cp's OEIS Frontend

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

%I A087278 #25 Sep 14 2022 02:02:10
%S A087278 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,
%T A087278 81,82,99,100,101,120,121,122,143,144,145,168,169,170,195,196,197,224,
%U A087278 225,226,255,256,257,288,289,290,323,324,325,360,361,362,399,400,401
%N A087278 Nonnegative integers whose distance to the nearest square is not greater than 1.
%H A087278 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A087278 a(3*k) = (k+1)^2 - 1 = A005563(k+1);
%F A087278 a(3*k+1) = (k+1)^2 = A000290(k+1);
%F A087278 a(3*k+2) = (k+1)^2 + 1 = A002522(k+1).
%F A087278 a(n) = floor(n/3)*(floor(n/3) + 2) + n mod 3.
%F A087278 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
%F A087278 From _Amiram Eldar_, Sep 14 2022: (Start)
%F A087278 Sum_{n>=1} 1/a(n) = coth(Pi)*Pi/2 + Pi^2/6 + 1/4.
%F A087278 Sum_{n>=1} (-1)^(n+1)/a(n) = cosech(Pi)*Pi/2 + Pi^2/12 - 1/4. (End)
%t A087278 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 *)
%t A087278 Table[n^2+{-1,0,1},{n,20}]//Flatten (* _Harvey P. Dale_, Jan 17 2022 *)
%o A087278 (Python)
%o A087278 def A087278(n):
%o A087278     a, b = divmod(n,3)
%o A087278     return a*(a+2)+b # _Chai Wah Wu_, Aug 03 2022
%Y A087278 Union of A005563, A000290 and A002522.
%Y A087278 Cf. A002264, A010872, A087279.
%K A087278 nonn,easy
%O A087278 0,3
%A A087278 _Reinhard Zumkeller_, Aug 28 2003