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 6.7 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.

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,changed

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, May 06 2010

Status

approved