This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A084058 a(n) = 2*a(n-1) + 7*a(n-2) for n>1, a(0)=1, a(1)=1. 11
 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; text; 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, Sep 25 2005 REFERENCES John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,7). FORMULA a(n) = ((1+sqrt(8))^n + (1-sqrt(8))^n)/2. G.f.: (1-x)/(1-2*x-7*x^2). E.g.f.: exp(x) * cosh(sqrt(8)*x). a(n) = Sum_{k=0..n} A098158(n,k)*8^(n-k). - Philippe Deléham, Dec 26 2007 G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013 Satisfies recurrence relation system a(n) = 2*b(n-1) - a(n-1), b(n) = 3*b(n-1) + 2*a(n-1), a(0)=1, b(0)=1. - Ilya Gutkovskiy, Apr 11 2017 MATHEMATICA a[n_]:= Simplify[((1 + Sqrt[8])^n + (1 - Sqrt[8])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-7x^2), {x, 0, 30}], x] (* Or *) LinearRecurrence[{2, 7}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *) PROG (MAGMA) Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((1+r8)^n+(1-r8)^n)/2: n in [0..30] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 16 2008 (MAGMA) I:=[1, 1]; [n le 2 select I[n] else 2*Self(n-1) +7*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-2*x-7*x^2)) \\ G. C. Greubel, Aug 01 2019 (Sage) ((1-x)/(1-2*x-7*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019 (GAP) a:=[1, 1];; for n in [3..30] do a[n]:=2*a[n-1]+7*a[n-2]; od; a; # G. C. Greubel, Aug 01 2019 CROSSREFS Essentially a duplicate of A083100. Sequence in context: A083672 A193644 A083100 * A108570 A092769 A263951 Adjacent sequences:  A084055 A084056 A084057 * A084059 A084060 A084061 KEYWORD easy,nonn AUTHOR Paul Barry, May 10 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.

Last modified December 15 14:37 EST 2019. Contains 329999 sequences. (Running on oeis4.)