A055588 a(n) = 3*a(n-1) - a(n-2) - 1 with a(0) = 1 and a(1) = 2.

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

Offset: 0

Author

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

Keywords

• nonn
• easy

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) is the 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.
• Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 20.
• Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• M. M. Mogbonju, I. A. Ogunleke, and 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(5) (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:1508.00318 [math.CO], 2015.
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A001906, A008346, A019827, A034943, A055587, A094790, A144955.

Partial sums of A001519.

Apart from the first term, same as A052925.

Programs

GAP

List([0..40], n-> Fibonacci(2*n)+1 ); # G. C. Greubel, Jun 06 2019

Magma

[Fibonacci(2*n)+1: n in [0..40]]; // Vincenzo Librandi, Sep 30 2017

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

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

PARI

vector(40, n, n--; fibonacci(2*n)+1) \\ G. C. Greubel, Jun 06 2019

Sage

[lucas_number1(n,3,1)+1 for n in range(40)] # Zerinvary Lajos, Jul 06 2008

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

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

From Paul Barry, Oct 26 2004: (Start)

a(n) = A001906(n) + 1.

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

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

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

E.g.f.: cosh(x) + sinh(x) + 2*exp(3*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, May 14 2024

Product_{n>=1} (1 - 1/a(n)) = sin(Pi/10) (A019827). - Amiram Eldar, Nov 28 2024