A271649 - OEIS

%I #59 Nov 27 2024 15:50:05

%S 4,8,16,28,44,64,88,116,148,184,224,268,316,368,424,484,548,616,688,

%T 764,844,928,1016,1108,1204,1304,1408,1516,1628,1744,1864,1988,2116,

%U 2248,2384,2524,2668,2816,2968,3124,3284,3448,3616,3788,3964,4144,4328,4516,4708,4904,5104,5308,5516

%N a(n) = 2*(n^2 - n + 2).

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

%C 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

%H Vincenzo Librandi, <a href="/A271649/b271649.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

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

%F a(n) = A137882(n), n > 1. - _R. J. Mathar_, Apr 12 2016

%F Sum_{n>=1} 1/a(n) = tanh(sqrt(7)*Pi/2)*Pi/(2*sqrt(7)). - _Amiram Eldar_, Jul 30 2024

%F From _Elmo R. Oliveira_, Nov 18 2024: (Start)

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

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

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

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

%p A271649:=n->2*(n^2-n+2): seq(A271649(n), n=1..60); # _Wesley Ivan Hurt_, Aug 31 2016

%t Table[2 (n^2 - n + 2), {n, 53}] (* or *)

%t Select[Range@ 5516, IntegerQ@ Sqrt[2 # - 7] &] (* or *)

%t Table[SeriesCoefficient[(-4 (1 - x + x^2))/(-1 + x)^3, {x, 0, n}], {n, 0, 52}] (* _Michael De Vlieger_, Apr 11 2016 *)

%t LinearRecurrence[{3,-3,1},{4,8,16},60] (* _Harvey P. Dale_, Jun 14 2022 *)

%o (Magma) [ 2*n^2 - 2*n + 4: n in [1..60]];

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

%o (PARI) a(n)=2*(n^2-n+2) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000124, A014206, A137882.

%Y 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).

%K nonn,easy

%O 1,1

%A _Juri-Stepan Gerasimov_, Apr 11 2016