|
| |
|
|
A099013
|
|
Sum_{ k=0..n} 3^(k-1)Fib(k).
|
|
1
| |
|
|
0, 1, 4, 22, 103, 508, 2452, 11929, 57856, 280930, 1363495, 6618856, 32128024, 155953777, 757013548, 3674624638, 17836995847, 86582609284, 420280790476, 2040085854985, 9902784679240, 48069126732586, 233332442310919
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Partial sums of A099012. Binomial transform of A063092 (with leading 0).
|
|
|
LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (4,6,-9).
|
|
|
FORMULA
| G.f.: x/((1-x)(1-3x-9x^2));
a(n)=4a(n-1)+6a(n-2)-9a(n-3);
a(n)=3^(n-1)*sum{k=0..n, Fib(n-k)3^(-k)};
a(n)=(3/2+3sqrt(5)/2)^n(1/22+7sqrt(5)/110)+(1/22-7sqrt(5)/110)(3/2-3sqrt(5)/2)^n-1/11.
|
|
|
MATHEMATICA
| Join[{a=0, b=1}, Table[c=3*b+9*a+1; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 18 2011*)
Table[Sum[3^(k-1) Fibonacci[k], {k, 0, n}], {n, 0, 30}] (* or *) LinearRecurrence[{4, 6, -9}, {0, 1, 4}, 30] (* From Harvey P. Dale, Dec 09 2011 *)
|
|
|
CROSSREFS
| Cf. A000045.
Sequence in context: A007901 A088581 A017970 * A184510 A184701 A001436
Adjacent sequences: A099010 A099011 A099012 * A099014 A099015 A099016
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 22 2004
|
|
|
EXTENSIONS
| Sign in second formula corrected by Harvey P. Dale, Dec 09 2011
|
| |
|
|
