A006277 a(n) = (a(n-1) + 1)*a(n-2). (Formerly M0888)

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

Offset

0,3

References

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

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 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.

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

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 *)

Prog

(Haskell)
a006277_list = 1 : scanl ((*) . (+ 1)) 2 a006277_list -- Jack Willis, Dec 22 2013
(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
(Magma) [n le 2 select 1 else (Self(n-1) + 1)*Self(n-2): n in [1..15]]; // Vincenzo Librandi, May 23 2019

Crossrefs

Cf. A001622, A242724, A243967, A243968.

Sequence in context: A093858 A080568 A091339 * A186927 A177010 A300484

Adjacent sequences: A006274 A006275 A006276 * A006278 A006279 A006280

Keyword

nonn

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, May 06 2010

Status

approved