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.
%I A107769 #31 May 14 2024 07:25:11
%S A107769 0,1,2,8,19,54,130,334,806,1995,4816,11746,28357,68748,165972,401388,
%T A107769 969036,2341141,5652014,13649228,32952151,79563330,192082870,
%U A107769 463752730,1119598130,2703006111,6525634012,15754412038,38034515209,91823775384,221682203880,535188986904,1292060510616,3119311948585
%N A107769 a(n) = (A001333(n+1) - 2*A005409(floor((n+3)/2)) - 1) / 4.
%C A107769 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
%H A107769 G. C. Greubel, <a href="/A107769/b107769.txt">Table of n, a(n) for n = 0..1000</a>
%H A107769 Gy. Tasi and F. Mizukami, <a href="https://doi.org/10.1023/A:1019163812482">Quantum algebraic-combinatoric study of the conformational properties of n-alkanes</a>, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63, eq 25).
%H A107769 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-7,3,-1,1,1).
%F A107769 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
%F A107769 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
%F A107769 4*a(n) = A001333(n+2) -2*A135153(n+4) +1. - _R. J. Mathar_, Jun 17 2020
%F A107769 From _G. C. Greubel_, May 24 2021: (Start)
%F A107769 a(n) = (1/4)*(A001333(n+2) - 2*A000129(floor(n/2)+2) + 1).
%F A107769 a(n) = (1/8)*(A002203(n+2) - 4*A000129(floor(n/2)+2) + 2). (End)
%t A107769 Table[(LucasL[n+2, 2] -4*Fibonacci[Floor[n/2]+2, 2] +2)/8, {n,0,40}] (* _G. C. Greubel_, May 24 2021 *)
%o A107769 (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
%Y A107769 Cf. A000129, A001333, A002203, A005409, A126877, A135153.
%K A107769 nonn,easy
%O A107769 0,3
%A A107769 _Emeric Deutsch_, Jun 12 2005
%E A107769 Entry revised by _N. J. A. Sloane_, Jul 29 2011