A271624 a(n) = 2*n^2 - 4*n + 4.

2, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234, 4420, 4610, 4804, 5002, 5204, 5410, 5620

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 11 2016

Keywords

• nonn
• easy

Comments

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

For n > 2, the number of square a(n)-gonal numbers is finite. - Muniru A Asiru, Oct 16 2016

Examples

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

Links

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

Crossrefs

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).

Cf. A005893, A160457.

Programs

Magma

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

Magma

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

Mathematica

Table[2 n^2 - 4 n + 4, {n, 54}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{2,4,10},60] (* Harvey P. Dale, Jul 18 2023 *)

PARI

x='x+O('x^99); Vec(2*x*(1-x+2*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

PARI

a(n)=2*n^2-4*n+4 \\ Charles R Greathouse IV, Apr 11 2016

Formula

a(n) = 2*A002522(n-1).

G.f.: 2*x*(1 - x + 2*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016

Sum_{n>=1} 1/a(n) = (1 + Pi*coth(Pi))/4 = 1.038337023734290587067... . - Vaclav Kotesovec, Apr 11 2016

a(n) = A005893(n-1), n > 1. - R. J. Mathar, Apr 12 2016

a(n) = 2 + 2*(n-1)^2. - Tyler Skywalker, Jul 21 2016

From Elmo R. Oliveira, Nov 17 2024: (Start)

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

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

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