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

0,5

Comments

In each row, two bisections count up.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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)

Example

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;

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

Mathematica

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

Prog

(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
(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

Crossrefs

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

Cf. A014682, A130472.

Sequence in context: A143151 A130106 A127093 * A333432 A333500 A182720

Adjacent sequences: A141540 A141541 A141542 * A141544 A141545 A141546

Keyword

sign,easy,tabl

Author

Paul Curtz, Aug 16 2008

Status

approved