A031878 Maximal number of edges in Hamiltonian path in complete graph on n nodes.

0, 1, 3, 5, 10, 13, 21, 25, 36, 41, 55, 61, 78, 85, 105, 113, 136, 145, 171, 181, 210, 221, 253, 265, 300, 313, 351, 365, 406, 421, 465, 481, 528, 545, 595, 613, 666, 685, 741, 761, 820, 841, 903, 925, 990, 1013, 1081, 1105, 1176, 1201, 1275, 1301, 1378

Offset

1,3

Comments

Given a regular polygon with n sides, a(n) is the number of circles that have an edge of the polygon as a diameter (5 for n=4, 10 for n=5, 13 for n=6, ...). - Ahmet Arduç, Jan 28 2017

Quasipolynomial of order 2. [Charles R Greathouse IV, Dec 07 2011]

Links

Table of n, a(n) for n=1..53.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = C(n, 2) if n odd, a(n) = C(n, 2)-n/2+1 if n even.

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

a(n) = ( n*n +n -(n-1)*(n mod 2) )/2. [Frank Ellermann]

Example

E.g. for n=4 [1:2][2:3][3:1][1:4][4:2], so a(4) = 5.

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 3, 5, 10}, 60] (* Harvey P. Dale, Mar 14 2015 *)
CoefficientList[ Series[-x (x^3 + 2x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 52}], x]  (* Robert G. Wilson v, Jul 30 2018 *)

Prog

(PARI) a(n)=if(n%2, n^2-n, n^2-2*n+2)/2  \\ Charles R Greathouse IV, Dec 07 2011

Crossrefs

Cf. A031940.

Sequence in context: A309270 A165718 A340528 * A345890 A265282 A160792

Adjacent sequences: A031875 A031876 A031877 * A031879 A031880 A031881

Keyword

nonn,easy

Author

Colin Mallows

Status

approved