cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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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]

Examples

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

Crossrefs

Cf. A031940.

Programs

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

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]