A276053 -id:A276053 - cp's OEIS frontend

A028495 Expansion of g.f. (1-x^2)/(1-x-2*x^2+x^3).

1, 1, 2, 3, 6, 10, 19, 33, 61, 108, 197, 352, 638, 1145, 2069, 3721, 6714, 12087, 21794, 39254, 70755, 127469, 229725, 413908, 745889, 1343980, 2421850, 4363921, 7863641, 14169633, 25532994, 46008619, 82904974, 149389218, 269190547, 485064009, 874055885

Offset: 0

N. J. A. Sloane

Form the graph with matrix A = [0,1,1; 1,0,0; 1,0,1] (P_3 with a loop at an extremity). Then A028495 counts closed walks of length n at the degree 3 vertex. - Paul Barry, Oct 02 2004
Equals INVERT transform of (1, 1, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 28 2009
From Johannes W. Meijer, May 29 2010: (Start)
a(n) is the number of ways White can force checkmate in exactly (n+1) moves, n>=0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6 and h6; Black Kc8, pawns b3, c7, d3, f7 and h7. (After Noam D. Elkies, see link; diagram 5).
Counts all paths of length n, n>=0, starting at the initial node on the path graph P_6, see the second Maple program. (End)
a(n) is the number of length n-1 binary words such that each maximal block of 1's has odd length.  a(4) = 6 because we have: 000, 001, 010, 100, 101, 111. - Geoffrey Critzer, Nov 17 2012
a(n) is the number of compositions of n where increments can only appear at every second position, starting with the second and third part, see example. Also, a(n) is the number of compositions of n where there is no fall between every second pair of parts, starting with the first and second part; see example. - Joerg Arndt, May 21 2013
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 0; 1, 0, 1; 0, 1, 0] or of the 3 X 3 matrix [1, 0, 1; 0, 0, 1; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
Range of row n of the circular Pascal array of order 7. - Shaun V. Ault, Jun 05 2014
a(n) is the number of compositions of n into parts from {1,2,4,6,8,10,...}. Example: a(4)= 6 because we have 4, 22, 211, 121, 112, and 1111. - Emeric Deutsch, Aug 17 2016
In general, a(n,m) = (2^n/(m+1))*Sum_{r=1..m} (1-(-1)^r)*cos(Pi*r/(m+1))^n*(1+cos(Pi*r/(m+1))) gives the number of paths of length n starting at the initial node on the path graph P_m. Here we have m=6. - Herbert Kociemba, Sep 15 2020
a(n-1) is the number of triangular dcc-polyominoes having area n (see Baril et al. at page 11). - Stefano Spezia, Oct 14 2023
a(n) is the number of permutations p of [n] with p(j)Alois P. Heinz, Mar 29 2024

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 19*x^6 + 33*x^7 + 61*x^8 + ...
From _Joerg Arndt_, May 21 2013: (Start)
There are a(6)=19 compositions of 6 where increments can only appear at every second position:
  01:  [ 1 1 1 1 1 1 ]
  02:  [ 1 1 1 1 2 ]
  03:  [ 1 1 2 1 1 ]
  04:  [ 1 1 2 2 ]
  05:  [ 1 1 3 1 ]
  06:  [ 1 1 4 ]
  07:  [ 2 1 1 1 1 ]
  08:  [ 2 1 2 1 ]
  09:  [ 2 1 3 ]
  10:  [ 2 2 1 1 ]
  11:  [ 2 2 2 ]
  12:  [ 3 1 1 1 ]
  13:  [ 3 1 2 ]
  14:  [ 3 2 1 ]
  15:  [ 3 3 ]
  16:  [ 4 1 1 ]
  17:  [ 4 2 ]
  18:  [ 5 1 ]
  19:  [ 6 ]
There are a(6)=19 compositions of 6 where there is no fall between every second pair of parts, starting with the first and second part:
  01:  [ 1 1 1 1 1 1 ]
  02:  [ 1 1 1 1 2 ]
  03:  [ 1 1 1 2 1 ]
  04:  [ 1 1 1 3 ]
  05:  [ 1 1 2 2 ]
  06:  [ 1 1 4 ]
  07:  [ 1 2 1 1 1 ]
  08:  [ 1 2 1 2 ]
  09:  [ 1 2 3 ]
  10:  [ 1 3 1 1 ]
  11:  [ 1 3 2 ]
  12:  [ 1 4 1 ]
  13:  [ 1 5 ]
  14:  [ 2 2 1 1 ]
  15:  [ 2 2 2 ]
  16:  [ 2 3 1 ]
  17:  [ 2 4 ]
  18:  [ 3 3 ]
  19:  [ 6 ]
(End)
19 = (1, 0, 1, 0, 1, 1) dot (1, 1, 2, 3, 6, 10) = (1 + 0 + 2 + 0 + 6 + 10). Cf. comment of Apr 28 2009. - _Gary W. Adamson_, Aug 10 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. V. Ault and C. Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics (2014).
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Alexandru Chirvasitu, Tara Hudson, and Aparna Upadhyay, Recursive sequences attached to modular representations of finite groups, arXiv:2105.04732 [math.RT], 2021. See (1) p. 27.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.
Noam D. Elkies, New Directions in Enumerative Chess Problems, The Electronic Journal of Combinatorics, 11 (2), 2004-2005; arXiv:math/0508645 [math.CO], 2005. - _Johannes W. Meijer_, May 29 2010
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 14.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1056
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1057
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See pp. 9, 15.
László Németh and Dragan Stevanović, Graph solution of system of recurrence equations, Research Gate, 2023. See Table 2 at p. 6.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A052534, A052547, A096976, A276053, A068911, A078038, A094790, A000045, A038754, A028495, A030436, A061551, A178381.

Cf. A006053, A052975, A060557, A094718.

spec := [S,{S=Sequence(Union(Prod(Sequence(Prod(Z,Z)),Z,Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
with(GraphTheory): P:=6: G:= PathGraph(P): A:=AdjacencyMatrix(G): nmax:=34; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k], k=1..P) od: seq(a(n), n=0..nmax); # Johannes W. Meijer, May 29 2010
a := (-1)^(3/7) - (-1)^(4/7):
b := (-1)^(5/7) - (-1)^(2/7):
c := (-1)^(1/7) - (-1)^(6/7):
f := n -> (a^n * (2 + a) + b^n * (2 + b) + c^n * (2 + c))/7:
seq(simplify(f(n)), n=0..36); # Peter Luschny, Sep 16 2020

LinearRecurrence[{1, 2, -1}, {1, 1, 2}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^3),{x,0,40}],x] (* Harvey P. Dale, Dec 23 2018 *)
a[n_,m_]:= 2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,6],{n,0,40}]//Round (* Herbert Kociemba, Sep 15 2020 *) (* Herbert Kociemba, Sep 14 2020 *)

{a(n) = if( n<0, n = -1-n; polcoeff( (1 - x^2) / (1 - 2*x - x^2 + x^3) + x * O(x^n), n), polcoeff( (1 - x^2) / (1 - x - 2*x^2 + x^3) + x * O(x^n), n))} /* Michael Somos, Apr 05 2012 */

a(n)=([0,1,0;0,0,1;-1,2,1]^n*[1;1;2])[1,1] \\ Charles R Greathouse IV, Aug 25 2016

Recurrence: {a(0)=1, a(1)=1, a(2)=2, a(n)-2*a(n+1)-a(n+2)+a(n+3)=0}.
a(n) = Sum_(1/7*(1+2*_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^3-2*_Z^2-_Z+1)).
a(n) = A094718(6, n). - N. J. A. Sloane, Jun 12 2004
a(n) = a(n-1) + Sum_{k=1..floor(n/2)} a(n-2*k). - Floor van Lamoen, Oct 29 2005
a(n) = 5*a(n-2) - 6*a(n-4) + a(n-6). - Floor van Lamoen, Nov 02 2005
a(n) = A006053(n+2) - A006053(n). - R. J. Mathar, Nov 16 2007
a(2*n) = A052975(n), a(2*n+1) = A060557(n). - Johannes W. Meijer, May 29 2010
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x / (1 - x))))). - Michael Somos, Apr 05 2012
a(-1 - n) = A052534(n). - Michael Somos, Apr 05 2012
a(n) = (2^n/7)*Sum_{r=1..6} (1-(-1)^r)*cos(Pi*r/7)^n*(1+cos(Pi*r/7)). - Herbert Kociemba, Sep 15 2020

More terms from James Sellers, Jun 05 2000

A276054 Sum of the asymmetry degrees of all compositions of n with parts in {1,2,4,6,8,10,...}.

Original entry on oeis.org

0, 0, 0, 2, 2, 8, 16, 34, 72, 146, 294, 590, 1156, 2278, 4422, 8572, 16510, 31682, 60558, 115398, 219190, 415348, 784996, 1480600, 2786818, 5236078, 9821222, 18393268, 34397388, 64241880, 119831316, 223266154, 415532226, 772587316, 1435082052, 2663283782

Offset: 0

Emeric Deutsch, Aug 17 2016

The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).

A sequence is palindromic if and only if its asymmetry degree is 0.

			a(4) = 2 because the compositions of 4 with parts in {1,2,4,6,8,...} are 4, 22, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 0 + 0 + 1 + 0 + 1 + 0 = 2.

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

Colin Barker, Table of n, a(n) for n = 0..1000
Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-5,-3,-1,0,5,2,-4,1).

Cf. A276053, A276055.

g:= 2*z^3*(1-z^2)*(1+z^3-z^4)/((1+z^2)*(1+z-z^3)*(1-z-2*z^2+z^3)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);

Table[Total@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; Nor[a == 1, EvenQ@ a]]], 1]]], {n, 0, 23}] // Flatten (* or *)
CoefficientList[Series[2 x^3*(1 - x^2) (1 + x^3 - x^4)/((1 + x^2) (1 + x - x^3) (1 - x - 2 x^2 + x^3)^2), {x, 0, 35}], x] (* Michael De Vlieger, Aug 28 2016 *)

concat(vector(3), Vec(2*x^3*(1-x^2)*(1+x^3-x^4)/((1+x^2)*(1+x-x^3)*(1-x-2*x^2+x^3)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016

G.f.: g(z) = 2*z^3*(1-z^2)*(1+z^3-z^4)/((1+z^2)*(1+z-z^3)*(1-z-2*z^2+z^3)^2). In the more general situation of compositions into a[1]=1} z^(a[j]), we have g(z) = (F(z)^2-F(z^2))/((1+F(z))*(1-F(z))^2).

a(n) = Sum_{k>0} k*A276053(n,k).

A276055 Number of palindromic compositions of n with parts in {1,2,4,6,8,10,...}.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 7, 3, 13, 6, 23, 10, 42, 19, 75, 33, 136, 61, 244, 108, 441, 197, 793, 352, 1431, 638, 2576, 1145, 4645, 2069, 8366, 3721, 15080, 6714, 27167, 12087, 48961, 21794, 88215, 39254, 158970, 70755, 286439, 127469, 516164, 229725, 930072

Offset: 0

Emeric Deutsch, Aug 17 2016

			a(6) = 7 because the palindromic compositions of 6 with parts in {1,2,4,6,8,...} are 6, 141, 222, 2112, 1221, 11211, and 111111.

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,2,0,-1).

Cf. A028495, A276053,

g := (1+z^2)*(1+z-z^3)/(1-z^2-2*z^4+z^6): gser:= series(g,z=0,55): seq(coeff(gser,z,n),n=0..50);

CoefficientList[Series[(1 + x^2) (1 + x - x^3)/(1 - x^2 - 2 x^4 + x^6), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2016 *)
LinearRecurrence[{0,1,0,2,0,-1},{1,1,2,1,4,2},50] (* Harvey P. Dale, Jul 03 2021 *)

G.f.: g(z) = (1+z^2)*(1+z-z^3)/(1-z^2-2*z^4+z^6). In the more general situation of compositions into a[1]=1} z^(a[j]), we have g(z) = (1+F(z))/(1-F(z^2)) (see Theorem 1.2 in the Hoggatt et al. reference).

a(2*n) = |A078038(n)|. a(2*n+1) = A028495(n). - R. J. Mathar, Jan 13 2023

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A028495 Expansion of g.f. (1-x^2)/(1-x-2*x^2+x^3).

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A276054 Sum of the asymmetry degrees of all compositions of n with parts in {1,2,4,6,8,10,...}.

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A276055 Number of palindromic compositions of n with parts in {1,2,4,6,8,10,...}.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula