A035253 -id:A035253 - cp's OEIS frontend

A035254 First differences of A035253.

-3, -1, 1, 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64

Offset: 0

Robet Bronson (bob(AT)bronsons.com)

Different from A030124. Cf. A002024, A014132, A014133. First differences are in A035214.

from math import isqrt
def A035254(n): return n+(isqrt(n<<3)-5>>1) # Chai Wah Wu, Jun 07 2025

a(n) = A002024(n)+n-3. - Chai Wah Wu, Jun 07 2025

A035214 2 followed by a run of n 1's.

Original entry on oeis.org

2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..10000
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Conjecture 4.5, p. 137.

Cf. A010054, A035253, A035254.

Table[(SquaresR[1, 8*n + 1] + 2)/2, {n, 0, 100}] (* or *) Table[Floor[(3 - Cos[Pi*Sqrt[8*n + 1]])/2], {n,0,100}] (* G. C. Greubel, May 14 2017 *)

for(n=0,100, print1(floor((3-cos(Pi*sqrt(8*n+1)))/2), ", ")) \\ G. C. Greubel, May 14 2017

a(n) = issquare(n<<3 + 1) + 1; \\ Kevin Ryde, Aug 03 2022

from sympy.ntheory.primetest import is_square
def A035214(n): return int(is_square((n<<3)|1))+1 # Chai Wah Wu, Jun 07 2025

a(n) = 2 if n is a triangular number, otherwise 1.
Equals A010054(n) + 1.
a(n) = floor((3-cos(Pi*sqrt(8*n+1)))/2). - Carl R. White, Mar 18 2006

Typo corrected by Neven Juric, Jan 10 2009

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A035254 First differences of A035253.

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A035214 2 followed by a run of n 1's.

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