A059116 The sequence lambda(4,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly four starting and/or finishing points.

0, 0, 6, 114, 978, 6810, 43746, 271194, 1653378, 9998970, 60229986, 362088474, 2174656578, 13054316730, 78345032226, 470127588954, 2820937720578, 16926142884090, 101558406986466, 609355090964634, 3656144492925378

Offset: 0

Author

Henry Bottomley, Jan 05 2001

Keywords

• nonn
• easy

Comments

For all n, a(n)=1*6^n-4*3^n+6*1^n-4*0^n+1*0^n [with 0^0=1] where powers are taken of triangular numbers and multiplied by binomial coefficients with alternating signs.

Examples

			a(2)=6 since intervals a-a and b-b can be combined as a-a-b-b, a-b-a-b, a-b-b-a, b-a-b-a, b-a-a-b, or b-a-a-b.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-27,18).

Crossrefs

Cf. A058809, A059117.

Programs

Magma

[1*6^n-4*3^n+6*1^n-4*0^n+1*0^n: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

Maple

A059116:=n->`if`(n<2, 0, 6^n-4*3^n+6): seq(A059116(n), n=0..20); # Wesley Ivan Hurt, Sep 14 2014

PARI

concat([0,0], Vec(-6*x^2*(9*x+1)/((x-1)*(3*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Sep 14 2014

Formula

For n>0, a(n) = 6^n-4*3^n+6.

a(n) = 10*a(n-1)-27*a(n-2)+18*a(n-3) for n>3. G.f.: -6*x^2*(9*x+1) / ((x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Sep 14 2014