A392243 a(n) = Sum_{i=1..n} i*(-1)^ceiling(sqrt(i)).

-1, 1, 4, 8, 3, -3, -10, -18, -27, -17, -6, 6, 19, 33, 48, 64, 47, 29, 10, -10, -31, -53, -76, -100, -125, -99, -72, -44, -15, 15, 46, 78, 111, 145, 180, 216, 179, 141, 102, 62, 21, -21, -64, -108, -153, -199, -246, -294, -343, -293, -242, -190, -137, -83, -28, 28, 85, 143, 202, 262, 323

Offset

1,3

Comments

If abs(a(n)) = abs(a(n+1)) then abs(a(n)) is a triangular number (A000217).

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Jake Foth, Boundary Dynamics of a Pseudo-Recamán Recurrence, Zenodo (2026).

Jake Foth, Boundary Dynamics of a Pseudo-Recamán Recurrence [PDF].

Formula

a(k^2) = (-1)^k * k^3.

a(n) = (-1)^k * (k^3 - r*(2*n-r+1)/2) where k = A000196(n) and r = A053186(n) are square root and remainder n = k^2 + r. - Kevin Ryde, Jan 13 2026

Let b(n) = abs(a(n)). Then b(1) = 1 and, for n >= 2, b(n) = b(n-1) + n if 2*(b(n-1) mod n) < n; otherwise b(n) = abs(b(n-1) - n). - Jake Foth, Jul 16 2026

Mathematica

Accumulate[Array[#*(-1)^Ceiling[Sqrt[#]] &, 100]] (* Paolo Xausa, Jan 12 2026 *)

Prog

(PARI) a(n) = sum(i=1, n, if (ceil(sqrt(i)) % 2, -i, i)); \\ Michel Marcus, Jan 06 2026

Crossrefs

Cf. A000290, A000578, A000217.

Cf. A000196, A392677.

Sequence in context: A132797 A309217 A217602 * A395214 A300690 A193077

Adjacent sequences: A392240 A392241 A392242 * A392244 A392245 A392246

Keyword

sign,easy,changed

Author

Dwight Boddorf, Jan 04 2026

Status

approved