login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A081704 Let f(0)=1, f(1)=t, f(n+1) = (f(n)^2+t^n)/f(n-1). f(t) is a polynomial with integer coefficients. Then a(n) = f(n) when t=3. 5
1, 3, 12, 51, 219, 942, 4053, 17439, 75036, 322863, 1389207, 5977446, 25719609, 110665707, 476169708, 2048851419, 8815747971, 37932185598, 163213684077, 702271863591, 3021718265724, 13001775737847, 55943723892063, 240713292246774, 1035735289557681 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

f satisfies the linear recursion f(n+1) = (t+2)f(n)-tf(n-1)). For t=3 this gives a(n+1) = 5*a(n)-3*a(n-1).

Given the 3 X 3 matrix [1,1,1; 1,1,2; 1,1,3] = M, a(n) = term (1,1) in M^(n+1). - Gary W. Adamson, Aug 06 2010

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-3).

FORMULA

a(n+1) = (a(n)^2 + 3^n) / a(n-1).

From Philippe Deléham, Nov 14 2008: (Start)

G.f.: (1-2*x)/(1-5*x+3*x^2).

a(n) = Sum_{k, 0<=k<=n} A147703(n,k)*2^k.

(End)

a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-1+sqrt(13)) + (1+sqrt(13))*(5+sqrt(13))^n))/sqrt(13). - Colin Barker, Nov 26 2016

MAPLE

f := proc(n) if n=0 then 1 elif n=1 then t else sort(simplify((f(n-1)^2+t^(n-1))/f(n-2)), t) fi end; a := i->subs(t=3, f(i));

MATHEMATICA

a[0]=1; a[1]=3; a[n_] := a[n]=5a[n-1]-3a[n-2]

LinearRecurrence[{5, -3}, {1, 3}, 30] (* Harvey P. Dale, Jul 28 2013 *)

PROG

(PARI) Vec((1-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 26 2016

CROSSREFS

Cf. A006012, A001519.

Equals 3*A018902(n-1) for n>0.

Sequence in context: A155179 A228770 A104268 * A166482 A007854 A151182

Adjacent sequences:  A081701 A081702 A081703 * A081705 A081706 A081707

KEYWORD

nonn,easy

AUTHOR

Victor Ufnarovski (ufn(AT)maths.lth.se), Apr 02 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 16 14:52 EST 2018. Contains 318167 sequences. (Running on oeis4.)