A006277 - OEIS

%I M0888 #53 Sep 08 2022 08:44:34

%S 1,1,2,3,8,27,224,6075,1361024,8268226875,11253255215681024,

%T 93044467205527772332546875,

%U 1047053135870867396062743192203958743681024,97422501162981936223682742789520433197690551802305989766350860546875

%N a(n) = (a(n-1) + 1)*a(n-2).

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.7.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, pp. 445-446.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Harvey P. Dale, <a href="/A006277/b006277.txt">Table of n, a(n) for n = 0..18</a>

%H J. L. Davison and Jeffrey Shallit, <a href="http://dx.doi.org/10.1007/BF01332350">Continued Fractions for Some Alternating Series</a>, Monatsh. Math., Vol. 111 (1991), pp. 119-126.

%F Sum_{n>=0} 1/a(n) = 3. - _Gerald McGarvey_, Jul 20 2004

%F a(n) = floor(A243967^(phi^n) * A243968^((1-phi)^n)), where phi is the golden ratio (1+sqrt(5))/2. - _Vaclav Kotesovec_, Jan 19 2015

%F Sum_{k>=0} (-1)^k/(a(k)*a(k+1)) = A242724. - _Amiram Eldar_, May 15 2021

%p A006277 := proc(n) options remember; if n <= 1 then RETURN(1) else A006277(n-2)*(A006277(n-1)+1); fi; end;

%t a=b=1;lst={a,b};Do[AppendTo[lst,c=a*b+a];a=b;b=c,{n,0,12}];lst (* _Vladimir Joseph Stephan Orlovsky_, May 06 2010 *)

%t RecurrenceTable[{a[n]==a[n-2]*(1+a[n-1]),a[0]==1,a[1]==1},a,{n,0,15}] (* _Vaclav Kotesovec_, Jan 19 2015 *)

%t nxt[{a_,b_}]:={b,a(b+1)}; NestList[nxt,{1,1},15][[All,1]] (* _Harvey P. Dale_, Jun 20 2021 *)

%o (Haskell)

%o a006277_list = 1 : scanl ((*) . (+ 1)) 2 a006277_list -- _Jack Willis_, Dec 22 2013

%o (Maxima) a(n) := if (n = 0 or n = 1) then 1 else a(n-2)*(a(n-1)+1) $

%o makelist(a(n),n,0,12); _Emanuele Munarini_, Mar 23 2017

%o (Magma) [n le 2 select 1 else (Self(n-1) + 1)*Self(n-2): n in [1..15]]; // _Vincenzo Librandi_, May 23 2019

%Y Cf. A001622, A242724, A243967, A243968.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, _Jeffrey Shallit_

%E More terms from _Vladimir Joseph Stephan Orlovsky_, May 06 2010