|
| |
|
|
A003665
|
|
2^(n-1)*( 2^n + (-1)^n ).
|
|
8
| |
|
|
1, 1, 10, 28, 136, 496, 2080, 8128, 32896, 130816, 524800, 2096128, 8390656, 33550336, 134225920, 536854528, 2147516416, 8589869056, 34359869440, 137438691328, 549756338176, 2199022206976, 8796095119360
(list; graph; refs; listen; history; 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 (hillcino368(AT)gmail.com), 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
a(n), n>0 = term (1,1) in the n-th power of the 2x2 matrix [1,3; 3,1]. [From 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. [From Milan R. Janjic (agnus(AT)blic.net), 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.
|
|
|
REFERENCES
| John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p16
M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.
Bill Sands, Problem 3257, CRUX MATH. 33 (2007), No.5, p. 298.
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
|
|
|
FORMULA
| a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1. a(n) = 4^n/2+(-2)^n/2. G.f. (1-x)/((1+2*x)*(1-4*x)). - Paul Barry, Mar 01 2003
a(n) = sum{k=0..floor(n/2), C(n, 2*k)*9^k}; E.g.f. exp(x)*cosh(3*x) - Paul Barry, Apr 05 2003
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 (hillcino368(AT)gmail.com), Sep 25 2005
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*9^(n-k). - Philippe DELEHAM, Dec 26 2007
a(n) = ((1+sqrt(9))^n+(1-sqrt(9))^n)/2. [From 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). [From Milan R. Janjic (agnus(AT)blic.net), Apr 29 2010]
|
|
|
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
|
|
|
CROSSREFS
| Sequence in context: A126364 A076712 A116973 * A185985 A066527 A103423
Adjacent sequences: A003662 A003663 A003664 * A003666 A003667 A003668
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2006
Corrected A-number in the first comment and added "aerated". - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2008
|
| |
|
|
