A271649 a(n) = 2*(n^2 - n + 2).

4, 8, 16, 28, 44, 64, 88, 116, 148, 184, 224, 268, 316, 368, 424, 484, 548, 616, 688, 764, 844, 928, 1016, 1108, 1204, 1304, 1408, 1516, 1628, 1744, 1864, 1988, 2116, 2248, 2384, 2524, 2668, 2816, 2968, 3124, 3284, 3448, 3616, 3788, 3964, 4144, 4328, 4516, 4708, 4904, 5104, 5308, 5516

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 11 2016

Keywords

• nonn
• easy

Comments

Numbers n such that 2*n - 7 is a perfect square.

Galois numbers for three-dimensional vector space, defined as the total number of subspaces in a three-dimensional vector space over GF(n-1), when n-1 is a power of a prime. - Artur Jasinski, Aug 31 2016, corrected by Robert Israel, Sep 23 2016

Examples

			a(1) = 2*(1^2 - 1 + 2) = 4.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000124, A014206, A137882.

Numbers h such that 2*h + k is a perfect square: no sequence (k=-9), A255843 (k=-8), this sequence (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)*A147973 (k=8), A139570 (k=9), A271625 (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

Programs

Magma

[ 2*n^2 - 2*n + 4: n in [1..60]];

Magma

[ n: n in [1..6000] | IsSquare(2*n-7)];

Maple

A271649:=n->2*(n^2-n+2): seq(A271649(n), n=1..60); # Wesley Ivan Hurt, Aug 31 2016

Mathematica

Table[2 (n^2 - n + 2), {n, 53}] (* or *)
Select[Range@ 5516, IntegerQ@ Sqrt[2 # - 7] &] (* or *)
Table[SeriesCoefficient[(-4 (1 - x + x^2))/(-1 + x)^3, {x, 0, n}], {n, 0, 52}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{4,8,16},60] (* Harvey P. Dale, Jun 14 2022 *)

PARI

a(n)=2*(n^2-n+2) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 4*A000124(n).

a(n) = 2*A014206(n).

a(n) = A137882(n), n > 1. - R. J. Mathar, Apr 12 2016

Sum_{n>=1} 1/a(n) = tanh(sqrt(7)*Pi/2)*Pi/(2*sqrt(7)). - Amiram Eldar, Jul 30 2024

From Elmo R. Oliveira, Nov 18 2024: (Start)

G.f.: 4*x*(1 - x + x^2)/(1 - x)^3.

E.g.f.: 2*(exp(x)*(x^2 + 2) - 2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)