*n + 3*(-1)^n*(n+1) + 2*(n + 1)^3). (End)

%e Triangle begins as:

%e   -1;

%e   -1, -1;

%e   -1,  2, -1;

%e   -1,  3,  3, -1;

%e   -1,  4, 14,  4,  -1;

%e   -1,  5, 17, 17,   5,  -1;

%e   -1,  6, 20, 41,  20,   6,  -1;

%e   -1,  7, 23, 47,  47,  23,   7, -1;

%e   -1,  8, 26, 53,  89,  53,  26,  8, -1;

%e   -1,  9, 29, 59,  99,  99,  59, 29,  9, -1;

%e   -1, 10, 32, 65, 109, 164, 109, 65, 32, 10, -1;

%p seq(print(seq((n + 1) * (if m <= n/2 then (m - 1) * (m + 2)\

%p  / 2 else (n - m + 2) * (n - (m + 1)) / 2 fi) + n, m=0..n)), n=0..10); # _Georg Fischer_, Oct 28 2023

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

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

%o (Magma)

%o function T(n,k) // A143199

%o   if k le Floor(n/2) then return n + (n+1)*(k-1)*(k+2)/2;

%o   else return T(n,n-k);

%o   end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 10 2024

%o (SageMath)

%o def A143199(n,k): return n +(n+1)*(k-1)*(k+2)//2 if (k<1+int(n//2)) else A143199(n,n-k)

%o flatten([[A143199(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jun 10 2024

%Y Cf. A000096, A132209, A142463.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 20 2008

%E Definition clarified and offset corrected by _Georg Fischer_, Oct 28 2023