A054886 -id:A054886 - cp's OEIS frontend

A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.

2, 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 0

Bill Jones (b92057(AT)yahoo.com), May 18 2006

Essentially the same as A006355, A047992, A054886, A055389, A068922, A078642, A090991. - Philippe Deléham, Sep 20 2006 and Georg Fischer, Oct 07 2018
Also the number of matchings in the (n-2)-pan graph. - Eric W. Weisstein, Jun 30 2016
Also the number of maximal independent vertex sets (and minimal vertex covers) in the (n-1)-ladder graph. - Eric W. Weisstein, Jun 30 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Ladder Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Vertex Cover
Eric Weisstein's World of Mathematics, Pan Graph
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A003714, A212804.

List([0..40],n->2*Fibonacci(n-1)); # Muniru A Asiru, Oct 07 2018

[Lucas(n) - Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Sep 14 2014

with(combinat): seq(2*fibonacci(n-1),n=0..40); # Muniru A Asiru, Oct 07 2018
a := n -> -2*I^n*ChebyshevU(n-2, -I/2):
seq(simplify(a(n)), n = 0..39);  # Peter Luschny, Dec 03 2023

LinearRecurrence[{1, 1}, {2, 0}, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 05 2011 *)
Table[LucasL[n] - Fibonacci[n], {n, 0, 40}] (* Vincenzo Librandi, Sep 14 2014 *)
Table[2 Fibonacci[n - 1], {n, 0, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
2 Fibonacci[Range[0, 20] - 1] (* Eric W. Weisstein, Jun 30 2017 *)
Subtract @@@ (Through[{LucasL, Fibonacci}[#]] & /@ Range[0, 20]) (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(2 (-1 + x))/(-1 + x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 30 2017 *)

a(n)=fibonacci(n-1)<<1 \\ Charles R Greathouse IV, Jun 05 2011

From Philippe Deléham, Sep 20 2006: (Start)
a(0)=2, a(1)=0; for n > 1, a(n) = a(n-1) + a(n-2).
G.f. (2 - 2*x)/(1 - x - x^2).
a(0)=2 and a(n) = 2*A000045(n-1) for n > 0. (End)
a(n) = A006355(n) + 0^n. - M. F. Hasler, Nov 05 2014
a(n) = Lucas(n-2) + Fibonacci(n-2). - Bruno Berselli, May 27 2015
a(n) = 3*Fibonacci(n-2) + Fibonacci(n-5). - Bruno Berselli, Feb 20 2017
a(n) = 2*A212804(n). - Bruno Berselli, Feb 21 2017
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Apr 18 2022

More terms from Philippe Deléham, Sep 20 2006

Corrected by T. D. Noe, Nov 01 2006

A063759 Spherical growth series for modular group.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152

Offset: 0

N. J. A. Sloane, Aug 14 2001

Also number of sequences S of length n with entries in {1,..,q} where q = 3, satisfying the condition that adjacent terms differ in absolute value by exactly 1, see examples. - W. Edwin Clark, Oct 17 2008

			For n = 2 the a(2) = 4 sequences are (1,2),(2,1),(2,3),(3,2). - _W. Edwin Clark_, Oct 17 2008
From _Joerg Arndt_, Nov 23 2012: (Start)
There are a(6) = 16 such words of length 6:
[ 1]   [ 1 2 1 2 1 2 ]
[ 2]   [ 1 2 1 2 3 2 ]
[ 3]   [ 1 2 3 2 1 2 ]
[ 4]   [ 1 2 3 2 3 2 ]
[ 5]   [ 2 1 2 1 2 1 ]
[ 6]   [ 2 1 2 1 2 3 ]
[ 7]   [ 2 1 2 3 2 1 ]
[ 8]   [ 2 1 2 3 2 3 ]
[ 9]   [ 2 3 2 1 2 1 ]
[10]   [ 2 3 2 1 2 3 ]
[11]   [ 2 3 2 3 2 1 ]
[12]   [ 2 3 2 3 2 3 ]
[13]   [ 3 2 1 2 1 2 ]
[14]   [ 3 2 1 2 3 2 ]
[15]   [ 3 2 3 2 1 2 ]
[16]   [ 3 2 3 2 3 2 ]
(End)

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for sequences related to modular groups
Index entries for linear recurrences with constant coefficients, signature (0,2)

Cf. A054886, A029744.

The sequence (ternary strings) seems to be related to A029744 and A090989.

import Data.List (transpose)
a063759 n = a063759_list !! n
a063759_list = concat $ transpose [a151821_list, a007283_list]
-- Reinhard Zumkeller, Dec 16 2013

CoefficientList[Series[(1+3*x+2*x^2)/(1-2*x^2),{x,0,40}],x](* Jean-François Alcover, Mar 21 2011 *)
Join[{1},Transpose[NestList[{Last[#],2First[#]}&,{3,4},40]][[1]]] (* Harvey P. Dale, Oct 22 2011 *)

a(n)=([0,1; 2,0]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Feb 09 2017

G.f.: (1+3*x+2*x^2)/(1-2*x^2).
a(n) = 2*a(n-2), n>2. - Harvey P. Dale, Oct 22 2011
a(2*n) = A151821(n+1); a(2*n+1) = A007283(n). - Reinhard Zumkeller, Dec 16 2013

Information from A145751 included by Joerg Arndt, Dec 03 2012

A090991 Number of meaningful differential operations of the n-th order on the space R^6.

Original entry on oeis.org

6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 1

Branko Malesevic, Feb 29 2004

Apparently a(n) = A054886(n+2) for n=1..1000. - Georg Fischer, Oct 06 2018

Indranil Ghosh, Table of n, a(n) for n = 1..4772
Tanya Khovanova, Recursive Sequences
Branko J. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
Branko J. Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A054886, A055389, A068922, A090989-A090995.

Essentially the same as A006355, A047992 and A078642.

a:=[6,10];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # Muniru A Asiru, Oct 06 2018

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(3+2*x)/(1-x-x^2) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 6; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:

CoefficientList[Series[2*(3+2z)/(1-z-z^2), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

my(x='x+O('x^40)); Vec(2*x*(3+2*x)/(1-x-x^2)) \\ G. C. Greubel, Feb 02 2019

(2*(3+2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 02 2019

a(k+4) = 3*a(k+2) - a(k).
a(k) = 2*Fibonacci(k+3).
From Philippe Deléham, Nov 19 2008: (Start)
a(n) = a(n-1) + a(n-2), n>2, where a(1)=6, a(2)=10.
G.f.: 2*x*(3+2*x)/(1-x-x^2). (End)
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 4. - Stefano Spezia, Apr 18 2022

A054889 Layer counting sequence for hyperbolic tessellation by regular pentagons of angle 2*Pi/5.

Original entry on oeis.org

1, 5, 20, 70, 245, 860, 3015, 10570, 37060, 129935, 455560, 1597225, 5599980, 19633910, 68837825, 241350100, 846189875, 2966799290, 10401800220, 36469419475, 127864266640, 448300820765, 1571773187140, 5510743762630

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

The layer sequence is the sequence of the cardinalities of the layers accumulating around a (finite-sided) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.

Georg Fischer, Table of n, a(n) for n = 1..500
Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
Index entries for linear recurrences with constant coefficients, signature (3,1,3,-1).

Cf. A054886, A054887, A054888, A054890.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4) )); // G. C. Greubel, Feb 08 2023

LinearRecurrence[{3,1,3,-1},{1,5,20,70,245},40] (* Georg Fischer, Apr 13 2020 *)

def A054889_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4) ).list()
a=A054889_list(40); a[1:] # G. C. Greubel, Feb 08 2023

G.f.: x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4).

a(21) inserted by Georg Fischer, Apr 13 2020

A054890 Layer counting sequence for hyperbolic tessellation by regular heptagons of angle Pi/3.

Original entry on oeis.org

1, 7, 42, 245, 1428, 8323, 48510, 282737, 1647912, 9604735, 55980498, 326278253, 1901689020, 11083855867, 64601446182, 376524821225, 2194547481168, 12790760065783, 74550012913530, 434509317415397

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

nonn

The layer sequence is the sequence of the cardinalities of the layers accumulating around a (finite-sided) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.

Michael De Vlieger, Table of n, a(n) for n = 1..1307
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A001109, A054886, A054887, A054888, A054889.

[n eq 1 select 1 else 7*Evaluate(ChebyshevSecond(n-1), 3): n in [1..40]]; // G. C. Greubel, Feb 08 2023

Rest@CoefficientList[Series[x*(1+x+x^2)/(1-6*x+x^2), {x,0,30}], x] (* Michael De Vlieger, Dec 29 2020 *)
LinearRecurrence[{6,-1},{1,7,42},20] (* Harvey P. Dale, Jun 06 2021 *)

[7*chebyshev_U(n-2, 3) + int(n==1) for n in range(1,41)] # G. C. Greubel, Feb 08 2023

a(n) = 7*A001109(n-1) + [n=1].
G.f.: x*(1+x+x^2)/(1-6*x+x^2).
a(n) = A001109(n) + A001109(n-1) + A001109(n-2), n>1. - Ralf Stephan, Apr 26 2003

A350529 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 1..k such that no iterated difference is zero, n, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 6, 2, 0, 0, 1, 5, 12, 10, 2, 0, 0, 1, 6, 20, 32, 16, 2, 0, 0, 1, 7, 30, 72, 86, 26, 2, 0, 0, 1, 8, 42, 138, 256, 232, 42, 2, 0, 0, 1, 9, 56, 234, 624, 906, 622, 68, 2, 0, 0

Offset: 0

Pontus von Brömssen, Jan 03 2022

For fixed n, T(n,k) is a quasi-polynomial of degree n in k. For example, T(4,k) = k^4 - (116/27)*k^3 + (25/3)*k^2 + b(k)*k + c(k), where b and c are periodic with period 6.

			  n\k|  0  1  2   3     4      5       6        7         8         9         10
  ---+--------------------------------------------------------------------------
   0 |  1  1  1   1     1      1       1        1         1         1          1
   1 |  0  1  2   3     4      5       6        7         8         9         10
   2 |  0  0  2   6    12     20      30       42        56        72         90
   3 |  0  0  2  10    32     72     138      234       368       544        770
   4 |  0  0  2  16    86    256     624     1278      2370      4030       6462
   5 |  0  0  2  26   232    906    2790     6900     15096     29536      53678
   6 |  0  0  2  42   622   3180   12366    36964     95494    215146     443464
   7 |  0  0  2  68  1662  11116   54572   197294    601986   1562274    3652850
   8 |  0  0  2 110  4426  38754  240278  1051298   3788268  11325490   30041458
   9 |  0  0  2 178 11774 134902 1056546  5595236  23814458  82024662  246853482
  10 |  0  0  2 288 31316 469306 4643300 29762654 149631992 593798912 2027577296
For n = 4 and k = 3, the following T(4,3) = 16 sequences are counted: 1212, 1213, 1312, 1313, 1323, 2121, 2131, 2132, 2312, 2313, 2323, 3121, 3131, 3132, 3231, 3232.

Pontus von Brömssen, Antidiagonals n = 0..20, flattened

Cf. A200154, A350364, A350530.
Rows: A000012 (n=0), A001477 (n=1), A002378 (n=2), A055232 (terms of row n=3 divided by 2).
Columns: A000007 (k=0), A019590 (k=1), A040000 (k=2), A054886 (k=3).

def A350529_col(k,nmax):
    d = []
    c = [0]*(nmax+1)
    while 1:
        if not d or all(d[-1]):
            c[len(d)] += 1 + (bool(d) and 2*d[0][0] != k+1)
            if len(d) < nmax:
                d.append([0])
                for i in range(len(d)-1):
                    d[-1].append(d[-1][-1]-d[-2][i])
        while d and d[-1][0] == k:
            d.pop()
        if not d or len(d) == 1 and 2*d[0][0] >= k: return c
        for i in range(len(d)):
            d[-1][i] += 1

cp's OEIS Frontend

A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.

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A063759 Spherical growth series for modular group.

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A090991 Number of meaningful differential operations of the n-th order on the space R^6.

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A054889 Layer counting sequence for hyperbolic tessellation by regular pentagons of angle 2*Pi/5.

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Magma

Mathematica

Sage

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A054890 Layer counting sequence for hyperbolic tessellation by regular heptagons of angle Pi/3.

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Magma

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A350529 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 1..k such that no iterated difference is zero, n, k >= 0.

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