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%I #167 Apr 26 2023 10:32:05
%S 1,3,10,35,125,450,1625,5875,21250,76875,278125,1006250,3640625,
%T 13171875,47656250,172421875,623828125,2257031250,8166015625,
%U 29544921875,106894531250,386748046875,1399267578125,5062597656250
%N Second binomial transform of F(n+1).
%C Binomial transform of F(2*n-1), index shifted by 1, where F is A000045. - corrected by _Richard R. Forberg_, Aug 12 2013
%C Case k=2 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.
%C Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2*n+1, s(0) = 3, s(2*n+1) = 4.
%C a(n+1) gives the number of periodic multiplex juggling sequences of length n with base state <2>. - _Steve Butler_, Jan 21 2008
%C a(n) is also the number of idempotent order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - _Abdullahi Umar_, Sep 14 2008
%C Counts all paths of length (2*n+1), n>=0, starting at the initial node on the path graph P_9, see the Maple program. - _Johannes W. Meijer_, May 29 2010
%C Given the 3 X 3 matrix M = [1,1,1; 1,1,0; 1,1,3], a(n) = term (1,1) in M^(n+1). - _Gary W. Adamson_, Aug 06 2010
%C Number of nonisomorphic graded posets with 0 and 1 of rank n+2, with exactly 2 elements of each rank level between 0 and 1. Also the number of nonisomorphic graded posets with 0 of rank n+1, with exactly 2 elements of each rank level above 0. (This is by Stanley's definition of graded, that all maximal chains have the same length.) - _David Nacin_, Feb 26 2012
%C a(n) = 3^n a(n;1/3) = Sum_{k=0..n} C(n,k) * F(k-1) * (-1)^k * 3^(n-k), which also implies the Deleham formula given below and where a(n;d), n=0,1,...,d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also the papers of Witula et al.). - _Roman Witula_, Jul 12 2012
%C The limiting ratio a(n)/a(n-1) is 1 + phi^2. - _Bob Selcoe_, Mar 17 2014
%C a(n) counts closed walks on K_2 containing 3 loops on the index vertex and 2 loops on the other. Equivalently the (1,1) entry of A^n where the adjacency matrix of digraph is A=(3,1; 1,2). - _David Neil McGrath_, Nov 18 2014
%D R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pages 96-100.
%H Vincenzo Librandi, <a href="/A081567/b081567.txt">Table of n, a(n) for n = 0..1000</a>
%H Santiago Alzate, Oscar Correa, Rigoberto Flórez, <a href="https://arxiv.org/abs/2009.02639">Fibonacci identities from Jordan Identities</a>, arXiv:2009.02639 [math.NT], 2020.
%H Carolina Benedetti, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, and Anthony Simpson, <a href="https://arxiv.org/abs/2001.03219">Kostant's partition function and magic multiplex juggling sequences</a>, arXiv:2001.03219 [math.CO], 2020. See Table 1 p. 12.
%H S. Butler and R. Graham, <a href="http://arxiv.org/abs/0801.2597">Enumerating (multiplex) juggling sequences</a>, arXiv:0801.2597 [math.CO], 2008.
%H P. E. Harris, E. Insko, and L. K. Williams, <a href="http://arxiv.org/abs/1401.0055">The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula</a>, arXiv preprint arXiv:1401.0055 [math.RT], 2013.
%H Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, <a href="https://doi.org/10.1515/math-2017-0047">Binomials Transformation Formulae of Scaled Fibonacci Numbers</a>, Open Mathematics, 15(1) (2017), 477-485.
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.10.023">A. Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra 278, (2004), 342-359.
%H A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial results for semigroups of order-decreasing partial transformations</a>, J. Integer Seq. 7 (2004), #04.3.8.
%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Note on Cosine Power Sums</a> J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
%H D. Nacin, <a href="http://arxiv.org/abs/1204.1534">The Minimal Non-Koszul A(Gamma)</a>, arXiv preprint arXiv:1204.1534 [math.QA], 2012. - From _N. J. A. Sloane_, Oct 05 2012
%H Roman Witula and Damian Slota, <a href="http://dx.doi.org/10.2298/AADM0902310W">delta-Fibonacci numbers</a>, Appl. Anal. Discr. Math 3 (2009) 310-329, <a href="http://www.ams.org/mathscinet-getitem?mr=2555042">MR2555042</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).
%F a(n) = 5*a(n-1) - 5*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 3.
%F a(n) = (1/2 - sqrt(5)/10) * (5/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2) * (sqrt(5)/2 + 5/2)^n.
%F G.f.: (1 - 2*x)/(1 - 5*x + 5*x^2).
%F a(n-1) = Sum_{k=1..n} binomial(n, k)*F(k)^2. - _Benoit Cloitre_, Oct 26 2003
%F a(n) = A090041(n)/2^n. - _Paul Barry_, Mar 23 2004
%F The sequence 0, 1, 3, 10, ... with a(n) = (5/2 - sqrt(5)/2)^n/5 + (5/2 + sqrt(5)/2)^n/5 - 2(0)^n/5 is the binomial transform of F(n)^2 (A007598). - _Paul Barry_, Apr 27 2004
%F From _Paul Barry_, Nov 15 2005: (Start)
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, j)*binomial(j+k, 2k);
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, k+j)*binomial(k, k-j)2^(n-k-j);
%F a(n) = Sum_{k=0..n} Sum_{j=0..n-k} binomial(n+k-j, n-k-j)*binomial(k, j)(-1)^j*2^(n-k-j). (End)
%F a(n) = A111776(n, n). - _Abdullahi Umar_, Sep 14 2008
%F a(n) = Sum_{k=0..n} A094441(n,k)*2^k. - _Philippe Deléham_, Dec 14 2009
%F a(n+1) = Sum_{k=-floor(n/5)..floor(n/5)} ((-1)^k*binomial(2*n, n+5*k)/2). -_Mircea Merca_, Jan 28 2012
%F a(n) = A030191(n) - 2*A030191(n-1). - _R. J. Mathar_, Jul 19 2012
%F G.f.: Q(0,u)/x - 1/x, where u=x/(1-2*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k+1 + u)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013
%F For n>=3: a(n) = a(n-1)*(3+(a(n-1) mod a(n-2) - a(n-2) mod a(n-3))/(a(n-2) - a(n-3))). - _Bob Selcoe_, Mar 17 2014
%F a(n) = sqrt(5)^(n-1)*(3*S(n-1, sqrt(5)) - sqrt(5)*S(n-2, sqrt(5))) with Chebyshev's S-polynomials (see A049310), where S(-1, x) = 0 and S(-2, x) = -1. This is the (1,1) entry of A^n with the matrix A=(3,1;1,2). See the comment by _David Neil McGrath_, Nov 18 2014. - _Wolfdieter Lang_, Dec 04 2014
%F Conjecture: a(n) = 2*a(n-1) + A039717(n). - _Benito van der Zander_, Nov 20 2015
%F a(n) = A189315(n+1) / 10. - _Tom Copeland_, Dec 08 2015
%F a(n) = A093129(n) + A030191(n-1). - _Gary W. Adamson_, Apr 24 2023
%e a(4)=125: 35*(3 + (35 mod 10 - 10 mod 3)/(10-3)) = 35*(3 + 4/7) = 125. - _Bob Selcoe_, Mar 17 2014
%p with(GraphTheory):G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=23; n2:=nmax*2+2: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..9); od: seq(a(2*n+1),n=0..nmax); # _Johannes W. Meijer_, May 29 2010
%t Table[MatrixPower[{{2,1},{1,3}},n][[2]][[2]],{n,0,44}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t LinearRecurrence[{5,-5},{1,3},30] (* _Vincenzo Librandi_, Feb 27 2012 *)
%o (Magma) I:=[1, 3]; [n le 2 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 27 2012
%o (Python)
%o def a(n, adict={0:1, 1:3}):
%o if n in adict:
%o return adict[n]
%o adict[n]=5*a(n-1) - 5*a(n-2)
%o return adict[n] # _David Nacin_, Mar 04 2012
%o (PARI) Vec((1-2*x)/(1-5*x+5*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 18 2014
%Y a(n) = 5*A052936(n-1), n > 1.
%Y Row sums of A114164.
%Y Cf. A000045, A007051 (INVERTi transform), A033191, A081568 (binomial transform), A086351 (INVERT transform), A111776, A147748, A178381, A189315.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 22 2003
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