A248825 a(n) = n^2 + 1 - (-1)^n.

0, 3, 4, 11, 16, 27, 36, 51, 64, 83, 100, 123, 144, 171, 196, 227, 256, 291, 324, 363, 400, 443, 484, 531, 576, 627, 676, 731, 784, 843, 900, 963, 1024, 1091, 1156, 1227, 1296, 1371, 1444, 1523, 1600, 1683, 1764, 1851, 1936, 2027, 2116

Offset: 0

Author

Paul Curtz, Oct 15 2014

Keywords

• nonn
• easy

Comments

Also, A016742 and A164897 interleaved.

See the spiral in Example field of A054552: after 0, the sequence is given by the terms of the semidiagonals 4, 16, 36, 64, 100, ... and 3, 11, 27, 51, 83, ... sorted into ascending order.

Primes of the sequence are in A056899.

Links

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

Crossrefs

Cf. A000290, A008586, A008602, A010673, A016742, A042963, A056899, A164897, A166519, A168277, A248800.

Programs

Magma

[n^2+1-(-1)^n: n in [0..60]]; // Vincenzo Librandi, Oct 16 2014

Mathematica

Table[n^2 + 1 - (-1)^n, {n, 0, 60}] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,4,11},60] (* Harvey P. Dale, Jun 30 2019 *)

PARI

vector(100,n,(n-1)^2+1+(-1)^n) \\ Derek Orr, Oct 15 2014

Sage

[n^2+1-(-1)^n for n in (0..60)] # Bruno Berselli, Oct 16 2014

Formula

a(n) = a(-n) = 2*a(n-1) - 2*(n-3) + a(n-4).

a(n) = n^2 + A010673(n) = (n+1)^2 - A168277(n+1).

a(n+1) = A248800(n) + A042963(n+1) = a(n) + A166519(n).

a(n+2) = a(n) + 4*n.

a(n+5) = a(n-5) + A008602(n).

G.f.: x*(3 - 2*x + 3*x^2)/((1 + x)*(1 - x)^3). - Bruno Berselli, Oct 15 2014

Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - Amiram Eldar, Aug 21 2022

Extensions

Edited by Bruno Berselli, Oct 16 2014