A135153 -id:A135153 - cp's OEIS frontend

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

Emeric Deutsch, Jun 12 2005

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

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).

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

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

[(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

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)

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

A152113 A001333 with terms repeated.

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 17, 17, 41, 41, 99, 99, 239, 239, 577, 577, 1393, 1393, 3363, 3363, 8119, 8119, 19601, 19601, 47321, 47321, 114243, 114243, 275807, 275807, 665857, 665857, 1607521, 1607521, 3880899, 3880899, 9369319, 9369319, 22619537, 22619537, 54608393

Offset: 1

N. J. A. Sloane, Sep 21 2009

Suggested by an email message from Hugo van der Sanden, Mar 23 2009, who says: Consider the partitions of a 2 X n rectangle into connected pieces consisting of unit squares cut along lattice lines. Then a(n) is the number of distinct pieces with rotational symmetry that extend to opposite corners.

a(n+2) is the number of palindromic words of length n on a 3-letter alphabet {a,b,c} which do not contain the "ab" subword. See A001906 for the words of length n on a 3-letter alphabet without "ab" subword but not necessarily palindromic. Example length 1: "a" or "b" or "c". Example length 2: "aa", "bb", "cc". Example length 3: There are 9 palindromic words but "aba" and "bab" are not admitted and only 7 remain. - R. J. Mathar, Jul 10 2019

			The pieces illustrating a(3) = 3 are:
 AAA BB. .CC
 AAA .BB CC.

Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A001333, A078469, A152124.

From Colin Barker, Jul 14 2013: (Start)
a(n) = 2*a(n-2) + a(n-4).
G.f.: -x*(x+1)*(x^2+1) / (x^4+2*x^2-1). (End)
a(n+1) = A135153(n) + A135153(n+2). - R. J. Mathar, Jul 10 2019

A238375 Row sums of triangle in A152719.

Original entry on oeis.org

1, 2, 4, 6, 11, 16, 28, 40, 69, 98, 168, 238, 407, 576, 984, 1392, 2377, 3362, 5740, 8118, 13859, 19600, 33460, 47320, 80781, 114242, 195024, 275806, 470831, 665856, 1136688, 1607520, 2744209, 3880898, 6625108, 9369318, 15994427, 22619536, 38613964, 54608392

Offset: 0

Philippe Deléham, Feb 25 2014

			Triangle A152719 and row sums:
  1;  ............................. sum =  1
  1, 1;  .......................... sum =  2
  1, 2, 1;  ....................... sum =  4
  1, 2, 2,  1;  ................... sum =  6
  1, 2, 5,  2,  1;  ............... sum = 11
  1, 2, 5,  5,  2, 1;  ............ sum = 16
  1, 2, 5, 12,  5, 2, 1;  ......... sum = 28
  1, 2, 5, 12, 12, 5, 2, 1;  ...... sum = 40

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,1,-1).

Cf. A000129, A002203, A005409, A048739, A135153 (first differences), A152719.

Table[Sum[Fibonacci[1+Min[k, n-k], 2], {k,0,n}], {n,0,45}] (* G. C. Greubel, May 21 2021 *)

my(x='x+O('x^44)); Vec((1+x)/((1-2*x^2-x^4)*(1-x))) \\ Joerg Arndt, May 22 2021

def Pell(n): return n if (n<2) else 2*Pell(n-1) + Pell(n-2)
def a(n): return sum(Pell(1+min(k, n-k)) for k  in (0..n))
[a(n) for n in (0..45)] # G. C. Greubel, May 21 2021

a(n) = Sum_{k=0..n} A152719(n,k).
G.f.: (1+x)/((1-2*x^2-x^4)*(1-x)).
a(2*n) = A005409(n+2).
a(2*n+1) = 2*A048739(n).
a(n) = (-4 + 2*(1+(-1)^n)*Pell((n+4)/2) + (1-(-1)^n)*Q((n+3)/2))/4, where Pell(n) = A000129(n) and Q(n) = A002203(n). - G. C. Greubel, May 21 2021
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)+a(n-4)-a(n-5). - Wesley Ivan Hurt, May 22 2021

A193530 Expansion of (1 - 2x - 2x^2 + 3x^3 + x^5)/((1-x)(1-2x-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

F. Chapoton and N. J. A. Sloane, Jul 29 2011

This sequence was initially confused with A003120, but they are different sequences. The g.f. used here as the definition was found by Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009.

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).

Cf. A000129, A001333, A002203, A135153.

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

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

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 *)

@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

a(n) = 1 + A005409(floor((n+3)/2)) + A107769(n).
From G. C. Greubel, May 21 2021: (Start)
a(n) = (1 + A001333(n) + A135153(n+2))/4.
a(n) = (2 + Q(n) + 2*(1+(-1)^n)*Pell((n+2)/2) + 2*(1-(-1)^n)*Pell((n+1)/2))/8.
a(2*n) = (2 + Q(2*n) + 4*Pell(n+1))/8.
a(2*n+1) = (2 + Q(2*n+1) + 4*Pell(n+1))/8, where Pell(n) = A000129(n), and Q(n) = A002203. (End)

cp's OEIS Frontend

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

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A152113 A001333 with terms repeated.

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A238375 Row sums of triangle in A152719.

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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)).

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A193530 Expansion of (1 - 2x - 2x^2 + 3x^3 + x^5)/((1-x)(1-2x-x^2)(1-2*x^2-x^4)).