A182097 -id:A182097 - 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

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A378716 Triangle read by rows: T(n,k) is the number of k-Fibonacci polyominoes with an area of n, with k > 1.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 3, 0, 1, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 5, 3, 1, 1, 0, 0, 0, 1, 7, 1, 1, 1, 0, 0, 0, 0, 1, 9, 5, 2, 0, 1, 0, 0, 0, 0, 1, 12, 5, 1, 1, 1, 0, 0, 0, 0, 0, 1, 16, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 21, 10, 3, 3, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 3

Stefano Spezia, Dec 05 2024

			The triangle begins as:
   1;
   1, 1;
   1, 0, 1;
   2, 1, 0, 1;
   2, 2, 0, 0, 1;
   3, 0, 1, 0, 0, 1;
   4, 2, 1, 0, 0, 0, 1;
   5, 3, 1, 1, 0, 0, 0, 1;
   7, 1, 1, 1, 0, 0, 0, 0, 1;
   9, 5, 2, 0, 1, 0, 0, 0, 0, 1;
  12, 5, 1, 1, 1, 0, 0, 0, 0, 0, 1;
  ...

Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, Generating Trees and Fibonacci Polyominoes, arXiv:2411.17812 [math.CO], 2024. See page 9.

Cf. A079957 (k=3), A182097 (k=2), A378704, A378706, A378707.

T[n_, k_]:=SeriesCoefficient[1/(1-Sum[x^((k+i)(k-i+1)/2), {i, k}]), {x, 0, n}]; Table[T[n, k], {n, 2, 14}, {k, 2,n}]//Flatten

T(n, k) = [x^n] 1/(1 - Sum_{i=1..k} x^((k+i)*(k-i+1)/2) ).

A099559 a(n) = Sum_{k=0..floor(n/5)} C(n-4k,k+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 14, 19, 25, 33, 44, 59, 79, 105, 139, 184, 244, 324, 430, 570, 755, 1000, 1325, 1756, 2327, 3083, 4084, 5410, 7167, 9495, 12579, 16664, 22075, 29243, 38739, 51319, 67984, 90060, 119304, 158044, 209364, 277349, 367410, 486715

Offset: 0

Paul Barry, Oct 22 2004

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Minerva Catral, P. L. Ford, P. E. Harris, S. J. Miller, et al. Legal Decompositions Arising from Non-positive Linear Recurrences, arXiv preprint arXiv:1606.09312 [math.CO], 2016. See Table 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-1).

Cf. A098578.

LinearRecurrence[{2,-1,0,0,1,-1},{0,1,2,3,4,5},50] (* Harvey P. Dale, Feb 20 2017 *)

a(n) = sum(k=0,n\5, binomial(n-4*k, k+1)); \\ Michel Marcus, Jul 11 2018

Partial sums of A003520 (with leading zero).
G.f.: x / ( (x-1)*(x^2-x+1)*(x^3+x^2-1) ).
a(n) = 2a(n-1)-a(n-2)+a(n-5)-a(n-6).
7*a(n) = A117373(n+2) -7 +10*b(n) +15*b(n-1) +9*b(n-2), where b(n) = A182097(n). - R. J. Mathar, Aug 07 2017
a(n) = A003520(n+4) -1. - R. J. Mathar, Aug 07 2017

Values from a(5) on corrected by R. J. Mathar, Jul 29 2008

A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 41, 88, 189, 406, 872, 1873, 4023, 8641, 18560, 39865, 85626, 183916, 395033, 848491, 1822473, 3914488, 8407925, 18059374, 38789712, 83316385, 178955183, 384377665, 825604416, 1773314929, 3808901426

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 07 2008

Essentially the same as A141015. - R. J. Mathar, Sep 14 2008

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

Subsequence of A000930.

Cf. A000931, A001906, A078027, A134816, A141015, A182097.

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x^2-x^3)/(1-x-2*x^2-x^3))); // G. C. Greubel, Jun 05 2018

(* b = A000931 *)
b[0]=0; b[1]=1; b[2]=1; b[n_]:= b[n]= b[n-2] + b[n-3];
a[1]=1; a[n_]:= a[n]= Sum[b[k]*a[n-k], {k,n-1}];
Table[a[n], {n, 35}]
(* or *)
LinearRecurrence[{1, 2, 1}, {1, 1, 2, 4}, 31] (* Georg Fischer, Mar 23 2019 *)

x='x+O('x^35); Vec(x*(1-x^2-x^3)/(1-x-2*x^2-x^3)) \\ G. C. Greubel, Jun 05 2018

a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1. [That is, b(n) = A000931(n+4) = A078027(n+6) = A134816(n) = A182097(n+1). - Petros Hadjicostas, Aug 09 2020]
From Colin Barker, Feb 01 2012: (Start)
a(n) = a(n-1) + 2*a(n-2) + a(n-3), n > 4.
G.f.: x*(1 - x^2 - x^3)/(1 - x - 2*x^2 - x^3). (End)
a(n) = A000930(2*n - 3) for n >= 3. - Georg Fischer, Mar 23 2019

A373740 Expansion of e.g.f. exp(x^2/2 * (1 + x)).

Original entry on oeis.org

1, 0, 1, 3, 3, 30, 105, 315, 2625, 11340, 57645, 467775, 2505195, 17027010, 142026885, 922296375, 7493911425, 65886420600, 503693415225, 4625660914875, 43369908657075, 379618464975750, 3824934458169825, 38406952928819475, 376103907454500225

Offset: 0

Seiichi Manyama, Jun 16 2024

nonn

Cf. A361567, A373741.

Cf. A182097.

a(n) = n!*sum(k=0, n\2, binomial(k, n-2*k)/(2^k*k!));

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(k,n-2*k)/(2^k * k!).

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

A236583 The number of tilings of an 8 X (3n) floor with 2 X 3 hexominoes.

Original entry on oeis.org

1, 1, 4, 11, 33, 96, 281, 821, 2400, 7015, 20505, 59936, 175193, 512089, 1496836, 4375251, 12788857, 37381824, 109267057, 319387565, 933569728, 2728823951, 7976351345, 23314871872, 68149361393

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

R. J. Mathar, Paving rectangular regions..., arXiv:1311.6135, Table 51.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], eq. (34).
Index entries for linear recurrences with constant coefficients, signature (3,0,-1,1).

Cf. A000079 (5 X n floor), A182097 (6 X n floor), A000244 (7 X n floor), A236584 (9 x 2n floor)

g := (-1+x)^2/(x^3-x^4+1-3*x) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

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

A236584 The number of tilings of a 9 X (2n) floor with 2 X 3 hexominoes.

Original entry on oeis.org

1, 1, 1, 5, 11, 19, 45, 105, 219, 475, 1061, 2313, 5027, 11035, 24173, 52793, 115499, 252827, 552981, 1209545, 2646419, 5789563, 12664925, 27706873, 60614235, 132602171, 290087749, 634616521, 1388325507, 3037181147

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

R. J. Mathar, Paving rectangular regions..., arXiv:1311.6135, Table 52.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], eq. (35).
Index entries for linear recurrences with constant coefficients, signature (2,-1,4,-2).

Cf. A000079 (5Xn floor), A182097 (6Xn floor), A000244 (7Xn floor), A236583 (8X3n floor)

g := (1-x)/(-4*x^3+1-2*x+x^2+2*x^4) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

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

A281791 Ways to tile a 5 X (2n+1) floor with tatami mats, including one monomer.

Original entry on oeis.org

3, 18, 10, 8, 18, 24, 32, 52, 68, 100, 142, 196, 280, 388, 542, 756, 1046, 1452, 2006, 2768, 3816, 5248, 7212, 9896, 13562, 18568, 25392, 34692, 47354, 64580, 88002, 119824, 163034, 221672, 301200, 409004, 555060, 752844, 1020550, 1382732, 1872520, 2534596, 3429206, 4637556, 6269070

Offset: 0

Yasutoshi Kohmoto, Jan 30 2017

Apart from a single 1 X 1 monomer, the area is tiled with 2 X 1 mats. No four mats are permitted to meet at a point.

			For n=0, the 5X1 floor allows the monomer to be placed at one of the two ends or in the middle: a(n=0)=3.

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

Cf. A271786 [3X(2n+1) floor]. 2nd column of A272474.

s1(n)=my(s); forstep(k=(n%4!=1),(n-1)\6,2, s+=((n+3)/4-k/2)*((n-1)/4-k/2)!/(k!*((n-1)/4-3*k/2)!)); 2*s
s3(n)=my(s); forstep(k=(n%4==1),(n-3)\6,2, s+=((n-3)/4-k/2)!/(k!*((n-3)/4-3*k/2)!)); 2*s
s5(n)=my(s); forstep(k=(n%4!=1),(n-5)\6,2, s+=((n+7)/4-k/2)*((n-5)/4-k/2)!/(k!*((n-5)/4-3*k/2)!)); 2*s
a(n)=s1(n) + s3(n) + s5(n) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = S_1(2n+1) + S_5(2n+1) + S_3(2n+1) for n>1 where
S_1(n) = 2* Sum_{k= 0<=k<=[(n-1)/6]} ((n+3)/4-1/2*k) *((n-1)/4-1/2*k)!/(k!*((n-1)/4-3/2*k)!). The sum is over even k if n==1 (mod 4), else over odd k.
S_5(n) = 2* Sum_{0<=k<=[(n-5)/6]} ((n+7)/4-1/2*k) *((n-5)/4-1/2*k)!/(k!*((n-5)/4-3/2*k)!). The sum is over even k if n==1 (mod 4) else over odd k.
S_3(n) = 2* Sum_{0<=k<=[(n-3)/6]} 2*((n-3)/4-1/2*k)!/(k!*((n-3)/4-3/2*k)!). The sum is over odd k if n==1 (mod 4), else over even k.
Where [m] is floor(m).
G.f. x +14*x^3 +2*x*(1 +2*x^2 +3*x^4 -2*x^6 -4*x^8 -2*x^10)/ (1-x^4-x^6)^2. (Includes zeros for even floor widths).- R. J. Mathar, Apr 10 2017
a(n) = 2*(A228577(n-1)+A228577(n+1))+4*(A182097(n-2)+A182097(n-1)), n>1. - R. J. Mathar, Apr 10 2017

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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A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A378716 Triangle read by rows: T(n,k) is the number of k-Fibonacci polyominoes with an area of n, with k > 1.

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A099559 a(n) = Sum_{k=0..floor(n/5)} C(n-4k,k+1).

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A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

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A373740 Expansion of e.g.f. exp(x^2/2 * (1 + x)).

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A236583 The number of tilings of an 8 X (3n) floor with 2 X 3 hexominoes.

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A236584 The number of tilings of a 9 X (2n) floor with 2 X 3 hexominoes.

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