A050271 Numbers k such that k = floor(sqrt(k)*ceiling(sqrt(k))).

1, 2, 3, 4, 7, 8, 9, 14, 15, 16, 23, 24, 25, 34, 35, 36, 47, 48, 49, 62, 63, 64, 79, 80, 81, 98, 99, 100, 119, 120, 121, 142, 143, 144, 167, 168, 169, 194, 195, 196, 223, 224, 225, 254, 255, 256, 287, 288, 289, 322, 323, 324, 359, 360, 361, 398, 399, 400

Offset: 1

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn
• easy

Comments

Is a(n) asymptotic to C*n^(3/2) where 1/2 < C < 1?

Consists exactly of numbers of the forms j^2 - 2, j^2 - 1, and j^2. As such, is asymptotic to (1/9)*n^2. - Ivan Neretin, Feb 08 2017

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A000290, A005563, A008865, A087278.

Programs

Maple

a:=n->floor((n+4)/3)^2+irem(n+1,3)-2:
seq(a(n),n=1..58); # Lorenzo Sauras Altuzarra, Jan 31 2023

Mathematica

Select[Range@400, Floor[(r = Sqrt@#)*Ceiling@r] == # &] (* Ivan Neretin, Feb 08 2017 *)
LinearRecurrence[{1,0,2,-2,0,-1,1},{1,2,3,4,7,8,9},60] (* Harvey P. Dale, Aug 10 2025 *)

PARI

isok(n) = floor(sqrt(n)*ceil(sqrt(n))) == n; \\ Michel Marcus, Nov 22 2013

PARI

Vec(x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^100)) \\ Colin Barker, Feb 09 2017

Python

def A050271(n):
    a, b = divmod(n+4,3)
    return a**2+b-2 # Chai Wah Wu, Aug 02 2022

Formula

a(n) = floor((n + 4)/3)^2 + ((n + 1) mod 3) - 2. - Ivan Neretin, Feb 08 2017

From Colin Barker, Feb 09 2017: (Start)

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7.

G.f.: x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2).

(End)

From Amiram Eldar, Sep 14 2022: (Start)

Sum_{n>=1} 1/a(n) = 2 + Pi^2/6 - cot(sqrt(2)*Pi)*Pi/(2*sqrt(2)).

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 + cosec(sqrt(2)*Pi)*Pi/(2*sqrt(2)). (End)

a(n) = A087278(n+1) - 1 if n > 0. - Lorenzo Sauras Altuzarra, Jan 31 2023

Extensions

Data corrected by Michel Marcus and Benoit Cloitre, Nov 22 2013