A006277 a(n) = (a(n-1) + 1)*a(n-2).

1, 1, 2, 3, 8, 27, 224, 6075, 1361024, 8268226875, 11253255215681024, 93044467205527772332546875, 1047053135870867396062743192203958743681024, 97422501162981936223682742789520433197690551802305989766350860546875

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.7.
• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.7 Cahen's Constant p. 435 and Section 6.10 Quadratic recurrence constants pp. 445-446.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..18
• J. L. Davison and Jeffrey Shallit, Continued Fractions for Some Alternating Series, Monatsh. Math., Vol. 111 (1991), pp. 119-126.

Crossrefs

Cf. A001622, A242724, A243967, A243968.

Programs

Haskell

a006277_list = 1 : scanl ((*) . (+ 1)) 2 a006277_list -- Jack Willis, Dec 22 2013

Magma

[n le 2 select 1 else (Self(n-1) + 1)*Self(n-2): n in [1..15]]; // Vincenzo Librandi, May 23 2019

Maple

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

Mathematica

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

Maxima

a(n) := if (n = 0 or n = 1) then 1 else a(n-2)*(a(n-1)+1) $
makelist(a(n),n,0,12); /* Emanuele Munarini, Mar 23 2017 */

Formula

Sum_{n>=0} 1/a(n) = 3. - Gerald McGarvey, Jul 20 2004

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

Sum_{k>=0} (-1)^k/(a(k)*a(k+1)) = A242724. - Amiram Eldar, May 15 2021

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, May 06 2010