login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A100295 Simple recursive sequence generated from a symmetric matrix. 2
1, 3, 14, 61, 269, 1184, 5213, 22951, 101046, 444873, 1958633, 8623232, 37965321, 167149115, 735903870, 3239948389, 14264452181, 62801801632, 276496162501, 1217323801087, 5359485727718, 23596094350545, 103886025056529, 457376803199488, 2013683168560465 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The characteristic polynomial of M = x^3 - 4*x^2 - 2*x + 1.
Limit_{n -> oo} a(n)/a(n-1) tends to 4.4026788295..., a root of the polynomial and an eigenvalue of M.
A100296 uses the alternative operation M^n * [1, 1, 1].
LINKS
FORMULA
a(n) = rightmost term in M^n * [1, 0, 0], where M = [{3, 2, 1}, {2, 1, 0}, {1, 0, 0}].
a(n) = 4*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: x*(1-x)/(1-4*x-2*x^2+x^3). - Colin Barker, May 25 2013
EXAMPLE
a(4) = 61 since M^4 * [1, 0, 0] = [269, 158, 61]. (leftmost term = a(5). M
MAPLE
a:= n-> (<<3|2|1>, <2|1|0>, <1|0|0>>^n)[1, 3]:
seq(a(n), n=1..30); # Alois P. Heinz, May 25 2013
MATHEMATICA
LinearRecurrence[{4, 2, -1}, {1, 3, 14}, 40] (* G. C. Greubel, Feb 05 2023 *)
PROG
(Magma) I:=[1, 3, 14]; [n le 3 select I[n] else 4*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 05 2023
(SageMath)
@CachedFunction
def a(n): # a = A100296
if (n<3): return (0, 1, 3)[n]
else: return 4*a(n-1) + 2*a(n-2) - a(n-3)
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 05 2023
CROSSREFS
Sequence in context: A281349 A307268 A237608 * A291025 A320499 A091701
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Nov 11 2004
EXTENSIONS
More terms from Colin Barker, May 25 2013
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 06:47 EDT 2024. Contains 370953 sequences. (Running on oeis4.)