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

Author

Franklin T. Adams-Watters, May 27 2008

Keywords

• easy
• nonn
• tabl

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)).

Examples

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

Links

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

Crossrefs

Cf. A002620, A140756.

Programs

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

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 *)

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

Python

from math import comb, isqrt
def A140757(n): return ((a-1)**2>>2)+(c>>1) if (c:=(a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))-(b:=n-comb(a,2)))&1 else ((a+1)**2>>2)-(c>>1) # Chai Wah Wu, Jun 09 2025

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

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.