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Triangle T(n,k) read by brows: T(n,2k)=k, T(n,2k+1) = k-n, for 0<=k<=n.
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%I #14 Sep 18 2024 11:23:52

%S 0,0,-1,0,-2,1,0,-3,1,-2,0,-4,1,-3,2,0,-5,1,-4,2,-3,0,-6,1,-5,2,-4,3,

%T 0,-7,1,-6,2,-5,3,-4,0,-8,1,-7,2,-6,3,-5,4,0,-9,1,-8,2,-7,3,-6,4,-5,0,

%U -10,1,-9,2,-8,3,-7,4,-6,5

%N Triangle T(n,k) read by brows: T(n,2k)=k, T(n,2k+1) = k-n, for 0<=k<=n.

%C In each row, two bisections count up.

%H G. C. Greubel, <a href="/A141543/b141543.txt">Rows n = 0..50 of the triangle, flattened</a>

%F From _G. C. Greubel_, Sep 17 2024: (Start)

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

%F T(n, n) = A130472(n).

%F T(2*n, n) = (-1)^n*A014682(n).

%F Sum_{k=0..n} T(n, k) = (-1)*A008794(n+1).

%F Sum_{k=0..n} (-1)^k*T(n, k) = A000217(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A159915(n+1).

%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/32)*(6*n^2 + 6*n - 5 + (-1)^n*(2*n + 1) + 2*(1 - i)*(-i)^n + 2*(1 + i)*i^n). (End)

%e Triangle begins as:

%e 0;

%e 0, -1;

%e 0, -2, 1;

%e 0, -3, 1, -2;

%e 0, -4, 1, -3, 2;

%e 0, -5, 1, -4, 2, -3;

%e 0, -6, 1, -5, 2, -4, 3;

%p A141543 := proc(n,k) if type(k,'even') then k/2; else (k-1)/2-n ; end if; end proc:

%p seq(seq(A141543(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Jul 07 2011

%t Flatten[Table[If[EvenQ[k],k/2,(k-1)/2-n],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Sep 24 2013 *)

%o (Magma)

%o A141543:= func< n,k | ((k+1) mod 2)*Floor(k/2) + (k mod 2)*(-n + Floor((k-1)/2)) >;

%o [A141543(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Sep 17 2024

%o (SageMath)

%o def A141543(n,k): return ((k+1)%2)*(k//2) + (k%2)*(-n + ((k-1)//2))

%o flatten([[A141543(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Sep 17 2024

%Y Sums include: A000217 (signed row), A008794 (row), A159915 (diagonal).

%Y Cf. A014682, A130472.

%K sign,easy,tabl

%O 0,5

%A _Paul Curtz_, Aug 16 2008