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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A055588 a(n) = 3a(n-1) - a(n-2) - 1 with a(0)=1, a(1)=2. 16
1, 2, 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369, 121394, 317812, 832041, 2178310, 5702888, 14930353, 39088170, 102334156, 267914297, 701408734, 1836311904, 4807526977, 12586269026, 32951280100, 86267571273 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of directed column-convex polyominoes with area n+2 and having two cells in the bottom row. - Emeric Deutsch, Jun 14 2001

a(n) = length of the list generated by the substitution: 3->3, 4->(3,4,6), 6->(3,4,6,6): {3, 4}, {3, 3, 4, 6}, {3, 3, 3, 4, 6, 3, 4, 6, 6}, {3, 3, 3, 3, 4, 6, 3, 4, 6, 6, 3, 3, 4, 6, 3, 4, 6, 6, 3, 4, 6, 6}, etc. - Wouter Meeussen, Nov 23, 2003

Equals row sums of triangle A144955. - Gary W. Adamson, Sep 27 2008

Equals the INVERT transform of A034943 and the INVERTi transform of A094790. - Gary W. Adamson,  Apr 01 2011

LINKS

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

E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

M. M. Mogbonju, I. A. Ogunleke, O. A. Ojo, Graphical Representation Of Conjugacy Classes In The Order-Preserving Full Transformation Semigroup, International Journal of Scientific Research and Engineering Studies (IJSRES), Volume 1 Issue 5, November 2014; ISSN: 2349-8862.

László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (1st line of Table 1 is 3*a(n-2)).

László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (1st line of Table 1 is a(n-2)).

Yan X Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

FORMULA

a(n) = (((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n)/sqrt(5) + 1.

a(n) = Sum_{m=0..n} A055587(n, m) = 1 + A001906(n).

G.f.: (1-2*x)/((1 - 3*x + x^2)*(1-x)).

From Paul Barry, Oct 07 2004: (Start)

a(n) = 4a(n-1) - 4a(n-2) + a(n-3);

a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2k)2^(n-3k). (End)

From Paul Barry, Oct 26 2004: (Start)

a(n) = Fibonacci(2n) + 1;

a(n) = Sum_{k=0..n} Fibonacci(2k+2)(2*0^(n-k)-1);

a(n) = A008346(2n). (End)

a(n) = Sum_{k=0..2n+1} ((-1)^(k+1))*fibonacci(k). - Michel Lagneau, Feb 03 2014

MAPLE

g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)+1), n=0..27); # Zerinvary Lajos, Mar 22 2009

MATHEMATICA

Join[{a=1, b=2}, Table[c=3*b-1*a-1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2011 *)

Table[Fibonacci[2 n] + 1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -4, 1}, {1, 2, 4}, 40] (* Vincenzo Librandi, Sep 30 2017 *)

PROG

(Sage) [lucas_number1(n, 3, 1)+1 for n in range(29)] # Zerinvary Lajos, Jul 06 2008

(MAGMA) [Fibonacci(2*n)+1: n in [0..30]]; // Vincenzo Librandi, Sep 30 2017

CROSSREFS

Cf. A055587, A001906. Partial sums of A001519.

Apart from first term, same as A052925.

Cf. A144955.

Cf. A034943, A094790.

Sequence in context: A115324 A196307 A107092 * A088456 A091561 A025265

Adjacent sequences:  A055585 A055586 A055587 * A055589 A055590 A055591

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 30 2000; Barry E. Williams, Jun 04 2000

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 20 12:58 EDT 2018. Contains 313917 sequences. (Running on oeis4.)