%I #35 Nov 18 2024 22:31:15

%S 3,13,27,45,67,93,123,157,195,237,283,333,387,445,507,573,643,717,795,

%T 877,963,1053,1147,1245,1347,1453,1563,1677,1795,1917,2043,2173,2307,

%U 2445,2587,2733,2883,3037,3195,3357,3523,3693,3867,4045,4227,4413,4603,4797,4995,5197,5403,5613,5827

%N a(n) = 2*n^2 + 4*n - 3.

%C Numbers n such that 2*n + 10 is a perfect square.

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

%F G.f.: x*(3 + 4*x - 3*x^2)/(1 - x)^3. - _Ilya Gutkovskiy_, Apr 11 2016

%F Sum_{n>=1} 1/a(n) = 13/30 - Pi*cot(sqrt(5/2)*Pi)/(2*sqrt(10)) = 0.5627678459924... . - _Vaclav Kotesovec_, Apr 11 2016

%F From _Elmo R. Oliveira_, Nov 17 2024: (Start)

%F E.g.f.: exp(x)*(2*x^2 + 6*x - 3) + 3.

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

%e a(1) = 2*1^2 + 4*1 - 3 = 3.

%t Table[2 n^2 + 4 n - 3, {n, 53}] (* _Michael De Vlieger_, Apr 11 2016 *)

%t LinearRecurrence[{3,-3,1},{3,13,27},60] (* _Harvey P. Dale_, Jun 08 2023 *)

%o (Magma) [ 2*n^2 + 4*n - 3: n in [1..60]];

%o (Magma) [ n: n in [1..6000] | IsSquare(2*n+10)];

%o (PARI) x='x+O('x^99); Vec(x*(3+4*x-3*x^2)/(1-x)^3) \\ _Altug Alkan_, Apr 11 2016

%Y Cf. A201713.

%Y Numbers h such that 2*h + k is a perfect square: A294774 (k=-9), A255843 (k=-8), A271649 (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), this sequence (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

%K nonn,easy,changed

%O 1,1

%A _Juri-Stepan Gerasimov_, Apr 11 2016