A228364 -id:A228364 - cp's OEIS frontend

A228577 The number of 1-length gaps in all possible covers of n-length line by 2-length segments.

0, 1, 0, 2, 2, 3, 6, 7, 12, 17, 24, 36, 50, 72, 102, 143, 202, 282, 394, 549, 762, 1057, 1462, 2019, 2784, 3832, 5268, 7232, 9916, 13581, 18580, 25394, 34674, 47303, 64478, 87819, 119520, 162549, 220920, 300060, 407302, 552552, 749186, 1015259, 1375134

Offset: 0

Philipp O. Tsvetkov, Aug 26 2013

2-gaps must be filled, so, for example, xxoo doesn't count for n=4. - Jon Perry, Nov 18 2014

			For n=6 we have xxoxxo, oxxxxo and oxxoxx, so a(6) = number of o's = 6. - _Jon Perry_, Nov 18 2014

A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, Volume 37:7 (2006), 825-831. See Eqn. (3.13). - N. J. A. Sloane, Jan 11 2022

D. Birmajer, J. Gil and M. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014 and J. Int. Seq. 18 (2015) # 15.1.2.
Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-2,-1).

Cf. A228361, A228364.

I:=[0,1,0,2,2,3]; [n le 6 select I[n] else 2*Self(n-2)+2*Self(n-3)-Self(n-4)-2*Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Nov 18 2014

A228577 := proc(n) coeftayl(x/(x^3+x^2-1)^2, x=0, n); end proc: seq(A228577(n), n=0..50); # Wesley Ivan Hurt, Nov 17 2014

CoefficientList[Series[x/(x^3 + x^2 - 1)^2, {x, 0, 100}], x]

For n>1, a(n) = n * A228361(n) - 2 * A228364(n).
G.f.: x/(x^3 + x^2 - 1)^2, convolution of A182097 by itself.
a(n) = 2*a(n-2) +2*a(n-3) -a(n-4) -2*a(n-5) -a(n-6) for n>5.
(n-1)*a(n) - (n+1)*a(n-2) - (n+2)*a(n-3) = 0 for n>2. - Michael D. Weiner, Nov 18 2014

A245963 Triangle read by rows: T(n,k) is the number of maximal hypercubes Q(p) in the Fibonacci cube Gamma(n) (i.e., Q(p) is an induced subgraph of Gamma(n) that is not a subgraph of a subgraph of Gamma(n) that is isomorphic to the hypercube Q(p+1)).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 3, 1, 0, 0, 1, 4, 0, 0, 0, 6, 1, 0, 0, 0, 4, 5, 0, 0, 0, 1, 10, 1, 0, 0, 0, 0, 10, 6, 0, 0, 0, 0, 5, 15, 1, 0, 0, 0, 0, 1, 20, 7, 0, 0, 0, 0, 0, 15, 21, 1, 0, 0, 0, 0, 0, 6, 35, 8, 0, 0, 0, 0, 0, 1, 35, 28, 1, 0, 0, 0, 0, 0, 0, 21, 56, 9, 0, 0, 0, 0, 0, 0, 7, 70, 36, 1

Offset: 0

Emeric Deutsch, Aug 13 2014

The nonzero entries in columns 0,1,2,... are rows 0,2,3,... of the Pascal triangle.
Row n contains 1+ceiling(n/2) entries.
Sum of entries in row n = A000931(n+6) (the Padovan sequence).
Sum_{k>=0}k*T(n,k) = A228364(n+1).

			Row 3 is 0,1,1. Indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; the pendant edge is a maximal Q(1) and the square is a maximal Q(2).
Triangle starts:
  1;
  0, 1;
  0, 2;
  0, 1, 1;
  0, 0, 3;
  0, 0, 3, 1;
  0, 0, 1, 4;
  0, 0, 0, 6, 1;

Michael De Vlieger, Table of n, a(n) for n = 0..10300 (Rows 1 <= n <= 200).
S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
M. Mollard, Maximal hypercubes in Fibonacci and Lucas cubes, arXiv:1201.1494 [math.CO], 2012.
M. Mollard, Maximal hypercubes in Fibonacci and Lucas cubes, Discrete Appl. Math., 160, 2012, 2479-2483.

Cf. A000931, A228364, A245964.

T := proc (n, k) options operator, arrow: binomial(1+k, n-2*k+1) end proc: for n from 0 to 20 do seq(T(n, k), k = 0 .. (n+1)*(1/2)) end do; # yields sequence in triangular form

Table[Binomial[k + 1, n - 2 k + 1], {n, 0, 17}, {k, 0, Ceiling[n/2]}] // Flatten (* Michael De Vlieger, Jul 16 2017 *)

T(n,k) = binomial(k+1,n-2*k+1).

G.f.: (1+t*z*(1+z))/(1-t*(1+z)*z^2).

A228494 The number of 3-length segments in all possible covers of L-length line by these segments with allowed gaps < 3.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 7, 12, 17, 24, 36, 54, 77, 108, 155, 222, 312, 436, 612, 858, 1194, 1656, 2298, 3184, 4397, 6060, 8346, 11480, 15762, 21612, 29607, 40518, 55385, 75632, 103197, 140692, 191647, 260856, 354814, 482290, 655131, 889364, 1206649, 1636218

Offset: 0

Philipp O. Tsvetkov, Aug 23 2013

Related with the number of all possible covers of L-length line segment by 3-length line segments with allowed gaps < 3 (A228362).

Index entries for linear recurrences with constant coefficients, signature (0,0,2,2,2,-1,-2,-3,-2,-1)

Cf. A228362, A228364.

c[k_, l_, m_] :=  Sum[(-1)^i Binomial[k - 1 - i*l, m - 1] Binomial[m, i], {i, 0,     Floor[(k - m)/l]}]; a[L_, l_, m_] :=  Sum[Binomial[m + 1, m + 1 - j]*c[L - l*m, l - 1, j], {j, 0, m + 1}]; sa[L_, l_] := Sum[j*a[L, l, j], {j, 1, Ceiling[L/l]}];Table[sa[j, 3], {j, 0, 100}]
CoefficientList[Series[x^3(x^2+x+1)^2/(x^5+x^4+x^3-1)^2,{x, 0, 100}], x]
LinearRecurrence[{0,0,2,2,2,-1,-2,-3,-2,-1},{0,0,0,1,2,3,4,7,12,17},50] (* Harvey P. Dale, May 21 2025 *)

concat([0,0,0], Vec(x^3*(x^2+x+1)^2/((x^2+1)*(x^3+x^2-1))^2+O(x^66))) \\ Joerg Arndt, Aug 23 2013

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

cp's OEIS Frontend

A228577 The number of 1-length gaps in all possible covers of n-length line by 2-length segments.

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A245963 Triangle read by rows: T(n,k) is the number of maximal hypercubes Q(p) in the Fibonacci cube Gamma(n) (i.e., Q(p) is an induced subgraph of Gamma(n) that is not a subgraph of a subgraph of Gamma(n) that is isomorphic to the hypercube Q(p+1)).

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Maple

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A228494 The number of 3-length segments in all possible covers of L-length line by these segments with allowed gaps < 3.

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