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

0,3

Links

Table of n, a(n) for n=0..59.

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

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)

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 *)

Prog

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

Crossrefs

Union of A005563, A000290 and A002522.

Cf. A002264, A010872, A087279.

Sequence in context: A190018 A217349 A329574 * A054220 A337801 A116214

Adjacent sequences: A087275 A087276 A087277 * A087279 A087280 A087281

Keyword

nonn,easy

Author

Reinhard Zumkeller, Aug 28 2003

Status

approved