A059517 a(n) = A059515(3,n). Number of ways of placing n identifiable nonnegative intervals with a total of exactly three starting and/or finishing points.

0, 0, 12, 138, 1056, 7050, 44472, 273378, 1659936, 10018650, 60289032, 362265618, 2175188016, 13055911050, 78349815192, 470141937858, 2820980767296, 16926272024250, 101558794406952, 609356253226098, 3656147979709776, 21936919259318250, 131621609699088312

Offset

0,3

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

IBM Ponder This, Counting segments overlays, Jan 01 2001.

Index entries for linear recurrences with constant coefficients, signature (10,-27,18).

Formula

a(n) = A058809(n) + A059116(n) = 6^n - 3*3^n + 3 (for n>0).

From Colin Barker, Sep 13 2014: (Start)

a(n) = 10*a(n-1) - 27*a(n-2) + 18*a(n-3) for n>3.

G.f.: -6*x^2*(3*x+2) / ((x-1)*(3*x-1)*(6*x-1)). (End)

E.g.f.: exp(x)*(3 - 3*exp(2*x) + exp(5*x)) - 1. - Elmo R. Oliveira, Jul 12 2026

Example

a(2)=12 since if aA indicates a zero length interval and a-A one of positive length the possibilities are: aA-b-B, b-aA-B, b-B-aA, bB-a-A, a-bB-A, a-A-bB, ab-A-B, ab-B-A, a-b-AB, b-a-AB, a-bA-B, b-a-AB.

Mathematica

LinearRecurrence[{10, -27, 18}, {0, 0, 12, 138}, 25] (* Paolo Xausa, Jul 14 2026 *)

Prog

(PARI) concat([0, 0], Vec(-6*x^2*(3*x+2)/((x-1)*(3*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Sep 13 2014

Crossrefs

Cf. A058809, A059515, A059516.

Sequence in context: A264503 A000467 A372733 * A243966 A377260 A377234

Adjacent sequences: A059514 A059515 A059516 * A059518 A059519 A059520

Keyword

nonn,easy,changed

Author

Henry Bottomley, Jan 19 2001

Extensions

More terms from Colin Barker, Sep 13 2014

Status

approved