A335711
The number of free polyominoes of width 2 and height n.
Original entry on oeis.org
2, 6, 12, 30, 65, 158, 362, 875, 2064, 4984, 11914, 28764, 69155, 166956, 402372, 971413, 2343518, 5657754, 13654968, 32966010, 79577189, 192116330, 463786190, 1119678911, 2703086892, 6525829036, 15754607062, 38034986040, 91824246215, 221683340568, 535190123592
Offset: 2
a(2)=2, bounding box 2 X 2, counts the L-shaped 3-omino and the full block 4-omino.
a(3)=6, bounding box 2 X 3, counts three 4-ominoes, two 5-omioes, and the full 2 X 3 block 6-omino.
a(4)=12, bounding box 2 X 4, counts three 5-ominoes, six 6-ominoes, two 7-ominoes, and the full 2 X 4 block 8-omino.
A126877
Numbers of unstrained alkane staggered conformers (acyclic) of the unbranched type for point group C1. See Table 4 of the Cyvin et al. reference for precise definition.
Original entry on oeis.org
1, 2, 8, 19, 54, 130, 334, 804, 1984, 4777, 11615, 27957, 67472, 162276, 390546
Offset: 5
A193530
Expansion of (1 - 2*x - 2*x^2 + 3*x^3 + x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)).
Original entry on oeis.org
1, 1, 2, 3, 7, 13, 31, 66, 159, 363, 876, 2065, 4985, 11915, 28765, 69156, 166957, 402373, 971414, 2343519, 5657755, 13654969, 32966011, 79577190, 192116331, 463786191, 1119678912, 2703086893, 6525829037, 15754607063, 38034986041, 91824246216, 221683340569, 535190123593, 1292063254826
Offset: 0
- Vincenzo Librandi, 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).
- Index entries for linear recurrences with constant coefficients, signature (3,1,-7,3,-1,1,1).
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m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-2*x-2*x^2 +3*x^3+x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)) )); // Vincenzo Librandi, Aug 28 2016
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f:=n->if n mod 2 = 0 then (1/4)*(A001333(n-2)+A001333((n-2)/2)+A001333((n-4)/2)+1) else (1/4)*(A001333(n-2)+A001333((n-1)/2)+A001333((n-3)/2)+1); fi; # produces the sequence with a different offset
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LinearRecurrence[{3,1,-7,3,-1,1,1}, {1,1,2,3,7,13,31}, 40] (* Vincenzo Librandi, Aug 28 2016 *)
Table[(2 +LucasL[n, 2] +2*(1+(-1)^n)*Fibonacci[(n+2)/2, 2] + 2*(1-(-1)^n)*Fibonacci[(n+1)/2, 2])/8, {n, 0, 40}] (* G. C. Greubel, May 21 2021 *)
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@CachedFunction
def Pell(n): return n if (n<2) else 2*Pell(n-1) + Pell(n-2)
def A193530(n): return (1 + Pell(n+1) - Pell(n) + (1 + (-1)^n)*Pell((n+2)/2) + (1-(-1)^n)*Pell((n+1)/2) )/4
[A193530(n) for n in (0..40)] # G. C. Greubel, May 21 2021
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