|
| |
|
|
A084058
|
|
a(n)=2a(n-1)+7a(n-2), a(0)=1, a(1)=1.
|
|
8
| |
|
|
1, 1, 9, 25, 113, 401, 1593, 5993, 23137, 88225, 338409, 1294393, 4957649, 18976049, 72655641, 278143625, 1064876737, 4076758849, 15607654857, 59752621657, 228758827313, 875786006225, 3352883803641, 12836269650857, 49142725927201
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Binomial transform of expansion of cosh(sqrt(8)x) (A001018 with interpolated zeros : 1, 0, 8, 0, 64, 0, 512, 0, ...); inverse binomial transform of A084128.
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005
|
|
|
REFERENCES
| John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
|
|
|
FORMULA
| a(n)=(1+sqrt(8))^n/2+(1-sqrt(8))^n/2; G.f.: (1-x)/(1-2x-7x^2); E.g.f.: exp(x)cosh(sqrt(8)x).
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*8^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007
|
|
|
PROG
| (MAGMA) Z<x>:= PolynomialRing(Integers()); N<r8>:=NumberField(x^2-8); S:=[ ((1+r8)^n+(1-r8)^n)/2: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 16 2008]
|
|
|
CROSSREFS
| Essentially a duplicate of A083100.
Sequence in context: A083672 A193644 A083100 * A108570 A092769 A139818
Adjacent sequences: A084055 A084056 A084057 * A084059 A084060 A084061
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), May 10 2003
|
| |
|
|
