login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A081567 Second binomial transform of F(n+1). 25
1, 3, 10, 35, 125, 450, 1625, 5875, 21250, 76875, 278125, 1006250, 3640625, 13171875, 47656250, 172421875, 623828125, 2257031250, 8166015625, 29544921875, 106894531250, 386748046875, 1399267578125, 5062597656250 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of F(2n-1), index shifted by 1, where F is A000045. - corrected by Richard R. Forberg, Aug 12 2013

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.

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,....,2n+1, s(0) = 3, s(2n+1) = 4.

a(n+1) gives the number of periodic multiplex juggling sequences of length n with base state <2>. - Steve Butler, Jan 21 2008

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

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

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

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

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

The limiting ratio a(n)/a(n-1) is 1 + phi^2. - Bob Selcoe, Mar 17 2014

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

REFERENCES

D. Chmiela, K. Kaczmarek and R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers (submitted to Fibonacci Quart. 2012, apparently unpublished Jan 2014).

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pages 96-100.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, arXiv:0801.2597 [math.CO], 2008.

P. E. Harris, E. Insko, L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055 [math.RT], 2013.

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8

Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.

D. Nacin, The Minimal Non-Koszul A(Gamma), arXiv preprint arXiv:1204.1534 [math.QA], 2012. - From N. J. A. Sloane, Oct 05 2012

R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042

Index entries for linear recurrences with constant coefficients, signature (5,-5).

FORMULA

a(n) = 5*a(n-1) - 5*a(n-2), a(0)=1, a(1)=3.

a(n) = (1/2 - sqrt(5)/10) * (5/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2) * (sqrt(5)/2 + 5/2)^n.

G.f.: (1-2*x)/(1-5*x+5*x^2).

a(n-1) = Sum_{k=1..n} binomial(n, k)*F(k)^2. - Benoit Cloitre, Oct 26 2003

a(n) = A090041(n)/2^n. - Paul Barry, Mar 23 2004

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

From Paul Barry, Nov 15 2005: (Start)

a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, j)*binomial(j+k, 2k);

a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, k+j)*binomial(k, k-j)2^(n-k-j);

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)

a(n) = Sum_{k=0..n} A094441(n,k)*2^k. - Philippe Deléham, Dec 14 2009

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

a(n) = A030191(n) - 2*A030191(n-1). - R. J. Mathar, Jul 19 2012

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

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

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

a(n) = 2*a(n-1) + A039717(n) (not proved). - Benito van der Zander, Nov 20 2015

a(n) = A189315(n+1) / 10. - Tom Copeland, Dec 08 2015

EXAMPLE

a(4)=125: 35*(3 + (35 mod 10 - 10 mod 3)/(10-3)) = 35*(3 + 4/7) = 125. - Bob Selcoe, Mar 17 2014

MAPLE

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

MATHEMATICA

Table[MatrixPower[{{2, 1}, {1, 3}}, n][[2]][[2]], {n, 0, 44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)

LinearRecurrence[{5, -5}, {1, 3}, 30] (* Vincenzo Librandi, Feb 27 2012 *)

PROG

(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

(Python)

def a(n, adict={0:1, 1:3}):

.if n in adict:

..return adict[n]

.adict[n]=5*a(n-1) - 5*a(n-2)

.return adict[n] # David Nacin, Mar 04 2012

(PARI) Vec((1-2*x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Mar 18 2014

CROSSREFS

Cf. A000045.

a(n) = 5*A052936(n-1), n>1.

Row sums of A114164.

a(n) = A111776(n, n). - Abdullahi Umar, Sep 14 2008

Cf. A033191, A147748 and A178381, A081568 (binomial transform), A086351 (INVERT transform), A007051 (INVERTi transform).

Cf. A189315.

Sequence in context: A187925 A094855 A243871 * A224509 A026026 A047037

Adjacent sequences:  A081564 A081565 A081566 * A081568 A081569 A081570

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 22 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 20 15:43 EDT 2019. Contains 321345 sequences. (Running on oeis4.)