A087278 Nonnegative integers whose distance to the nearest square is not greater than 1.

0, 1, 2, 3, 4, 5, 8, 9, 10, 15, 16, 17, 24, 25, 26, 35, 36, 37, 48, 49, 50, 63, 64, 65, 80, 81, 82, 99, 100, 101, 120, 121, 122, 143, 144, 145, 168, 169, 170, 195, 196, 197, 224, 225, 226, 255, 256, 257, 288, 289, 290, 323, 324, 325, 360, 361, 362, 399, 400, 401

Offset: 0

Author

Reinhard Zumkeller, Aug 28 2003

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Union of A005563, A000290 and A002522.

Cf. A002264, A010872, A087279.

Programs

Mathematica

dnsQ[n_]:=Module[{x=Floor[Sqrt[n]]},Min[n-x^2,(x+1)^2-n]<=1]; Select[Range[0,450],dnsQ] (* Harvey P. Dale, May 25 2011 *)
Table[n^2+{-1,0,1},{n,20}]//Flatten (* Harvey P. Dale, Jan 17 2022 *)

Python

def A087278(n):
    a, b = divmod(n,3)
    return a*(a+2)+b # Chai Wah Wu, Aug 03 2022

Formula

a(3*k) = (k+1)^2 - 1 = A005563(k+1);

a(3*k+1) = (k+1)^2 = A000290(k+1);

a(3*k+2) = (k+1)^2 + 1 = A002522(k+1).

a(n) = floor(n/3)*(floor(n/3) + 2) + n mod 3.

G.f.: -x*(1+x)*(x^4-2*x^3+x^2+1) / ( (1+x+x^2)^2*(x-1)^3 ). - R. J. Mathar, May 22 2019

From Amiram Eldar, Sep 14 2022: (Start)

Sum_{n>=1} 1/a(n) = coth(Pi)*Pi/2 + Pi^2/6 + 1/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = cosech(Pi)*Pi/2 + Pi^2/12 - 1/4. (End)