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A007482 a(n) = number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.
(Formerly M2893)
38
1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647, 283667, 1010295, 3598219, 12815247, 45642179, 162557031, 578955451, 2061980415, 7343852147, 26155517271, 93154256107, 331773802863, 1181629920803, 4208437368135 (list; graph; refs; listen; history; text; internal format)
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(n-1) + 2*a(n-2)) 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 1x1 squares, 2x2 squares and 2x1 (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

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

R. K. Guy, William O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly, 34, No. 2, 152-155 (1996). Math. Rev. 97d:11017.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 442

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/(1-3*x-2*x^2).

a(n) = 3*a(n-1) + 2*a(n-2).

a(n) = (ap^(n+1)-am^(n+1))/(ap-am), where ap = (3+sqrt(17))/2 and am = (3-sqrt(17))/2.

Let b(0) = 1, b(k) = floor(b(k-1)) + 2/b(k-1); then, for n>0, b(n) = a(n)/a(n-1). - 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(n-k, k)2^k*3^(n-2k). - Paul Barry, Apr 23 2005

a(n) = Sum_{k=0..n} A112906(n,k). - Philippe Deléham, Nov 21 2007

a(n) = - a(-2-n) * (-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(n-1)) / 2. - Michael Somos, Oct 13 2016

EXAMPLE

G.f. = 1 + 3*x + 11*x^2 + 39*x^3 + 139*x^4 + 495*x^5 + 1763*x^6 + ...

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 xrange(1, 25)] # Zerinvary Lajos, Apr 22 2009

(PARI) {a(n) = n++; 2 * imag(( (3 + quadgen(68)) / 2 )^n)}; /* Michael Somos, Jun 03 2015 */

(Haskell)

a007482 n = a007482_list !! (n-1)

a007482_list = 1 : 3 : zipWith (+)

               (map (* 3) $ tail a007482_list) (map (* 2) a007482_list)

-- Reinhard Zumkeller, Oct 21 2015

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: A227638 A166336 A002783 * A134760 A257290 A281482

Adjacent sequences:  A007479 A007480 A007481 * A007483 A007484 A007485

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 23 21:40 EST 2017. Contains 281216 sequences.