A141543 Triangle T(n,k) read by brows: T(n,2k)=k, T(n,2k+1) = k-n, for 0<=k<=n.
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, 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, -10, 1, -9, 2, -8, 3, -7, 4, -6, 5
Offset: 0
Examples
Triangle begins as: 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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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Magma
A141543:= func< n,k | ((k+1) mod 2)*Floor(k/2) + (k mod 2)*(-n + Floor((k-1)/2)) >; [A141543(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 17 2024
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Maple
A141543 := proc(n,k) if type(k,'even') then k/2; else (k-1)/2-n ; end if; end proc: seq(seq(A141543(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Jul 07 2011
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Mathematica
Flatten[Table[If[EvenQ[k],k/2,(k-1)/2-n],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Sep 24 2013 *) -
SageMath
def A141543(n,k): return ((k+1)%2)*(k//2) + (k%2)*(-n + ((k-1)//2)) flatten([[A141543(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Sep 17 2024
Formula
From G. C. Greubel, Sep 17 2024: (Start)
T(n, k) = (1/2)*((1+(-1)^k)*floor(k/2) + (1-(-1)^k)*(-n + floor((k - 1)/2)) ).
T(n, n) = A130472(n).
T(2*n, n) = (-1)^n*A014682(n).
Sum_{k=0..n} T(n, k) = (-1)*A008794(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000217(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A159915(n+1).
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)
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