A140757 Cumulative sums of A140756.

1, 0, 2, 3, 1, 4, 3, 5, 2, 6, 7, 5, 8, 4, 9, 8, 10, 7, 11, 6, 12, 13, 11, 14, 10, 15, 9, 16, 15, 17, 14, 18, 13, 19, 12, 20, 21, 19, 22, 18, 23, 17, 24, 16, 25, 24, 26, 23, 27, 22, 28, 21, 29, 20, 30, 31, 29, 32, 28, 33, 27, 34, 26, 35, 25, 36, 35, 37, 34, 38, 33, 39, 32, 40, 31, 41

Offset

1,3

Comments

Each positive integer occurs exactly twice in this sequence, with 0 occurring only once. In particular, if A002620(n) < m < A002620(n+1), then m occurs in rows n-1 and n; and A002620(n) occurs in rows in rows n-1 (as T(n-1,n-1)) and n+1 (as T(n+1,n)).

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

T(n,k) = floor((n+(-1)^{n-k})^2/4) - (-1)^{n-k}*floor((n-k)/2), as a triangle, with n >= 1, 1 <= k <= n.

Example

As a triangle:
  1;
  0,  2;
  3,  1,  4;
  3,  5,  2,  6;
  7,  5,  8,  4,  9;
  8, 10,  7, 11,  6, 12;
  ...

Mathematica

A140756[n_]:= With[{t=Floor[(-1+Sqrt[8*n-7])/2]}, (-1)^(Binomial[t+2, 2] -n)*(n -Binomial[t+1, 2])];
A140757[n_]:= Sum[A140756[j], {j, n}];
Table[A140757[n], {n, 100}] (* G. C. Greubel, Oct 21 2023 *)

Prog

(PARI) T(n, k)=if((n-k)%2==0, ((n+1)^2\4)-((n-k)\2), ((n-1)^2\4)+((n-k)\2) ) - Paul D. Hanna
(Magma)
A140756:=[(-1)^(n+k)*k: k in [1..n], n in [1..40]];
A140757:= func< n | (&+[A140756[j]: j in [1..n]]) >;
[A140757(n): n in [1..100]]; // G. C. Greubel, Oct 21 2023
(SageMath)
A140756=flatten([[(-1)^(n+k)*k for k in range(1, n+1)] for n in range(1, 41)])
def A140757(n): return sum(A140756[j] for j in range(n))
[A140757(n) for n in range(1, 101)] # G. C. Greubel, Oct 21 2023

Crossrefs

Cf. A002620, A140756.

Sequence in context: A067992 A317024 A354803 * A258254 A100035 A201927

Adjacent sequences: A140754 A140755 A140756 * A140758 A140759 A140760

Keyword

easy,nonn,tabl

Author

Franklin T. Adams-Watters, May 27 2008

Status

approved