A239909 Arises from a construction of equiangular lines in complex space of dimension 2.

1, 1, 2, 3, 5, 9, 15, 26, 45, 77, 133, 229, 394, 679, 1169, 2013, 3467, 5970, 10281, 17705, 30489, 52505, 90418, 155707, 268141, 461761, 795191, 1369386, 2358197, 4061013, 6993405, 12043229, 20739450, 35715071, 61504345, 105915637, 182395603, 314100514

Offset: 1

Author

N. J. A. Sloane, Apr 09 2014

Keywords

• nonn
• easy

Comments

a(n+2) is the number of binary words of length n in which every run of zeros has length congruent to 1 modulo 3. - Ira M. Gessel, Jan 22 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• G. McConnell, Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3, arXiiv preprint arXiv:1402.7330, 2014, p. 4.
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1).

Crossrefs

Cf. A116732.

Programs

Magma

I:=[1,1,2,3]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-4): n in [1..50]];

Mathematica

LinearRecurrence[{1, 1, 1, -1}, {1, 1, 2, 3}, 40] (* or *) CoefficientList[Series[(1 - x^3)/(x^4 - x^3 - x^2 - x + 1), {x, 0, 100}], x] (* Vincenzo Librandi, Apr 09 2014 *)

Formula

From Vincenzo Librandi Apr 09 2014: (Start)

G.f.: x*(1-x^3)/(x^4-x^3-x^2-x+1).

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) for n>4.

a(n) = a(n-1) + 2*a(n-3) + A116732(n-5) for n>4. (End)