A275446 -id:A275446 - cp's OEIS frontend

A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).

1, 1, 2, 4, 7, 14, 26, 50, 95, 181, 345, 657, 1252, 2385, 4544, 8657, 16493, 31422, 59864, 114051, 217286, 413966, 788674, 1502555, 2862617, 5453761, 10390321, 19795288, 37713313, 71850128, 136886433, 260791401, 496850954, 946583628

Offset: 0

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

a(n) = number of compositions of n with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1.1 of the Hoggatt et al. reference. Example: a(4)= 7 because we have 22, 31, 13, 211, 121, 112, and 1111. - Emeric Deutsch, Aug 17 2016.
Diagonal sums of A054142. - Paul Barry, Jan 21 2005
Equals INVERT transform of (1, 1, 1, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 27 2009
Number of tilings of a 4 X 2n rectangle by 4 X 1 tetrominoes. - M. Poyraz Torcuk, Dec 10 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024. See p. 18.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 465
Todd Mullen, On Variants of Diffusion, Dalhousie University (Halifax, NS Canada, 2020).
Todd Mullen, Richard Nowakowski, and Danielle Cox, Counting Path Configurations in Parallel Diffusion, arXiv:2010.04750 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Cf. A275446.
Bisection of A003269 (odd part),
Cf. A236582, A251074, A236580.

a:=[1,1,2,4];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, May 09 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)/( 1-x-2*x^2+x^4) )); // G. C. Greubel, May 09 2019

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

CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^4), {x, 0, 40}], x] (* or *)
Table[Length@ Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; And[EvenQ@ a, a != 2]]], 1], {n, 0, 40}]  (* Michael De Vlieger, Aug 17 2016 *)
LinearRecurrence[{1,2,0,-1},{1,1,2,4},40] (* Harvey P. Dale, Apr 12 2018 *)

my(x='x+O('x^40)); Vec((1-x^2)/(1-x-2*x^2+x^4)) \\ G. C. Greubel, May 09 2019

((1-x^2)/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
a(n) = a(n-1) + 2*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=4.
a(n) = Sum_{alpha = RootOf(1-x-2*x^2+x^4)} (1/283)*(27 + 112*alpha + 9*alpha^2 -48*alpha^3)*alpha^(-n-1).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k). - Paul Barry, Jan 21 2005
a(n) = A158943(n) -A158943(n-2). - R. J. Mathar, Jan 13 2023

More terms from James Sellers, Jun 05 2000

A103632 Expansion of (1 - x + x^2)/(1 - x - x^4).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 76, 105, 145, 200, 276, 381, 526, 726, 1002, 1383, 1909, 2635, 3637, 5020, 6929, 9564, 13201, 18221, 25150, 34714, 47915, 66136, 91286, 126000, 173915, 240051, 331337, 457337, 631252, 871303, 1202640

Offset: 0

Paul Barry, Feb 11 2005

Diagonal sums of A103631.
The Kn11 sums, see A180662, of triangle A065941 equal the terms of this sequence without a(0) and a(1). - Johannes W. Meijer, Aug 11 2011
For n >= 2, a(n) is the number of palindromic compositions of n-2 with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1,2 of the Hoggatt et al. reference. Example: a(9) = 8 because we have 7, 151, 11311, 232, 313, 12121, 21112, and 1111111. - Emeric Deutsch, Aug 17 2016
Essentially the same as A003411. - Georg Fischer, Oct 07 2018

Muniru A Asiru, Table of n, a(n) for n = 0..1000
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 (1,0,0,1).

Cf. A275446.

a:=[1,0,1,1];;  for n in [5..50] do a[n]:=a[n-1]+a[n-4]; od; a; # Muniru A Asiru, Oct 07 2018

I:=[1,0,1,1]; [n le 4 select I[n] else Self(n-1) + Self(n-4): n in [1..50]]; // G. C. Greubel, Mar 10 2019

A103632 := proc(n): add( binomial(floor((2*n-3*k-1)/2), n-2*k), k=0..floor(n/2)) end: seq(A103632(n), n=0..46); # Johannes W. Meijer, Aug 11 2011

LinearRecurrence[{1,0,0,1}, {1,0,1,1}, 50] (* G. C. Greubel, Mar 10 2019 *)

my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^4)) \\ G. C. Greubel, Mar 10 2019

((1-x+x^2)/(1-x-x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Mar 10 2019

G.f.: (1 - x + x^2)/(1 - x - x^4).
a(n) = a(n-1) + a(n-4) with a(0)=1, a(1)=0, a(2)=1 and a(3)=1.
a(n) = Sum_{k=0..floor(n/2)} binomial(floor((2*n-3*k-1)/2), n-2*k).
a(n) = A003269(n+1) - A003269(n-4), n > 4.

Formula corrected by Johannes W. Meijer, Aug 11 2011

A275442 Triangle read by rows: T(n,k) is the number of compositions without 2's and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 8, 5, 16, 7, 26, 4, 9, 44, 12, 12, 70, 32, 16, 108, 76, 21, 166, 156, 8, 28, 248, 308, 32, 37, 368, 572, 104, 49, 540, 1020, 288, 65, 784, 1768, 696, 16, 86, 1132, 2976, 1568, 80, 114, 1622, 4908, 3304, 304, 151, 2312, 7944, 6624, 960, 200, 3280, 12652, 12768, 2640, 32

Offset: 0

Emeric Deutsch, Aug 16 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).
Number of entries in row n is 1 + floor(n/4).
Sum of entries in row n is A005251(n+1).
T(n,0) = A000931(n+5) (= number of palindromic compositions of n without 2's).
Sum_{k >= 0} k*T(n,k) = A275443(n).

			Row 5 is [3,4] because the compositions of 5 without 2's are 5, 113, 131, 311, 14, 41, and 11111, having asymmetry degrees 0, 1, 0, 1, 1, 1, and 0, respectively.
Triangle starts:
  1;
  1;
  1;
  2;
  2,2;
  3,4;
  4,8;
  5,16.

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

John Tyler Rascoe, Rows n = 0..200, flattened
Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A000931, A005251, A180177, A275433, A275443, A275446.

G := (1-z^2)/(1-z-z^2+z^4-2*t*z^4): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form

Table[BinCounts[#, {0, 1 + Floor[n/4], 1}] &@ 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_, _} /; a == 2]], 1]]], {n, 0, 20}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)

T_zt(max_row) = {my(N = max_row+1, z='z+O('z^N), h=(1-z^2)/(1-z-z^2+z^4-2*t*z^4)); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_zt(10) \\ John Tyler Rascoe, May 09 2025

G.f.: G(t,z) = (1-z^2)/(1-z-z^2+z^4-2*t*z^4). In the more general situation of compositions into a[1]=1} z^(a[j]), we have G(t,z) = (1 + F(z))/(1 - F(z^2) - t*(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.

A275447 Sum of the asymmetry degrees of all compositions of n with parts in {2,1,3,5,7,9,...}.

Original entry on oeis.org

0, 0, 0, 2, 4, 10, 24, 54, 120, 258, 552, 1164, 2432, 5042, 10384, 21268, 43344, 87962, 177840, 358358, 719964, 1442584, 2883504, 5751020, 11447164, 22743262, 45110096, 89334192, 176658732, 348875904, 688122336, 1355674528, 2667921660, 5245033102

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) = 4 because the compositions of 4 with parts in {2,1,3,5,7,...} are 22, 31, 13, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 0 + 1 + 1 + 1 + 0 + 1 + 0 = 4.

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 (2,2,-2,-3,-2,-2,2,3,0,-1).

Cf. A275446.

g := 2*z^3*(1-z^2)/((1+z^2)*(1-z-2*z^2+z^4)^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_, _} /; And[EvenQ@ a, a != 2]]], 1]]], {n, 0, 21}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)

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

G.f.: g(z) = 2*z^3*(1-z^2)/((1+z^2)*(1-z-2*z^2+z^4)^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*A275446(n,k).

cp's OEIS Frontend

A052535 Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).

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A103632 Expansion of (1 - x + x^2)/(1 - x - x^4).

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A275442 Triangle read by rows: T(n,k) is the number of compositions without 2's and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).

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A275447 Sum of the asymmetry degrees of all compositions of n with parts in {2,1,3,5,7,9,...}.

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A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).