A271624 - OEIS

%I #87 Nov 17 2024 15:47:53

%S 2,4,10,20,34,52,74,100,130,164,202,244,290,340,394,452,514,580,650,

%T 724,802,884,970,1060,1154,1252,1354,1460,1570,1684,1802,1924,2050,

%U 2180,2314,2452,2594,2740,2890,3044,3202,3364,3530,3700,3874,4052,4234,4420,4610,4804,5002,5204,5410,5620

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

%C Numbers n such that 2*n - 4 is a perfect square.

%C For n > 2, the number of square a(n)-gonal numbers is finite. - _Muniru A Asiru_, Oct 16 2016

%H Muniru A Asiru, <a href="/A271624/b271624.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) = 2*A002522(n-1).

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

%F Sum_{n>=1} 1/a(n) = (1 + Pi*coth(Pi))/4 = 1.038337023734290587067... . - _Vaclav Kotesovec_, Apr 11 2016

%F a(n) = A005893(n-1), n > 1. - _R. J. Mathar_, Apr 12 2016

%F a(n) = 2 + 2*(n-1)^2. - _Tyler Skywalker_, Jul 21 2016

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

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

%F a(n) = 2*A160457(n).

%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 + 4 = 2.

%t Table[2 n^2 - 4 n + 4, {n, 54}] (* _Michael De Vlieger_, Apr 11 2016 *)

%t LinearRecurrence[{3,-3,1},{2,4,10},60] (* _Harvey P. Dale_, Jul 18 2023 *)

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

%o (Magma) [ n: n in [1..6000] | IsSquare(2*n-4)];

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

%o (PARI) a(n)=2*n^2-4*n+4 \\ _Charles R Greathouse IV_, Apr 11 2016

%Y Cf. A002522, numbers n such that 2*n + k is a perfect square: no sequence (k = -9), A255843 (k = -8), A271649 (k = -7), A093328 (k = -6), A097080 (k = -5), this sequence (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).

%Y Cf. A005893, A160457.

%K nonn,easy

%O 1,1

%A _Juri-Stepan Gerasimov_, Apr 11 2016