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

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003480 a(n) = 4a(n-1) - 2a(n-2) (n >= 3).
(Formerly M1763)
32
1, 2, 7, 24, 82, 280, 956, 3264, 11144, 38048, 129904, 443520, 1514272, 5170048, 17651648, 60266496, 205762688, 702517760, 2398545664, 8189147136, 27959497216, 95459694592, 325919783936, 1112759746560, 3799199418368 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Gives the number of L-convex polyominoes with n cells, that is convex polyominoes where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientation of the L). - Simone Rinaldi (rinaldi(AT)unisi.it), Feb 19 2007

Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 2) is "size of raises in pot-limit poker, one blind, maximum raising".

Dimensions of the graded components of the Hopf algebra of noncommutative multi-symmetric functions of level 2. For level r, the sequence would be the INVERT transform of binomial(n+r-1,n). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

The sum of the numbers in the n-th row of the summatory Pascal triangle (A059576). - Ron R. King, Jan 22 2009

(1 + 2x + 7x^2 + 24x^3 + ...) = 1 / (1 - 2x - 3x^2 - 4x^3 - ...). - Gary W. Adamson, Jul 27 2009

Let M = a triangle with the odd indexed Fibonacci numbers (1, 2, 5, 13,...) in every column, with the leftmost column shifted upwards one row. A003480 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence. The analogous operation using the even indexed Fibonacci numbers generates A001835 starting with offset 1. - Gary W. Adamson, Jul 27 2010

a(n) is the number of generalized compositions of n when there are i+1 different types of the part i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t+3*t^2+4*t^3+...)),

  an o.g.f. for A003480, then

  A001003(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. - Tom Copeland, Sep 06 2011

Excluding initial 1, a(n) is the 2nd sub-diagonal of A228405. - Richard R. Forberg, Sep 02 2013

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

D. Battaglino, J. M. Fedou, S. Rinaldi and S. Socci, The number of k-parallelogram polyominoes, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1143-1154.

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.

E. Duchi, S. Rinaldi and G. Schaeffer, The number of Z-convex polyominoes

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 418

J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

Index entries for sequences related to poker

Index to sequences with linear recurrences with constant coefficients, signature(4,-2).

FORMULA

a(n) = n*a(1) + (n-1)*a(2) + ...3*a(n-2) + 2*a(n-1). - Amarnath Murthy, Aug 17 2002

G.f.: (1-x)^2/(1-4*x+2*x^2).

G.f.: 1/( 1 - sum(k>=1, (k+1)*x^k ) ).

a(n+1)*a(n+1) - a(n+2)*a(n) = 2^n, n > 0. - Douglas Rogers, Jul 12 2004

For n>0, a(n)=((2+sqrt(2))^(n+1)-(2-sqrt(2))^(n+1))/(4*sqrt(2)). - Rolf Pleisch, Aug 03 2009

If the leading 1 is removed, 2, 7, 24, ... is the binomial transform of 2, 5, 12, 29 ... which is A000129 without its first 2 terms, and the second binomial transform of 2, 3, 4, 6, .. which is A029744, again without its leading 1. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

a(n)=Sum((1+p_1)(1+p_2)...)(1+p_m)), summation being over all compositions (p_1, p_2, ..., p_m) of n. Example: a(3)=24; indeed, the compositions of 3 are (1,1,1), (1,2),(2,1), (3) and we have 2*2*2+2*3+3*2+4=24. - Emeric Deutsch, Oct 17 2010

a(n) = sum(k>=0, C(n+2*k-1,n) / 2^(k+1)). - Vaclav Kotesovec, Dec 31 2013

MAPLE

A003480:=(z-1)**2/(1-4*z+2*z**2); # Simon Plouffe in his 1992 dissertation

INVERT([seq(n+1, n=1..20)]); # Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

MATHEMATICA

a[0]=1; a[1]=2; a[2]=7; a[n_]:=a[n]=4*a[n-1] - 2*a[n-2]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 22 2011 *)

Join[{1}, LinearRecurrence[{4, -2}, {2, 7}, 40]] (* Harvey P. Dale, Oct 23 2011 *)

PROG

(PARI) a(n)=polcoeff((1-x)^2/(1-4*x+2*x^2)+x*O(x^n), n)

(PARI) a(n)=local(x); if(n<1, n==0, x=(2+quadgen(8))^n; imag(x)+real(x)/2)

(Haskell)

a003480 n = a003480_list !! n

a003480_list = 1 : 2 : 7 : (tail $ zipWith (-)

   (tail $ map (* 4) a003480_list) (map (* 2) a003480_list))

-- Reinhard Zumkeller, Jan 16 2012, Oct 03 2011

CROSSREFS

Row sums of A059576. Cf. A007052, A126764.

Equals (1/2) A007070, n>0.

Cf. A001835, A006012, A145839, A145840, A145841.

Sequence in context: A027128 A099463 A021000 * A020727 A088854 A000777

Adjacent sequences:  A003477 A003478 A003479 * A003481 A003482 A003483

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified September 14 21:54 EDT 2014. Contains 246769 sequences.