cp's OEIS Frontend

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.

%I A059116 #21 Mar 09 2024 13:01:06
%S A059116 0,0,6,114,978,6810,43746,271194,1653378,9998970,60229986,362088474,
%T A059116 2174656578,13054316730,78345032226,470127588954,2820937720578,
%U A059116 16926142884090,101558406986466,609355090964634,3656144492925378
%N 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.
%C A059116 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.
%H A059116 Vincenzo Librandi, <a href="/A059116/b059116.txt">Table of n, a(n) for n = 0..1000</a>
%H A059116 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,-27,18).
%F A059116 For n>0, a(n) = 6^n-4*3^n+6.
%F A059116 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
%e A059116 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.
%p A059116 A059116:=n->`if`(n<2, 0, 6^n-4*3^n+6): seq(A059116(n), n=0..20); # _Wesley Ivan Hurt_, Sep 14 2014
%o A059116 (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
%o A059116 (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
%Y A059116 Cf. A058809, A059117.
%K A059116 nonn,easy
%O A059116 0,3
%A A059116 _Henry Bottomley_, Jan 05 2001