A270889 Integers n such that the circular graph C_n has a square size deficiency.

3, 6, 27, 150, 867, 5046, 29403, 171366, 998787, 5821350, 33929307, 197754486, 1152597603, 6717831126, 39154389147, 228208503750, 1330096633347, 7752371296326, 45184131144603, 263352415571286, 1534930362283107, 8946229758127350, 52142448186480987, 303908459360758566, 1771308307978070403

Offset

0,1

Comments

Define the size deficiency of a graph G as the number of edges needed to complete G. If G is a cycle graph C_n, this sequence gives the values of n for which C_n has a size deficiency which is a perfect square.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

J. R. M. Antalan and I. F. Callano, On the Size Deficiency of Cycle Graphs and Some Integer Sequences, Asian Journal of Mathematics and Computer Research, 11 (3) (2016),192-200.

Mathematics Stack Exchange, Solving the nonlinear Diophantine equation x2-3x=2y^2

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Formula

a(n+2) = 6*a(n+1) - a(n) - 6; a(0) = 3 , a(1) = 6.

G.f.: 3*(1-5*x+2*x^2)/((1-x)*(1-6*x+x^2)). - Joerg Arndt, Mar 25 2016

a(n) = 3 * A055997(n+1). - Joerg Arndt, Mar 25 2016

a(n) = 7*a(n-1)-7*a(n-2)+a(n-3) for n>2. - Colin Barker, Apr 03 2016

a(n) = 3*(2+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/4. - Colin Barker, Apr 03 2016

Mathematica

a[0] = 3; a[1] = 6; a[n_] := a[n] = 6 a[n - 1] - a[n - 2] - 6; Table[a@ n, {n, 0, 24}] (* Michael De Vlieger, Mar 25 2016 *)
LinearRecurrence[{7, -7, 1}, {3, 6, 27}, 30] (* Harvey P. Dale, Jan 23 2019 *)

Prog

(PARI) is(n)=issquare(n*(n-3)/2) \\ Charles R Greathouse IV, Mar 25 2016
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -7, 7]^n*[3; 6; 27])[1, 1] \\ Charles R Greathouse IV, Mar 25 2016
(PARI) Vec(3*(1-5*x+2*x^2)/((1-x)*(1-6*x+x^2)) + O(x^50)) \\ Colin Barker, Apr 03 2016

Crossrefs

Sequence in context: A060170 A223143 A372039 * A097678 A251609 A366560

Adjacent sequences: A270886 A270887 A270888 * A270890 A270891 A270892

Keyword

nonn,easy

Author

John Rafael M. Antalan, Mar 25 2016

Status

approved