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%I M3875 #87 Jan 22 2024 06:20:26
%S 1,5,17,61,217,773,2753,9805,34921,124373,442961,1577629,5618809,
%T 20011685,71272673,253841389,904069513,3219891317,11467812977,
%U 40843221565,145465290649,518082315077,1845177526529,6571697209741
%N a(n) = 3*a(n-1) + 2*a(n-2), with a(0)=1, a(1)=5.
%C Number of subsequences of [1,...,2n+1] in which each odd number has an even neighbor. The even neighbor must differ from the odd number by exactly 1.
%C From _Gary W. Adamson_, Aug 06 2016: (Start)
%C a(n) is the upper left term in the (n+1)-th matrix power of [(1,4); (1,2)] and is the INVERT transform of (1, 4, 4*2, 4*2^2, 4*2^3, 4*2^4, ...), i.e. of (1, 4, 8, 16, 32, 64, 128, ...).
%C The sequence is equal to row sums of an eigentriangle generated as follows: Let matrix A = an infinite lower triangle with (1, 4, 8, 16, ...) in every column and B = a triangle with (1, 1, 5, 17, 61, ...) as the rightmost diagonal and the rest zeros. Then the eigentriangle is A * B as follows: (1; 4, 1; 8, 4, 5; 16, 8, 20, 17; ...) with sums (1, 5, 17, 61, ...). Individual rows can be recovered by taking the dot product of (1, 4, 8, 16, ...) reversed and equal numbers of terms of(1, 1, 5, 17, ...). For example, 61 = (16, 8, 4, 1) dot (1, 1, 5, 17) = (16 + 8 + 20 + 17). (End)
%C The sequence is equal to A007482 convolved with (1, 2, 0, 0, 0, ...); i.e. (1 + 5x + 17x^2 + ...) = (1 + 3x + 11x^2 + 39x^3 + ...) * (1 + 2x). - _Gary W. Adamson_, Aug 08 2016
%C a(n) is the number of edge covers of the fan graph F(1,n+1) with a pendant attached to the vertex of degree n+1. - _Feryal Alayont_, Dec 08 2023
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007483/b007483.txt">Table of n, a(n) for n = 0..500</a>
%H C. Bautista-Ramos and C. Guillen-Galvan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Bautista/bautista4.html">Fibonacci numbers of generalized Zykov sums</a>, J. Integer Seq., 15 (2012), Article 12.7.8.
%H A. Burstein, S. Kitaev, and T. Mansour, <a href="https://arxiv.org/abs/math/0310379">Independent sets in certain classes of (almost) regular graphs</a>, arXiv:math/0310379 [math.CO], 2003.
%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.
%H R. K. Guy and William O. J. Moser, <a href="http://www.fq.math.ca/Scanned/34-2/guy.pdf">Numbers of subsequences without isolated odd members</a>, Fibonacci Quarterly, 34, No. 2, 152-155 (1996). Math. Rev. 97d:11017.
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,2).
%F G.f.: (1 + 2*x)/(1 - 3*x - 2*x^2).
%F a(n) = ((17 + 7*sqrt(17))/34)*((3 + sqrt(17))/2)^n* + ((17 - 7*sqrt(17))/34)*((3 - sqrt(17))/2)^n. - _Paul Barry_, Dec 08 2004
%F a(n-1) = Sum_{k=0..n} 2^(n-k)*A122542(n,k), n>=1. - _Philippe Deléham_, Oct 08 2006
%F a(n) = upper left term in the 2 X 2 matrix [1,2; 2,2]^(n+1). Also [a(n), a(n+1)] = the 2 X 2 matrix [0,1; 2,3]^(n+1) * [1,1]. Example: [0,1; 2,3]^4 * [1,1] = [61, 217]. - _Gary W. Adamson_, Mar 16 2008
%F Also, for n>=2, a(n)=[1,2;2,2]^(n-1)*[1,2]*[1,2]. - _John M. Campbell_, Jul 09 2011
%F a(n) = A007482(n) + 2*A007482(n-1). - _R. J. Mathar_, Sep 21 2012
%F a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + 2*ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - _G. C. Greubel_, Jun 28 2021
%e a(2) = 17 = (8, 4, 1) dot (1, 1, 5) = 8 + 4 + 5. - _Gary W. Adamson_, Aug 06 2016
%e From _Feryal Alayont_, Dec 08 2023: (Start)
%e a(2) counts the edge covers of F(1,3) with a pendant attached at the vertex of degree 3. This is the graph:
%e * -- *
%e / | \
%e / | \
%e *---*---*
%e An edge cover is a subset of the edges where each vertex is an endpoint of at least one edge. We show a one-to-one correspondence between subsequences of [1,...,5] and edge covers. Label edges connecting the top left vertex to the bottom vertices with odd numbers, 1, 3, 5, left to right. Label bottom edges with 2 (left) and 4 (right). An odd number in the subsequence means that edge is not in the edge cover. An even number means that edge is in. All bottom vertices are covered because if an odd edge is missing, an even edge covers the vertex attached to that odd. The pendant edge must be in every cover, so that edge covers both top vertices. (End)
%t LinearRecurrence[{3, 2}, {1, 5}, 24] (* _Jean-François Alcover_, Sep 26 2017 *)
%t a[0]=1;a[1]=5;a[n_]:= a[n]= 3*a[n-1]+2*a[n-2];Table[a[n],{n,0,23}] (* _James C. McMahon_, Dec 17 2023 *)
%o (Magma) [Floor((3/2+Sqrt(17)/2)^n*(1/2+7*Sqrt(17)/34)+(1/2-7*Sqrt(17)/34)*(3/2-Sqrt(17)/2)^n)+1: n in [0..30]]; // _Vincenzo Librandi_, Jul 09 2011
%o (PARI) a(n)=([1,2;2,2]^n*[1,2]~*[1,2])[1,1] \\ _Charles R Greathouse IV_, Jul 10 2011
%o (Haskell)
%o a007483 n = a007483_list !! n
%o a007483_list = 1 : 5 : zipWith (+)
%o (map (* 3) $ tail a007483_list) (map (* 2) a007483_list)
%o -- _Reinhard Zumkeller_, Nov 02 2015
%o (Sage)
%o @CachedFunction
%o def a(n): return 5^n if (n<2) else 3*a(n-1) + 2*a(n-2)
%o [a(n) for n in (0..40)] # _G. C. Greubel_, Jun 28 2021
%Y Cf. A007482, A072272.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Sep 19 1994
%E Definition simplified by _N. J. A. Sloane_, Aug 25 2014
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