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A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ). 10
1, 1, 10, 28, 136, 496, 2080, 8128, 32896, 130816, 524800, 2096128, 8390656, 33550336, 134225920, 536854528, 2147516416, 8589869056, 34359869440, 137438691328, 549756338176, 2199022206976, 8796095119360, 35184367894528, 140737496743936, 562949936644096, 2251799847239680 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Binomial transform of expansion of cosh(3*x), the aerated version of A001019, 1,0,9,0,81,0,729,... - Paul Barry, Apr 05 2003

Alternatively: start with the fraction 1/1, take the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top. The limit of the sequence of fractions used to generate this sequence is sqrt(9). - Cino Hilliard, Sep 25 2005

This sequence also gives the number of ordered pairs of subsets (A, B) of {1, 2, ..., n} such that |A u B| is even. (Here "u" stands for the set-theoretic union.) The special case n = 13 can be found as in CRUX Problem 3257. - Walther Janous (walther.janous(AT)tirol.com), Mar 01 2008

For n > 0, a(n) is term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 3,1]. - Gary W. Adamson, Aug 06 2010

a(n) is the number of compositions of n when there are 1 type of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010

a(n) = ((1+3)^n+(1-3)^n)/2. In general, if b(0),b(1),... is the k-th binomial transform of the sequence ((3^n+(-3)^n)/2), then b(n) = ((k+3)^n+(k-3)^n)/2. More generally, if b(0),b(1),... is the k-th binomial transform of the sequence ((m^n+(-m)^n)/2), then b(n) = ((k+m)^n+(k-m)^n)/2. See A034494, A081340-A081342, A034659. - Charlie Marion, Jun 25 2011.

Pisano period lengths: 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 5, 1, 12, 6, 4, 1, 8, 1, 9, 4, ... - R. J. Mathar, Aug 10 2012

Starting with offset 1 the sequence is the INVERT transform of (1, 9, 9, 9, ...). - Gary W. Adamson, Aug 06 2016

REFERENCES

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)

Bill Sands, Problem 3257, Crux Math. 33 (2007), No.5, p. 298.

Index entries for linear recurrences with constant coefficients, signature (2,8).

FORMULA

From Paul Barry, Mar 01 2003: (Start)

a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1.

a(n) = (4^n + (-2)^n)/2.

G.f.: (1-x)/((1+2*x)*(1-4*x)). (End)

From Paul Barry, Apr 05 2003: (Start)

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*9^k.

E.g.f. exp(x)*cosh(3*x). (End)

a(n) = (A078008(n) + A001045(n+1))*2^(n-1) = A014551(n)*2^(n-1). - Paul Barry, Feb 12 2004

Given a(0)=1, b(0)=1 then for i=1, 2, ... a(i)/b(i) = (a(i-1)+ 9*b(i-1)) / (a(i-1) + b(i-1)). - Cino Hilliard, Sep 25 2005

a(n) = Sum_{k=0..n} A098158(n,k)*9^(n-k). - Philippe Deléham, Dec 26 2007

a(n) = ((1+sqrt(9))^n + (1-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008

If p[1]=1, and p[i]=9, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Apr 29 2010

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(9*k-1)/(x*(9*k+8) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

MAPLE

A003665:=n->2^(n-1)*( 2^n + (-1)^n ): seq(A003665(n), n=0..30); # Wesley Ivan Hurt, Apr 28 2017

MATHEMATICA

CoefficientList[Series[(1+8x)/(1-2x-8x^2), {x, 0, 30}], x] (* or *)

LinearRecurrence[{2, 8}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

PROG

(PARI) a(n)=2^(n-1)*( 2^n + (-1)^n );

(MAGMA) [2^(n-1)*( 2^n + (-1)^n ): n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

(Sage) [2^(n-1)*(2^n +(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 02 2019

(GAP) List([0..30], n-> 2^(n-1)*(2^n +(-1)^n)); # G. C. Greubel, Aug 02 2019

CROSSREFS

Cf. A001019, A001045, A014551, A078008, A098158.

Cf. A034494, A081340-A081342, A034659.

Sequence in context: A076712 A116973 A264579 * A219812 A185985 A239477

Adjacent sequences:  A003662 A003663 A003664 * A003666 A003667 A003668

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry revised by N. J. A. Sloane, Nov 22 2006

STATUS

approved

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Last modified December 11 12:33 EST 2019. Contains 329916 sequences. (Running on oeis4.)