A276123 - OEIS

%I #46 Jul 04 2024 14:12:49

%S 1,1,1,4,10,55,154,868,2449,13825,39025,220324,621946,3511351,9912106,

%T 55961284,157971745,891869185,2517635809,14213945668,40124201194,

%U 226531261495,639469583290,3610286238244,10191389131441,57538048550401,162422756519761

%N a(0) = a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1) + 1)*(a(n-2) + 1) / a(n-3).

%H Colin Barker, <a href="/A276123/b276123.txt">Table of n, a(n) for n = 0..1000</a>

%H Matthew Christopher Russell, <a href="https://doi.org/doi:10.7282/T3MC926D">Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences"</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,17,0,-17,0,1).

%F a(n) = (9-3*(-1)^n)/2*a(n-1) - a(n-2) - 1.

%F From _Colin Barker_, Aug 21 2016: (Start)

%F a(n) = 17*a(n-2) - 17*a(n-4) + a(n-6) for n > 5.

%F G.f.: (1 + x - 16*x^2 - 13*x^3 + 10*x^4 + 4*x^5) / ((1-x)*(1+x)*(1 - 16*x^2 + x^4)). (End)

%F a(2n+1) = A073352(n). a(2n) = A048907(n). - _R. J. Mathar_, Jul 04 2024

%t LinearRecurrence[{0, 17, 0, -17, 0, 1}, {1, 1, 1, 4, 10, 55}, 40] (* _Vincenzo Librandi_, Aug 27 2016 *)

%t nxt[{a_,b_,c_}]:={b,c,((c+1)(b+1))/a}; NestList[nxt,{1,1,1},30][[All,1]] (* _Harvey P. Dale_, Oct 01 2021 *)

%o (PARI) Vec((1+x-16*x^2-13*x^3+10*x^4+4*x^5)/((1-x)*(1+x)*(1-16*x^2+x^4)) + O(x^30)) \\ _Colin Barker_, Aug 21 2016

%o (Magma) I:=[1,1,1,4,10,55]; [n le 6 select I[n] else 17*Self(n-2)-17*Self(n-4)+Self(n-6): n in [1..30]]; // _Vincenzo Librandi_, Aug 27 2016

%Y Cf. A072881, A076839, A276175.

%K nonn,easy

%O 0,4

%A _Bruno Langlois_, Aug 21 2016

%E More terms from _Colin Barker_, Aug 21 2016