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

0,2

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

Table of n, a(n) for n=0..46.

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

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

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 *)

Prog

(PARI) vector(100, n, (n-1)^2+1+(-1)^n) \\ Derek Orr, Oct 15 2014
(Magma) [n^2+1-(-1)^n: n in [0..60]]; // Vincenzo Librandi, Oct 16 2014
(Sage) [n^2+1-(-1)^n for n in (0..60)] # Bruno Berselli, Oct 16 2014

Crossrefs

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

Sequence in context: A041527 A290493 A266384 * A001641 A374712 A007382

Adjacent sequences: A248822 A248823 A248824 * A248826 A248827 A248828

Keyword

nonn,easy

Author

Paul Curtz, Oct 15 2014

Extensions

Edited by Bruno Berselli, Oct 16 2014

Status

approved