cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A003480 a(0) = 1, a(1) = 2, for n > 1, a(n) = 4*a(n-1) - 2*a(n-2).

Original entry on oeis.org

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, 12971278180352, 44286713884672, 151204299177984
Offset: 0

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Author

N. J. A. Sloane

Keywords

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 be 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->oo} 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 the initial 1, a(n) is the 2nd subdiagonal of A228405. - Richard R. Forberg, Sep 02 2013

References

  • G. Castiglione and A. Restivo, L-convex polyominoes: a survey, Chapter 2 of K. G. Subranian et al., eds., Formal Models, Languages and Applications, World Scientific, 2015.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row sums of A059576 and of A181289. Second differences of A007070.
Cf. A007052, A126764.
Cf. A000129, A001003, A001835, A006012, A029744, A145839, A145840, A145841, A228405.
Column k=2 of A261780.

Programs

  • 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
  • Maple
    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 *)
  • 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)

Formula

a(n) = (n+1)*a(0) + n*a(1) + ... + 3*a(n-2) + 2*a(n-1). - Amarnath Murthy, Aug 17 2002
G.f.: (1-x)^2/(1-4*x+2*x^2). - Simon Plouffe in his 1992 dissertation
a(n) = A007070(n)/2, n > 0.
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. - D. G. 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} binomial(n+2*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013
E.g.f.: (1 + exp(2*x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)))/2. - Stefano Spezia, May 20 2024

A126765 a(n) = number of L-convex polyominoes inscribed in an (n+1) X (n+1) box.

Original entry on oeis.org

1, 5, 42, 402, 4070, 42510, 452900, 4891988, 53376966, 586921790, 6493225772, 72192371100, 805935279708, 9028253155628, 101433497725320, 1142504966609512, 12897113121607750, 145870996300613406, 1652690392388658012, 18753389068268792780, 213091273336786301940
Offset: 0

Views

Author

N. J. A. Sloane, Feb 23 2007

Keywords

Comments

See A126764 for definition of L-convex.

References

  • 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.

Links

Crossrefs

Cf. A000105, A126764.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 4*n+1, (12*(4*n-1)*(2*n-1)*(n-1)^2*
      a(n-1)-4*(n-2)*(2*n-3)*n*(4*n+1)*a(n-2))/((4*n-3)*(2*n-1)*n*(n-1)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 23 2023
  • Mathematica
    CoefficientList[Series[(1/2)Sqrt[(2+5x-2x^2+(2-x)Sqrt[1-12x+4x^2])/ (1-12x+4x^2)],{x,0,20}],x] (* Harvey P. Dale, Jun 14 2011 *)

Formula

G.f.: (1/2) * sqrt( (2+5*x-2*x^2+(2-x)*sqrt(1-12*x+4*x^2) )/ (1-12*x+4*x^2) ).
a(n) ~ 2^(n-9/4) * (3+2*sqrt(2))^(n+1) / sqrt((1+sqrt(2))*Pi*n). - Vaclav Kotesovec, Feb 16 2015

A360055 Number of L-connected free polyominoes with n cells (see comments for definition).

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 24, 48, 83, 155, 265, 472, 793, 1356, 2235, 3700, 5977, 9636, 15262, 24068, 37439, 57920, 88628, 134838, 203264, 304688, 453126, 670238, 984556, 1438878, 2089996, 3021162, 4343229, 6215668, 8851151, 12550775, 17716075, 24907961, 34873541, 48644474
Offset: 0

Views

Author

Allan C. Wechsler and John Mason, Jan 23 2023

Keywords

Comments

By "L-connected" we mean that every pair of cells in the polyomino can be connected by a chain of cells in the polyomino, consisting of a vertical chain and a horizontal chain that share one of their end cells.
We consider the empty polyomino, or "nullomino" to be L-connected because there are no pairs of cells, so the universal quantifier in the definition is trivially satisfied.
The entries through n = 6 were hand-counted by Allan C. Wechsler, and John Mason confirmed those entries and extended the data through n = 18.

Examples

			For n = 4 the 4 solutions are the straight tetromino, the L tetromino, the T tetromino, and the square tetromino. The skew or S tetromino is not L-connected, because there is no qualifying path between the two most distant cells.

Links

Crossrefs

Main entry at A000105; see the cross-references there.
Cf. A126764 (the fixed case).

Programs

Extensions

Terms a(19) and beyond from Andrew Howroyd, Jan 24 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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