A107769 a(n) = (A001333(n+1) - 2*A005409(floor((n+3)/2)) - 1) / 4.

0, 1, 2, 8, 19, 54, 130, 334, 806, 1995, 4816, 11746, 28357, 68748, 165972, 401388, 969036, 2341141, 5652014, 13649228, 32952151, 79563330, 192082870, 463752730, 1119598130, 2703006111, 6525634012, 15754412038, 38034515209, 91823775384, 221682203880, 535188986904, 1292060510616, 3119311948585

Offset: 0

Author

Emeric Deutsch, Jun 12 2005

Keywords

• nonn
• easy

Comments

a(n) is the number of free polyominoes of width 2 and height n+1 which have no symmetry, i.e., rotations by 180 degrees, flips along the short or long axis generate a different free polyomino. The three elements t, g+ and g- of sequences by Tasi et al. represent a domino in the short cross-section where either both, only the "upper" or only the "lower" square of the domino is occupied. E.g., a(3) = 8 represents 3 5-ominoes of shape the 2x4, 3 6-ominoes of shape 2x4, and 2 7-ominoes of shape 2x4. - R. J. Mathar, Jun 17 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63, eq 25).
• Index entries for linear recurrences with constant coefficients, signature (3,1,-7,3,-1,1,1).

Crossrefs

Cf. A000129, A001333, A002203, A005409, A126877, A135153.

Programs

Mathematica

Table[(LucasL[n+2, 2] -4*Fibonacci[Floor[n/2]+2, 2] +2)/8, {n,0,40}] (* G. C. Greubel, May 24 2021 *)

Sage

[(lucas_number2(n+2,2,-1) -4*lucas_number1(2+(n//2),2,-1) +2)/8 for n in (0..40)] # G. C. Greubel, May 24 2021

Formula

4*a(n) = Pell(n+3) - Pell(n+2) - 2*Pell(floor((n+4)/2)) + 1, with Pell(n) = A000129(n). - Ralf Stephan, Jun 02 2007

G.f.: x*(1-x+x^2)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)). - Colin Barker, Apr 08 2013

4*a(n) = A001333(n+2) -2*A135153(n+4) +1. - R. J. Mathar, Jun 17 2020

From G. C. Greubel, May 24 2021: (Start)

a(n) = (1/4)*(A001333(n+2) - 2*A000129(floor(n/2)+2) + 1).

a(n) = (1/8)*(A002203(n+2) - 4*A000129(floor(n/2)+2) + 2). (End)

Extensions

Entry revised by N. J. A. Sloane, Jul 29 2011