A006054 -id:A006054 - cp's OEIS frontend

A106805 Expansion of g.f.: 1/(1 - 2*x - x^2 + x^3).

1, 2, 5, 11, 25, 56, 126, 283, 636, 1429, 3211, 7215, 16212, 36428, 81853, 183922, 413269, 928607, 2086561, 4688460, 10534874, 23671647, 53189708, 119516189, 268550439, 603427359, 1355888968, 3046654856, 6845771321, 15382308530, 34563733525, 77664004259

Offset: 0

Roger L. Bagula, May 17 2005

Essentially the same as A006054. - Joerg Arndt, Nov 08 2022

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

A006054 shifted left twice.

I:=[1,2,5]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) -Self(n-3): n in [1..36]]; // G. C. Greubel, Sep 11 2021

LinearRecurrence[{2,1,-1}, {1,2,5}, 35] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

Vec( 1/(1-2*x-x^2+x^3) + O(x^66) )  /* Joerg Arndt, Sep 30 2012 */

def A106805_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2+x^3) ).list()
A106805_list(35) # G. C. Greubel, Sep 11 2021

G.f. for sequence with 1 prepended: 1/( 1 - Sum_{k>=0} x*(x+x^2-x^3)^k ). - Joerg Arndt, Sep 30 2012

Edited by the Associate Editors of the OEIS, Apr 09 2009

Name corrected by Joerg Arndt, Sep 30 2012

A122514 Expansion of x/(1 - 2*x^2 - x^3 + x^4).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 4, 5, 10, 11, 21, 27, 43, 64, 92, 144, 205, 316, 462, 693, 1035, 1532, 2301, 3406, 5099, 7581, 11303, 16855, 25088, 37432, 55728, 83097, 123800, 184490, 274969, 409683, 610628, 909845, 1355970, 2020635, 3011157, 4487395, 6686979

Offset: 0

Roger L. Bagula, Sep 16 2006

a(n) is the number of compositions of n+2 such that: i) the first part is odd, ii) the last part is even, and iii) no two consecutive parts have the same parity. - Geoffrey Critzer, Mar 04 2012

			a(7) = 5 because there are 5 such compositions of the integer 9: 1+8, 7+2, 3+6, 5+4, 1+2+1+2+1+2. - _Geoffrey Critzer_, Mar 04 2012

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

Cf. A006054, A006053.

nn = 44; a = x/(1 - x^2); b = x^2/(1 - x^2); Drop[ CoefficientList[Series[1/(1 - a b), {x, 0, nn}], x], 2] (* Geoffrey Critzer, Mar 04 2012 *)
CoefficientList[Series[x/(1-2x^2-x^3+x^4),{x,0,50}],x] (* Harvey P. Dale, Jul 17 2019 *)

A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries in the top row. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 5, 11, 7, 1, 11, 31, 29, 10, 1, 25, 83, 102, 56, 13, 1, 56, 217, 329, 245, 92, 16, 1, 126, 556, 1000, 938, 487, 137, 19, 1, 283, 1403, 2917, 3292, 2180, 855, 191, 22, 1, 636, 3498, 8247, 10865, 8740, 4406, 1376, 254, 25, 1, 1429, 8636, 22756, 34248

Offset: 0

Emeric Deutsch, Oct 14 2010

The sum of entries in row n is A003480(n).
T(n,0)=A006054(n+1) (n>=1).
Sum(k*T(n,k), k>=0)=A181337(n).
For the statistic "number of odd entries in the top row" see A181304.

			T(2,1)=4 because we have (0/2), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
2,4,1;
5,11,7,1;
11,31,29,10,1;
25,83,102,56,13,1;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.

Cf. A003480, A006054, A181304, A181337,

G := (1+z)*(1-z)^2/(1-2*z-z^2+z^3-s*z*(1+z-z^2)): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], s, k), k = 0 .. n) end do; # yields sequence in triangular form

G.f.=G(s,z)=(1+z)(1-z)^2/[1-2z-z^2+z^3-sz(1+z-z^2)].
The g.f. of column k is z^k*(1+z)(1-z)^2*(1+z-z^2)^k/(1-2z-z^2+z^3)^{k+1} (we have a Riordan array).
The g.f. H=H(t,s,z), where z marks size and t (s) marks odd (even) entries in the top row, is given by H = (1+z)(1-z)^2/[(1+z)(1-z)^2-(t+s)z-sz^2*(1-z)].

A189426 Expansion of (x^2)/(1-2*x-x^2+x^3)^2.

Original entry on oeis.org

0, 0, 1, 4, 14, 42, 119, 322, 847, 2180, 5521, 13804, 34160, 83818, 204204, 494494, 1191227, 2856666, 6823334, 16240714, 38534657, 91175154, 215179125, 506670394, 1190534467, 2792076392, 6536567296, 15278103876, 35656587624, 83101366684

Offset: 0

L. Edson Jeffery, Apr 22 2011

Convolution of A006054={0,0,1,2,5,11,25,56,126,...} with itself.

For n=0,1,2,..., partial sums are given by Sum_{k=0..n} a(k)=A189427(n), where A189427={0,0,1,5,19,61,180,...}.

Index entries for linear recurrences with constant coefficients, signature (4, -2, -6, 3, 2, -1).

Cf. A006054, A189427.

CoefficientList[Series[x^2/(1-2x-x^2+x^3)^2,{x,0,40}],x] (* or *) LinearRecurrence[{4,-2,-6,3,2,-1},{0,0,1,4,14,42},40] (* Harvey P. Dale, Feb 29 2012 *)

Vec((x^2)/(1-2*x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

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

a(n)=4*a(n-1)-2*a(n-2)-6*a(n-3)+3*a(n-4)+2*a(n-5)-a(n-6), n>=6.

A052658 Expansion of e.g.f. (1-x^2)*(1-x)/(1-2x-x^2+x^3).

Original entry on oeis.org

1, 1, 4, 30, 264, 3000, 40320, 635040, 11410560, 230791680, 5185555200, 128172844800, 3455996544000, 100952461209600, 3175730791833600, 107037070043904000, 3848161361780736000, 146994587721805824000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 605

spec := [S,{S=Sequence(Prod(Z,Sequence(Z),Sequence(Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[((1-x^2)(1-x))/(1-2x-x^2+x^3),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, May 16 2012 *)

E.g.f.: (-1+x^2)*(-1+x)/(x^3-x^2-2*x+1)
Recurrence: {a(1)=1, a(0)=1, a(2)=4, (n^3+6*n^2+11*n+6)*a(n)+(-n^2-5*n-6)*a(n+1)+(-2*n-6)*a(n+2)+a(n+3)=0, a(3)=30}
a(n) = Sum(-1/7*(_alpha+_alpha^2-2)*_alpha^(-1-n), _alpha=RootOf(_Z^3-_Z^2-2*_Z+1))*n!.
a(n) = n!*A006054(n+1),n>0. - R. J. Mathar, Jun 03 2022

A091594 Triangle read by rows: T(n,m) := Sum_{k=0..floor((n-m)/2)} binomial(n-2k,m) * binomial(n-m-k,k).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 5, 8, 7, 4, 1, 8, 15, 16, 11, 5, 1, 13, 28, 34, 28, 16, 6, 1, 21, 51, 70, 66, 45, 22, 7, 1, 34, 92, 140, 148, 116, 68, 29, 8, 1, 55, 164, 274, 320, 281, 190, 98, 37, 9, 1, 89, 290, 527, 672, 651, 494, 295, 136, 46, 10, 1, 144, 509, 999, 1379, 1456, 1219, 819, 439, 183, 56, 11, 1

Offset: 0

Paul Barry, Jan 23 2004

A Fibonacci related number triangle.

			Rows begin:
   1,
   1,  1,
   2,  2,  1,
   3,  4,  3,  1,
   5,  8,  7,  4,  1,
   8, 15, 16, 11,  5,  1,
  13, 28, 34, 28, 16,  6, 1,
  21, 51, 70, 66, 45, 22, 7, 1,
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Columns include A000045, A029907, A054455. Row sums are A006054.

k-th column has g.f. 1/(1-x-x^2) * ( x*(1-x^2)/(1-x-x^2) )^k.

A095310 a(n+3) = 2a(n+2) + 3(n+1) - a(n).

Original entry on oeis.org

1, 5, 12, 38, 107, 316, 915, 2671, 7771, 22640, 65922, 191993, 559112, 1628281, 4741905, 13809541, 40216516, 117119750, 341079507, 993301748, 2892722267, 8424270271, 24533405595, 71446899736, 208069745986, 605946785585

Offset: 1

Gary W. Adamson, Jun 02 2004

Let M = the 3 X 3 matrix [1 1 1 / 3 1 0 / 1 0 0], then M^n * [1 0 0] = [a(n) q a(n-1)] where q is another sequence with the same recursion rule.

			a(6) = 316 = 2*107 + 3*38 - 12.
a(5) = 107 since M^5 * [1 0 0] = [107 q 38].

Index entries for linear recurrences with constant coefficients, signature (2, 3, -1).

Cf. A001263, A006356, A006054, A061646, A061647, A095125, A095126.

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {0}, {0}})[[1, 1]]; Table[ a[n], {n, 27}] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{2,3,-1},{1,5,12},30] (* Harvey P. Dale, Jan 25 2014 *)

G.f.: (-x^2+3*x+1)/(x^3-3*x^2-2*x+1). - Harvey P. Dale, Jan 25 2014

Corrected and extended by Robert G. Wilson v, Jun 05 2004

Edited by N. J. A. Sloane, Jun 07 2004

A179542 Trajectory of 1 under the morphism 1->(1,2,3), 2->(1,2), 3->(1) related to the heptagon and A006356.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2

Offset: 0

Gary W. Adamson, Jul 18 2010

nonn

Given M = the generating matrix for the heptagon shown in A006356:
[1,1,1; 1,1,0; 1,0,0] take powers of M, extracting top row getting:
(1,1,1), (3,2,1), (6,5,3), (14,11,6), where left and right columns (offset) =
A006356, and middle column = A006054. n-th iterate of the sequence is
composed of A006356(n) terms parsed into a frequency of 1's, 2's, and 3's
matching the 3-termed vectors with appropriate sums.

			Starting with 1, the next two iterates are:
(1, 2, 3) -> (1, 2, 3, 1, 2, 1) -> (1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3).
The 3rd iterate has 14 terms composed of six 1's, five 2's, and three 3's; matching the top row of M^3 = (6, 5, 3), sum = 14 = A006356(3).

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for sequences that are fixed points of mappings

Cf. A006356, A006054

NestList[ Flatten[ # /. {1 -> {1, 2, 3}, 2 -> {1, 2}, 3 -> 1}] &, {1}, 5] // Flatten (* Robert G. Wilson v, Jul 23 2010 *)

More terms from Robert G. Wilson v, Jul 23 2010

A215139 a(n) = (a(n-1) - a(n-3))*7^((1+(-1)^n)/2) with a(6)=5, a(7)=4, a(8)=22.

Original entry on oeis.org

5, 4, 22, 17, 91, 69, 364, 273, 1428, 1064, 5537, 4109, 21315, 15778, 81683, 60368, 312130, 230447, 1190553, 878423, 4535832, 3345279, 17267992, 12732160, 65708167, 48440175, 249956105, 184247938, 950654341, 700698236, 3615152086, 2664497745, 13746596563, 10131444477

Offset: 6

Roman Witula, Aug 04 2012

The Ramanujan-type sequence the number 9 for the argument 2*Pi/7. The sequence is connecting with the following decomposition: (s(4)/s(1))^(1/3)*s(1)^n + (s(1)/s(2))^(1/3)*s(2)^n + (s(2)/s(4))^(1/3)*s(4)^n = x(n)*(4-3*7^(1/3))^(1/3) + y(n)*(11-3*49^(1/3))^(1/3), where s(j) := sin(2*Pi*j/7), x(0)=1, x(1)=-7^(1/6)/2, x(2)=y(0)=y(1)=0, y(2)=7^(1/3)/4 and X(n)=sqrt(7)*(X(n-1)-X(n-3)) for every n=3,4,..., and X=x or X=y. It could be deduced the formula 4*y(n) = a(n)*7^(1/3 + (3+(-1)^n)/4), which implies a(0)=0, a(1)= 0, a(2)= 1/7, a(3)=1/7, a(4)=1, a(5)=6/7, i.e., A163260(n)=7*a(n) for every n=0,1,...,5. The sequence a(n) is discussed in third Witula paper.

			From values of x(2),y(2) and the identity 2*sin(t)^2=1-cos(2*t) we obtain (s(4)/s(1))^(1/3)*c(1) + (s(1)/s(2))^(1/3)*c(4) + (s(2)/s(4))^(1/3)*c(1) = (4-3*7^(1/3))^(1/3) - (1/2)*(7*(11-3*49^(1/3)))^(1/3), where c(j):=cos(2*Pi*j/7). Further, from values of x(1),x(3),y(1),y(3) and the identity 4*sin(t)^3=3*sin(t)-sin(3*t) we obtain (s(4)/s(1))^(1/3)*s(4) + (s(1)/s(2))^(1/3)*s(1) + (s(2)/s(4))^(1/3)*s(2) = (-3*7^(1/6)/2 +4*7^(1/2))*(4-3*7^(1/3))^(1/3) - 7^(5/6)*(11-3*49^(1/3))^(1/3).

R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

G. C. Greubel, Table of n, a(n) for n = 6..1005
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (0,7,0,-14,0,7).

Cf. A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560, A215569, A215572, A214699.

I:=[5,4,22,17,91,69]; [n le 6 select I[n] else 7*Self(n-2) - 14*Self(n-4) + 7*Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 19 2018

LinearRecurrence[{0,7,0,-14,0,7}, {5,4,22,17,91,69}, {1,50}] (* G. C. Greubel, Apr 19 2018 *)

Vec(-x*(1+x)*(6*x^4+x^3-12*x^2-x+5)/(-1+7*x^2-14*x^4+7*x^6) + O(x^50)) \\ Michel Marcus, Apr 20 2016

G.f.: -x*(1+x)*(6*x^4+x^3-12*x^2-x+5) / ( -1+7*x^2-14*x^4+7*x^6 ). - R. J. Mathar, Sep 14 2012

More terms from Michel Marcus, Apr 20 2016

A219788 Consider the succession rule (x, y, z) -> (z, y+z, x+y+z). Sequence gives z values starting at (0, 1, 2).

Original entry on oeis.org

2, 3, 8, 17, 39, 87, 196, 440, 989, 2222, 4993, 11219, 25209, 56644, 127278, 285991, 642616, 1443945, 3244515, 7290359, 16381288, 36808420, 82707769, 185842670, 417584689, 938304279, 2108350577, 4737420744, 10644887786, 23918845739, 53745158520, 120764274993

Offset: 1

Andrew Pharo, Nov 27 2012

The rule can be generalized for any number of starting terms s: (xs, ..., x2, x1) -> (x1, x1 + x2, ..., x1 + x2 + ... + xs), using (0, 1, ..., s-1) as seed values. This sequence is s=3, and s=2 yields the Fibonacci series.
For s=3 the ratio of S1 (the first in the sub-series) to S3 (the 3rd in the sub-series) converges on 2.2469796 and the ration of S2 (the 2nd in the sub-series) to S3 converges on 1.2469796 thus the difference, S2-S3, converges on 1 regardless of the seed values used.
For s=20 the series is: 19, 190, 2660, 33915, 445949, ....
a(n-2) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 0, 1] or of the 3 X 3 matrix [0, 1, 1; 1, 1, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
From Andrew Pharo, Jun 02 2014: (Start)
For s=2 the ratio of successive terms is 1.6180339887... or phi (or phi(2));
for s=3 this ratio is 2.24697960412319..., phi(3) = 4*cos(Pi/7)^2-1 (see Falbo link);
for s=4 this ratio is 3.5133370918694...;
for s=20 this ratio is 13.0538985560545... and so on.
We can define a function phi(s) which approximates to:
phi(s) ~ phi(2) + theta*(s-2) where theta ~ 0.636264133.
(End)

			The seed values are (0,1,2), giving a(1) = 2. (2, 2+1, 2+1+0) is the next triple, giving a(2) = 2+1+0 = 3. (3, 6, 8) is next, yielding a(3) = 8. The triples that follow begin (8,14,17), (17,31,39), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..2844
Clement Falbo, The Golden Ratio - A Contrary Viewpoint, Vol. 36, No. 2, March 2005, The College Mathematics Journal.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (10) and Theorem 8.
Brian Hopkins, Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
R. Sachdeva and A. K. Agarwal, Combinatorics of certain restricted n-color composition functions, Discrete Mathematics, 340, (2017), 361-372.
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Rest@ CoefficientList[Series[-x (-2 + x)/(1 - 2 x - x^2 + x^3), {x, 0, 32}], x] (* Michael De Vlieger, Jun 17 2020 *)
sr[{x_,y_,z_}]:={z,y+z,x+y+z}; NestList[sr,{0,1,2},40][[All,3]] (* Harvey P. Dale, Aug 18 2020 *)

first(n)=my(x=0,y=1,z=2,v=List([z])); for(i=2, n, [x,y,z]=[z, y+z, x+y+z]; listput(v,c)); Vec(v) \\ Charles R Greathouse IV, Nov 28 2012

a(n) = 2a(n-1) + a(n-2) - a(n-3). - Charles R Greathouse IV, Nov 28 2012
The essentially identical sequence 1,0,2,3,8,17,39,... with offset 0 is defined by a(n) = 2a(n-1) + a(n-2) - a(n-3) with initial terms a(0)=1, a(1)=0, a(2)=2. - N. J. A. Sloane, Jan 16 2017
G.f.: -x*(-2+x) / ( 1-2*x-x^2+x^3 ). - R. J. Mathar, Feb 03 2014
a(n) = 2*A006054(n+1)-A006054(n). - R. J. Mathar, Aug 22 2016

cp's OEIS Frontend

A106805 Expansion of g.f.: 1/(1 - 2*x - x^2 + x^3).

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A122514 Expansion of x/(1 - 2*x^2 - x^3 + x^4).

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A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries in the top row. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

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A189426 Expansion of (x^2)/(1-2*x-x^2+x^3)^2.

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A052658 Expansion of e.g.f. (1-x^2)*(1-x)/(1-2x-x^2+x^3).

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A091594 Triangle read by rows: T(n,m) := Sum_{k=0..floor((n-m)/2)} binomial(n-2k,m) * binomial(n-m-k,k).

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A095310 a(n+3) = 2*a(n+2) + 3*(n+1) - a(n).

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A179542 Trajectory of 1 under the morphism 1->(1,2,3), 2->(1,2), 3->(1) related to the heptagon and A006356.

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A095310 a(n+3) = 2a(n+2) + 3(n+1) - a(n).