A220122 -id:A220122 - cp's OEIS frontend

A017907 Expansion of 1/(1 - x^13 - x^14 - ...).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 19, 23, 28, 34, 41, 49, 58, 68, 79, 91, 104, 118, 134, 153, 176, 204, 238, 279, 328, 386, 454, 533, 624, 728, 846, 980, 1133, 1309

Offset: 0

N. J. A. Sloane

a(n) = number of compositions of n in which each part is >= 13. - Milan Janjic, Jun 28 2010

a(n+25) equals the number of binary words of length n having at least 12 zeros between every two successive ones. - Milan Janjic, Feb 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Column k=12 of A141539, k=13 of A220122. - Alois P. Heinz, Dec 09 2012

a:= n-> (Matrix(13, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$11, 1][i] else 0 fi)^n)[13,13]: seq(a(n), n=0..80); # Alois P. Heinz, Aug 04 2008

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
CoefficientList[Series[(x-1)/(x-1+x^13),{x,0,70}],x] (* Harvey P. Dale, Feb 07 2015 *)

Vec((x-1)/(x-1+x^13)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: (x-1)/(x-1+x^13). - Alois P. Heinz, Aug 04 2008
For positive integers n and k such that k <= n <= 13*k, and 12 divides n-k, define c(n,k) = binomial(k,(n-k)/12), and c(n,k) = 0, otherwise. Then, for n>=1,  a(n+13) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=0, a(n)=a(n-1)+a(n-13). - Harvey P. Dale, Feb 07 2015

A220123 Number of tilings of a 4 X n rectangle using integer-sided rectangular tiles of area 4.

Original entry on oeis.org

1, 1, 2, 3, 9, 16, 35, 65, 143, 281, 590, 1174, 2440, 4925, 10142, 20563, 42178, 85819, 175632, 357875, 731536, 1491966, 3047879, 6218844, 12699982, 25919176, 52922491, 108022099, 220541999, 450186874, 919074255, 1876149465, 3830134125, 7818778884, 15961716918

Offset: 0

Alois P. Heinz, Dec 05 2012

			a(3) = 3, because there are 3 tilings of a 4 X 3 rectangle using integer-sided rectangular tiles of area 4:
._._._.   ._.___.   .___._.
| | | |   | |   |   |   | |
| | | |   | |___|   |___| |
| | | |   | |   |   |   | |
|_|_|_|   |_|___|   |___|_|

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Caleb Wagner, Number of tilings of a 4 X n rectangle using integer sided rectangular tiles of area 4, Nov 2013
Index entries for linear recurrences with constant coefficients, signature (1,1,0,5,-1,1,0,-1).

Column k=4 of A220122. Cf. A005178.

gf:= -(x-1)*(x+1)*(x^2+1)/(x^8-x^6+x^5-5*x^4-x^2-x+1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: -(x-1)*(x+1)*(x^2+1) / (x^8 - x^6 + x^5 - 5*x^4 - x^2 - x + 1).
a(n) = a(n-1) + a(n-2) + 5*a(n-4) - a(n-5) + a(n-6) - a(n-8). - Caleb Wagner, Nov 06 2013
a(2*n+1) = Sum_{k=0..n} A005178(k+1)*a(2*n-2*k). - Shravan Haribalaraman, Aug 29 2022

A220124 Number of tilings of a 6 X n rectangle using integer-sided rectangular tiles of area 6.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 46, 88, 209, 473, 1002, 2120, 5197, 11085, 25384, 57234, 126959, 279883, 640387, 1412460, 3182794, 7138463, 15963680, 35593612, 80143244, 178644947, 400681480, 897313564, 2008904252, 4492651815, 10074017008, 22526314856, 50459193392

Offset: 0

Alois P. Heinz, Dec 05 2012

nonn

			a(4) = 7, because there are 7 tilings of a 6 X 4 rectangle using integer-sided rectangular tiles of area 6:
._._._._. ._._._._. ._._._._. ._._._._. ._._._._. ._._._._. ._._._._.
| | | | | |   | | | | |   | | | | |   | |   |   | |     | | | |     |
| | | | | |   | | | | |   | | | | |   | |   |   | |     | | | |     |
| | | | | |___| | | | |___| | | | |___| |___|___| |_____| | | |_____|
| | | | | |   | | | | |   | | | | |   | |   |   | |     | | | |     |
| | | | | |   | | | | |   | | | | |   | |   |   | |     | | | |     |
|_|_|_|_| |___|_|_| |_|___|_| |_|_|___| |___|___| |_____|_| |_|_____|

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=6 of A220122.

A220125 Number of tilings of an 8 X n rectangle using integer-sided rectangular tiles of area 8.

Original entry on oeis.org

1, 1, 2, 3, 9, 16, 35, 65, 250, 495, 1209, 2412, 6510, 13707, 32467, 68285, 176843, 387518, 926118, 2022259, 4928293, 11009067, 25938745, 57797488, 137808762, 311753120, 730878789, 1651849769, 3880235364, 8842052707, 20602970440, 46930887843, 109406749213

Offset: 0

Alois P. Heinz, Dec 06 2012

nonn

			a(3) = 3, because there are 3 tilings of an 8 X 3 rectangle using integer-sided rectangular tiles of area 8:
._._._.   .___._.   ._.___.
| | | |   |   | |   | |   |
| | | |   |   | |   | |   |
| | | |   |   | |   | |   |
| | | |   |___| |   | |___|
| | | |   |   | |   | |   |
| | | |   |   | |   | |   |
| | | |   |   | |   | |   |
|_|_|_|   |___|_|   |_|___|

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=8 of A220122.

A220126 Number of tilings of a 9 X n rectangle using integer-sided rectangular tiles of area 9.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 13, 37, 64, 95, 188, 326, 513, 875, 1449, 2302, 4289, 7422, 12072, 21561, 37559, 62571, 107806, 184027, 306628, 532661, 915705, 1537768, 2654311, 4574383, 7752577, 13292546, 22778306, 38625110, 66213867, 113475693, 192821642, 330086669

Offset: 0

Alois P. Heinz, Dec 06 2012

nonn

			a(5) = 4, because there are 4 tilings of a 9 X 5 rectangle using integer-sided rectangular tiles of area 9:
._._._._._.  ._____._._.  ._._____._.  ._._._____.
| | | | | |  |     | | |  | |     | |  | | |     |
| | | | | |  |     | | |  | |     | |  | | |     |
| | | | | |  |_____| | |  | |_____| |  | | |_____|
| | | | | |  |     | | |  | |     | |  | | |     |
| | | | | |  |     | | |  | |     | |  | | |     |
| | | | | |  |_____| | |  | |_____| |  | | |_____|
| | | | | |  |     | | |  | |     | |  | | |     |
| | | | | |  |     | | |  | |     | |  | | |     |
|_|_|_|_|_|  |_____|_|_|  |_|_____|_|  |_|_|_____|

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=9 of A220122.

A220127 Number of tilings of a 10 X n rectangle using integer-sided rectangular tiles of area 10.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 254, 470, 1130, 2150, 4369, 8889, 17112, 33235, 62619, 117687, 259213, 500978, 1073742, 2114851, 4334588, 8817832, 17673956, 35420952, 69871967, 137274169, 281650158, 556043329, 1139084899, 2264971273, 4589226181, 9256382600

Offset: 0

Alois P. Heinz, Dec 06 2012

nonn

			a(4) = 5, because there are 5 tilings of a 10 X 4 rectangle using integer-sided rectangular tiles of area 10:
._._._._.   .___._._.   ._.___._.   ._._.___.   .___.___.
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
| | | | |   |___| | |   | |___| |   | | |___|   |___|___|
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
| | | | |   |   | | |   | |   | |   | | |   |   |   |   |
|_|_|_|_|   |___|_|_|   |_|___|_|   |_|_|___|   |___|___|

Alois P. Heinz, Table of n, a(n) for n = 0..100

Column k=10 of A220122.

A220128 1 followed by period 6: (1, 3, 2, 3, 1, 4) repeated; offset 0.

Original entry on oeis.org

1, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3

Offset: 0

Alois P. Heinz, Dec 06 2012

Also the number of tilings of an n X 3 rectangle using integer-sided rectangular tiles of area n.

Also decimal expansion of 12443/109890 = 0.1132314132314... .

			a(6) = 4, because there are 4 tilings of a 6 X 3 rectangle using integer-sided rectangular tiles of area 6:
._._._.  .___._.  ._.___.  ._____.
| | | |  |   | |  | |   |  |     |
| | | |  |   | |  | |   |  |_____|
| | | |  |___| |  | |___|  |     |
| | | |  |   | |  | |   |  |_____|
| | | |  |   | |  | |   |  |     |
|_|_|_|  |___|_|  |_|___|  |_____|

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Row n=3 of A220122.

[1] cat &cat [[1, 3, 2, 3, 1, 4]^^20]; // Wesley Ivan Hurt, Jun 20 2016

a:=n-> `if`(n=0, 1, [4, 1, 3, 2, 3, 1][irem(n, 6)+1]): seq(a(n), n=0..100);

PadRight[{1}, 120, {4,1,3,2,3,1}] (* Harvey P. Dale, Jan 06 2016 *)

G.f.: (-3*x^4-4*x^3-4*x^2-2*x-1) / (x^4+x^3-x-1).
From Wesley Ivan Hurt, Jun 20 2016: (Start)
a(n) + a(n-1) = a(n-3) + a(n-4) for n>4.
a(0) = 1, a(n) = (7 + 3*cos(n*Pi) + 2*cos(2*n*Pi/3))/3 for n>0. (End)
E.g.f.: 2*(-9/2 + cos(sqrt(3)*x/2)*exp(-x/2) + 2*sinh(x) + 5*cosh(x))/3. - Ilya Gutkovskiy, Jun 21 2016

A220129 1 followed by period 12: (1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11) repeated; offset 0.

Original entry on oeis.org

1, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11

Offset: 0

Alois P. Heinz, Dec 06 2012

Also the number of tilings of an n X 4 rectangle using integer-sided rectangular tiles of area n.

			a(6) = 7, because there are 7 tilings of a 6 X 4 rectangle using integer-sided rectangular tiles of area 6:
._._._._.  ._._._._.  ._._____.  .___._._.
| | | | |  |     | |  | |     |  |   | | |
| | | | |  |_____| |  | |_____|  |   | | |
| | | | |  |     | |  | |     |  |___| | |
| | | | |  |_____| |  | |_____|  |   | | |
| | | | |  |     | |  | |     |  |   | | |
|_|_|_|_|  |_____|_|  |_|_____|  |___|_|_|
._.___._.  ._._.___.  .___.___.
| |   | |  | | |   |  |   |   |
| |   | |  | | |   |  |   |   |
| |___| |  | | |___|  |___|___|
| |   | |  | | |   |  |   |   |
| |   | |  | | |   |  |   |   |
|_|___|_|  |_|_|___|  |___|___|

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=4 of A220122.

a:= n-> `if`(n=0, 1, [11, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1][irem(n, 12)+1]):
seq(a(n), n=0..100);

PadRight[{1},120,{11,1,5,3,9,1,7,1,9,3,5,1}] (* Harvey P. Dale, Aug 24 2025 *)

G.f.: -(10*x^6+11*x^5+16*x^4+9*x^3+7*x^2+2*x+1) / (x^6+x^5+x^4-x^2-x-1).

A220130 Number of tilings of an n X 5 rectangle using integer-sided rectangular tiles of area n.

Original entry on oeis.org

1, 1, 8, 4, 16, 2, 13, 1, 16, 4, 9, 1, 21, 1, 8, 5, 16, 1, 13, 1, 17, 4, 8, 1, 21, 2, 8, 4, 16, 1, 14, 1, 16, 4, 8, 2, 21, 1, 8, 4, 17, 1, 13, 1, 16, 5, 8, 1, 21, 1, 9, 4, 16, 1, 13, 2, 16, 4, 8, 1, 22, 1, 8, 4, 16, 2, 13, 1, 16, 4, 9, 1, 21, 1, 8, 5, 16, 1

Offset: 0

Alois P. Heinz, Dec 06 2012

1 followed by period 60: (1, 8, ..., 22) repeated; offset 0.

			a(3) = 4, because there are 4 tilings of a 3 X 5 rectangle using integer-sided rectangular tiles of area 3:
._._._._._.   ._____._._.   ._._____._.   ._._._____.
| | | | | |   |_____| | |   | |_____| |   | | |_____|
| | | | | |   |_____| | |   | |_____| |   | | |_____|
|_|_|_|_|_|   |_____|_|_|   |_|_____|_|   |_|_|_____|

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=5 of A220122.

gf:= -(21*x^12 +22*x^11 +51*x^10 +56*x^9 +80*x^8 +65*x^7 +72*x^6 +45*x^5 +40*x^4 +16*x^3 +11*x^2 +2*x +1) / (x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 -x^5 -2*x^4 -2*x^3 -2*x^2 -x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);

G.f.: see Maple program.

A220131 Number of tilings of an n X 6 rectangle using integer-sided rectangular tiles of area n.

Original entry on oeis.org

1, 1, 13, 6, 35, 3, 46, 1, 35, 6, 15, 1, 88, 1, 13, 8, 35, 1, 46, 1, 37, 6, 13, 1, 88, 3, 13, 6, 35, 1, 48, 1, 35, 6, 13, 3, 88, 1, 13, 6, 37, 1, 46, 1, 35, 8, 13, 1, 88, 1, 15, 6, 35, 1, 46, 3, 35, 6, 13, 1, 90, 1, 13, 6, 35, 3, 46, 1, 35, 6, 15, 1, 88, 1, 13

Offset: 0

Alois P. Heinz, Dec 06 2012

1 followed by period 60: (1, 13, ..., 90) repeated; offset 0.

			a(3) = 6, because there are 6 tilings of a 3 X 6 rectangle using integer-sided rectangular tiles of area 3:
._._._._._._.   ._____._._._.   ._._____._._.
| | | | | | |   |_____| | | |   | |_____| | |
| | | | | | |   |_____| | | |   | |_____| | |
|_|_|_|_|_|_|   |_____|_|_|_|   |_|_____|_|_|
._._._____._.   ._._._._____.   ._____._____.
| | |_____| |   | | | |_____|   |_____|_____|
| | |_____| |   | | | |_____|   |_____|_____|
|_|_|_____|_|   |_|_|_|_____|   |_____|_____|

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=6 of A220122.

gf:= -(89*x^16 +90*x^15 +103*x^14 +109*x^13 +144*x^12 +58*x^11 +103*x^10 +91*x^9 +120*x^8 +91*x^7 +103*x^6 +58*x^5 +56*x^4 +21*x^3 +15*x^2 +2*x +1) / (x^16 +x^15 +x^14 +x^13 +x^12 -x^4 -x^3 -x^2 -x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);

G.f.: see Maple program.

cp's OEIS Frontend

A017907 Expansion of 1/(1 - x^13 - x^14 - ...).

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A220123 Number of tilings of a 4 X n rectangle using integer-sided rectangular tiles of area 4.

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A220124 Number of tilings of a 6 X n rectangle using integer-sided rectangular tiles of area 6.

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A220125 Number of tilings of an 8 X n rectangle using integer-sided rectangular tiles of area 8.

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A220126 Number of tilings of a 9 X n rectangle using integer-sided rectangular tiles of area 9.

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A220127 Number of tilings of a 10 X n rectangle using integer-sided rectangular tiles of area 10.

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A220128 1 followed by period 6: (1, 3, 2, 3, 1, 4) repeated; offset 0.

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A220129 1 followed by period 12: (1, 5, 3, 9, 1, 7, 1, 9, 3, 5, 1, 11) repeated; offset 0.

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A220130 Number of tilings of an n X 5 rectangle using integer-sided rectangular tiles of area n.

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