

A007482


a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.
(Formerly M2893)


44



1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647, 283667, 1010295, 3598219, 12815247, 45642179, 162557031, 578955451, 2061980415, 7343852147, 26155517271, 93154256107, 331773802863, 1181629920803, 4208437368135
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OFFSET

0,2


COMMENTS

The even neighbor must differ from the odd number by exactly one.
If we defined this sequence by the recurrence (a(n) = 3*a(n1) + 2*a(n2)) that it satisfies, we could prefix it with an initial 0.
a(n) equals term (1,2) in M^n, M = the 3 X 3 matrix [1,1,2; 1,0,1; 2,1,1].  Gary W. Adamson, Mar 12 2009
a(n) equals term (2,2) in M^n, M = the 3 X 3 matrix [0,1,0; 1,3,1; 0,1,0].  Paul Barry, Sep 18 2009
From Gary W. Adamson, Aug 06 2010: (Start)
Starting with "1" = INVERT transform of A002605: (1, 2, 6, 16, 44, ...).
Example: a(3) = 39 = (16, 6, 2, 1) dot (1, 1, 3, 11) = (16 + 6 + 6 + 11). (End)
Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 2, 12, 24, 30, 4, 12, 48, 24, 4,272, 12, 18, 24, ... .  R. J. Mathar, Aug 10 2012
A007482 is also the number of ways of tiling a 3 X n rectangle with 1 X 1 squares, 2 X 2 squares and 2 X 1 (vertical) dominoes.  R. K. Guy, May 20 2015
With offset 1 (a(0) = 0, a(1) = 1) this is a divisibility sequence.  Michael Somos, Jun 03 2015
Number of elements of size 2^(n) in a fractal generated by the second order reversible cellular automaton, rule 150R (see the reference and the link).  Yuriy Sibirmovsky, Oct 04 2016
a(n) is the number of compositions (ordered partitions) of n into parts 1 (of three kinds) and 2 (of two kinds).  Joerg Arndt, Oct 05 2016
a(n) equals the number of words of length n over {0,1,2,3,4} in which 0 and 1 avoid runs of odd lengths.  Milan Janjic, Jan 08 2017
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have two '1's in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.833.  Peter Karpov, Apr 20 2017
This is the Lucas sequence U(P=3,Q=2), and hence for n>=0, a(n+2)/a(n+1) equals the continued fraction 3 + 2/(3 + 2/(3 + 2/(3 + ... + 2/3))) with n 2's.  Greg Dresden, Oct 06 2019


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 439


LINKS

T. D. Noe, Table of n, a(n) for n = 0..200
Alexander Burstein and Opel Jones, Enumeration of Dumont permutations avoiding certain fourletter patterns, arXiv:2002.12189 [math.CO], 2020.
R. K. Guy and William O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly, 34, No. 2, 152155 (1996). Math. Rev. 97d:11017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 442
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Yuriy Sibirmovsky, A fractal with number of elements described by a(n)
Index entries for linear recurrences with constant coefficients, signature (3,2).


FORMULA

G.f.: 1/(13*x2*x^2).
a(n) = 3*a(n1) + 2*a(n2).
a(n) = (ap^(n+1)am^(n+1))/(apam), where ap = (3+sqrt(17))/2 and am = (3sqrt(17))/2.
Let b(0) = 1, b(k) = floor(b(k1)) + 2/b(k1); then, for n>0, b(n) = a(n)/a(n1).  Benoit Cloitre, Sep 09 2002
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...].  Philippe Deléham, Nov 21 2007
a(n) = Sum_{k=0..floor(n/2)} binomial(nk, k)2^k*3^(n2k).  Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..n} A112906(n,k).  Philippe Deléham, Nov 21 2007
a(n) =  a(2n) * (2)^(n+1) for all n in Z.  Michael Somos, Jun 03 2015
If c = (3 + sqrt(17))/2, then c^n = (A206776(n) + sqrt(17)*a(n1)) / 2.  Michael Somos, Oct 13 2016
a(n) = 3^n*hypergeom([(1n)/2,n/2], [n], 8/9)) for n>=1.  Peter Luschny, Jun 28 2017
a(n) = round(((sqrt(17) + 3)/2)^(n+1)/sqrt(17)). The distance of the argument from the nearest integer is about 1/2^(n+3).  M. F. Hasler, Jun 16 2019
E.g.f.: (1/17)*exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)).  Stefano Spezia, Oct 07 2019
a(n) = (sqrt(2)*i)^n * ChebyshevU(n, 3*i/(2*sqrt(2))).  G. C. Greubel, Dec 24 2021


EXAMPLE

G.f. = 1 + 3*x + 11*x^2 + 39*x^3 + 139*x^4 + 495*x^5 + 1763*x^6 + ...
From M. F. Hasler, Jun 16 2019: (Start)
For n = 0, (1, ..., 2n) = () is the empty sequence, which is equal to its only subsequence, which satisfies the condition voidly, whence a(0) = 1.
For n = 1, (1, ..., 2n) = (1, 2); among the four subsequences {(), (1), (2), (1,2)} only (1) does not satisfy the condition, whence a(1) = 3.
For n = 2, (1, ..., 2n) = (1, 2, 3, 4); among the sixteen subsequences {(), ..., (1,2,3,4)}, the 5 subsequences (1), (3), (1,3), (2,3,4) and (1,2,3,4) do not satisfy the condition, whence a(2) = 16  5 = 11.
(End)


MAPLE

a := n > `if`(n=0, 1, 3^n*hypergeom([(1n)/2, n/2], [n], 8/9)):
seq(simplify(a(n)), n = 0..23); # Peter Luschny, Jun 28 2017


MATHEMATICA

a[n_]:=(MatrixPower[{{1, 4}, {1, 2}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3, 2}, {1, 3}, 30] (* Harvey P. Dale, May 25 2013 *)
a[ n_] := Module[ {m = n + 1, s = 1}, If[ m < 0, {m, s} = {m, (2)^m}]; s SeriesCoefficient[ x / (1  3 x  2 x^2), {x, 0, m}]]; (* Michael Somos, Jun 03 2015 *)
a[ n_] := With[{m = n + 1}, If[ m < 0, (2)^m a[ m], Expand[((3 + Sqrt[17])/2)^m  ((3  Sqrt[17])/2)^m ] / Sqrt[17]]]; (* Michael Somos, Oct 13 2016 *)


PROG

(Sage) [lucas_number1(n, 3, 2) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009
(PARI) {a(n) = 2*imag(( (3 + quadgen(68)) / 2)^(n+1))}; /* Michael Somos, Jun 03 2015 */
(Haskell)
a007482 n = a007482_list !! (n1)
a007482_list = 1 : 3 : zipWith (+)
(map (* 3) $ tail a007482_list) (map (* 2) a007482_list)
 Reinhard Zumkeller, Oct 21 2015
(Maxima) a(n) := if n=0 then 1 elseif n=1 then 3 else 3*a(n1)+2*a(n2);
makelist(a(n), n, 0, 12); /* Emanuele Munarini, Jun 28 2017 */
(Magma) I:=[1, 3]; [n le 2 select I[n] else 3*Self(n1) + 2*Self(n2): n in [1..30]]; // G. C. Greubel, Jan 16 2018


CROSSREFS

Row sums of triangle A073387.
Cf. A000045, A000129, A001045, A007455, A007481, A007483, A007484, A201000 (prime subsequence), A052913 (binomial transform), A026597 (inverse binomial transform).
Cf. A206776.
Sequence in context: A166336 A002783 A289834 * A134760 A257290 A281482
Adjacent sequences: A007479 A007480 A007481 * A007483 A007484 A007485


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



