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

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

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 December 12 15:01 EST 2017. Contains 295939 sequences.