A003520 -id:A003520 - cp's OEIS frontend

A017899 Expansion of 1/(1 -x^5 -x^6 -x^7 - ...).

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, 140, 185, 245, 325, 431, 571, 756, 1001, 1326, 1757, 2328, 3084, 4085, 5411, 7168, 9496, 12580, 16665, 22076, 29244, 38740, 51320, 67985, 90061, 119305, 158045, 209365, 277350

Offset: 0

N. J. A. Sloane

a(n) is the number of compositions of n into parts >=5. - Joerg Arndt, Jun 22 2011
a(n+5) equals the number of binary words such that 0 appears only in runs whose lengths are a multiple of 5. - Milan Janjic, Feb 17 2015
a(n-5) equals the number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 2 or 3. - Ethan Patrick White, Jun 24 2020

Alois P. Heinz, Table of n, a(n) for n = 0..1000
J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
J. D. Opdyke, A unified approach to algorithms generating unrestricted.., J. Math. Model. Algor. 9 (2010) 53-97.
Ethan P. White, Richard K. Guy, Renate Scheidler, Difference Necklaces, arXiv:2006.15250 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1).

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904.

Apart from initial terms, same as A003520.

f := proc(r) local t1,i; t1 := []; for i from 1 to r do t1 := [op(t1),0]; od: for i from 1 to r+1 do t1 := [op(t1),1]; od: for i from 2*r+2 to 50 do t1 := [op(t1),t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order
a:= n-> (Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$3, 1][i] else 0 fi)^n)[5,5]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 04 2008

CoefficientList[ Series[(1 - x)/(1 - x - x^5), {x, 0, 50}], x] (* Adi Dani, Jun 25 2011 *)
LinearRecurrence[{1,0,0,0,1},{1,0,0,0,0},60] (* Harvey P. Dale, Jun 07 2015 *)

Vec((1-x)/(1-x-x^5)+O(x^99)) \\ Charles R Greathouse IV, Jun 21 2011

G.f.: (1-x)/(1-x-x^5) = 1/(1-Sum_{k>=5} x^k).

For positive integers n and k such that k <= n <= 5*k, and 4 divides n-k, define c(n,k) = binomial(k,(n-k)/4), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+5) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011

A264878 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,1 0,-1 -2,0 or 1,1.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 5, 0, 3, 0, 7, 4, 20, 0, 4, 1, 20, 65, 12, 56, 0, 5, 0, 49, 228, 572, 36, 137, 0, 6, 1, 175, 1101, 2348, 3613, 108, 295, 0, 8, 0, 323, 4832, 22152, 22400, 19372, 324, 709, 0, 11, 1, 1085, 18501, 129230, 356692, 207424, 103585, 972, 1983, 0

Offset: 1

R. H. Hardin, Nov 27 2015

Table starts
..1.0....1....0........1..........0............1..............0
..1.0....1....1........7.........20...........49............175
..1.0....5....4.......65........228.........1101...........4832
..2.0...20...12......572.......2348........22152.........129230
..3.0...56...36.....3613......22400.......356692........3303808
..4.0..137..108....19372.....207424......4747695.......78535556
..5.0..295..324...103585....1946752.....68488297.....1924357508
..6.0..709..972...629654...18265856...1050281271....47123513432
..8.0.1983.2916..3930725..171168256..16268725036..1152731721920
.11.0.5280.8748.23940621.1602206720.247512489984.28078658475952

			Some solutions for n=4 k=4
..1..2..3..4.14...10..2.12..4.14....1..2.12..4.14...10..2..3..4.14
.15..0..8..9.19....6..0..1..9..3...15..0..8..9..3....6..0..1..9.19
.20..5..6..7.24...20..5.22..7..8...20..5..6..7.24...20..5.22..7..8
.16.10.11.12.13...16.17.11.19.13...16.10.11.19.13...16.17.11.12.13
.21.22.23.17.18...21.15.23.24.18...21.22.23.17.18...21.15.23.24.18

R. H. Hardin, Table of n, a(n) for n = 1..179

Column 1 is A003520(n+1).

Column 4 is A003946(n-2).

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-5)
k=2: a(n) = a(n-1)
k=3: [order 45]
k=4: a(n) = 3*a(n-1) for n>3
Empirical for row n:
n=1: a(n) = a(n-2)
n=2: [order 20]
n=3: [order 46]

A278657 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape and monominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 2, 25, 50, 25, 2, 1, 1, 3, 50, 311, 311, 50, 3, 1, 1, 4, 155, 1954, 4101, 1954, 155, 4, 1, 1, 5, 508, 11914, 56864, 56864, 11914, 508, 5, 1, 1, 6, 1343, 76003, 728857, 1532496, 728857, 76003, 1343, 6, 1

Offset: 0

Alois P. Heinz, Nov 25 2016

			A(2,3) = A(3,2) = 7:
  .___.  .___.  .___.  .___.  .___.  .___.  .___.
  |_|_|  |   |  |   |  | |_|  |_| |  | ._|  |_. |
  |_|_|  | ._|  |_. |  |   |  |   |  | |_|  |_| |
  |_|_|  |_|_|  |_|_|  |___|  |___|  |___|  |___| .
.
Square array A(n,k) begins:
  1, 1,   1,     1,      1,        1,          1, ...
  1, 1,   1,     1,      1,        2,          3, ...
  1, 1,   1,     7,     25,       50,        155, ...
  1, 1,   7,    50,    311,     1954,      11914, ...
  1, 1,  25,   311,   4101,    56864,     728857, ...
  1, 2,  50,  1954,  56864,  1532496,   42238426, ...
  1, 3, 155, 11914, 728857, 42238426, 2492016728, ...

Liang Kai, Antidiagonals n = 0..23, flattened (Antidiagonals n = 0..15 from Alois P. Heinz)
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Pentomino

Columns (or rows) k=0-7 give: A000012, A003520, A278874, A278875, A278876, A278456, A278877, A278878.

Cf. A174249, A233427.

A014097 a(n) = a(n-1)+a(n-4).

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 8, 13, 19, 26, 34, 47, 66, 92, 126, 173, 239, 331, 457, 630, 869, 1200, 1657, 2287, 3156, 4356, 6013, 8300, 11456, 15812, 21825, 30125, 41581, 57393, 79218, 109343, 150924, 208317, 287535

Offset: 1

David Broadhurst

Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 4 sites wide.

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

Indranil Ghosh, Table of n, a(n) for n = 1..7130
D. J. Broadhurst, Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, arXiv:hep-th/9612012, 1996.
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Cf. A020999, A000204, A001609, A000079, A003269, A003520, A005708, A005709, A005710.

LinearRecurrence[{1,0,0,1},{1,1,1,5},40] (* Harvey P. Dale, Mar 06 2016 *)

a(n):=sum(binomial(n-3*j,n-4*j)*n/(n-3*j),j,0,(n-1)/3); /* Vladimir Kruchinin, Mar 25 2016 */

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,1]^(n-1)*[1;1;1;5])[1,1] \\ Charles R Greathouse IV, Sep 09 2016

G.f.: -x*(1+4*x^3)/(-1+x+x^4). a(n)= 4*A003269(n)-3*A003269(n-1). - R. J. Mathar, Nov 16 2007
a(n) = Sum_{j=0..(n-1)/3}(binomial(n-3*j,n-4*j)*n/(n-3*j)). - Vladimir Kruchinin, Mar 25 2016
From Greg Dresden, Aug 23 2019: (Start)
a(n) = r1^n + r2^n + r3^n + r4^n, where {r1,r2,r3,r4} are the four roots of x^4-x^3-1=0, see A086106, A230151.
a(n) = round(r^n) for n>21 and r the positive real root of x^4-x^3-1.
(End)

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

A079675 a(1)=1; a(n)=sum(u=1,n-1,sum(v=1,u,sum(w=1,v,sum(x=1, w,sum(y=1,x,a(y)))))).

Original entry on oeis.org

1, 1, 6, 26, 106, 431, 1757, 7168, 29244, 119305, 486716, 1985603, 8100456, 33046585, 134816705, 549997641, 2243767969, 9153665985, 37343255690, 152345382480, 621507555626, 2535499503900, 10343812679475, 42198572937400

Offset: 1

Benoit Cloitre, Jan 26 2003

Row sums of Riordan array (1,1/(1-x)^5). A quintisection of A003520. - Paul Barry, Feb 02 2006

Michael De Vlieger, Table of n, a(n) for n = 1..1639
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 16.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (6,-10,10,-5,1).

Cf. A055991, A052529, A001906.

LinearRecurrence[{6,-10,10,-5,1},{1,1,6,26,106,431},40] (* Harvey P. Dale, Aug 21 2017 *)

a(1)=1, a(2)=1, a(3)=6, a(4)=26, a(5)=106, a(6)=431; for n>=7, a(n)=5*u(n-1)-4*u(n-2)+u(n-3)+b(n) where b(n) is the 6 periodic sequence (0, 1, 1, 0, -1, -1)

G.f.: (1-x)^5/((1-x)^5-x); a(n)=sum{k=0..n, binomial(5n-4k-1,k)}; - Paul Barry, Feb 02 2006

A058368 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 5 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 6, 7, 8, 9, 10, 16, 23, 31, 40, 50, 66, 89, 120, 160, 210, 276, 365, 485, 645, 855, 1131, 1496, 1981, 2626, 3481, 4612, 6108, 8089, 10715, 14196, 18808, 24916, 33005, 43720, 57916, 76724, 101640, 134645, 178365, 236281, 313005, 414645

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(5) = 6 because there is one way to put zero molecule to the necklace and 5 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1).

Cf. A000204, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710.

LinearRecurrence[{1,0,0,0,1},{1,1,1,1,6},50] (* Harvey P. Dale, Aug 14 2020 *)

a(n) = 1 + n*Sum_{i=1..n/5} binomial(n-4*i-1, i-1)/i.
a(n) = a(n-1) + a(n-5) for n >= 6.
G.f.: (x+5*x^5)/(1-x-x^5).

A058364 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 9 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 39, 51, 64, 78, 93, 109, 126, 144, 172, 211, 262, 326, 404, 497, 606, 732, 876, 1048, 1259, 1521, 1847, 2251, 2748, 3354, 4086, 4962, 6010, 7269, 8790, 10637, 12888, 15636, 18990, 23076, 28038

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(9) = 10 because there is one way to put zero molecule to the necklace and 9 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-8*i, i-1)/i, i=1..n/9). a(n) = a(n-1) + a(n-9), a(n) = 1 for n = 1..8, a(9) = 10. generating function = (x+9*x^9)/(1-x-x^9).

A058365 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 8 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 10, 11, 12, 13, 14, 15, 16, 25, 35, 46, 58, 71, 85, 100, 116, 141, 176, 222, 280, 351, 436, 536, 652, 793, 969, 1191, 1471, 1822, 2258, 2794, 3446, 4239, 5208, 6399, 7870, 9692, 11950, 14744, 18190, 22429, 27637, 34036, 41906

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(8) = 9 because there is one way to put zero molecule to the necklace and 8 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-7*i, i-1)/i, i=1..n/8). a(n) = a(n-1) + a(n-8), a(n) = 1 for n = 1..7, a(8) = 9. generating function = (x+8*x^8)/(1-x-x^8).

A058366 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 7 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 9, 10, 11, 12, 13, 14, 22, 31, 41, 52, 64, 77, 91, 113, 144, 185, 237, 301, 378, 469, 582, 726, 911, 1148, 1449, 1827, 2296, 2878, 3604, 4515, 5663, 7112, 8939, 11235, 14113, 17717, 22232, 27895, 35007, 43946, 55181, 69294, 87011

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(7) = 8 because there is one way to put zero molecule to the necklace and 7 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-6*i, i-1)/i, i=1..n/7). a(n) = a(n-1) + a(n-7), a(n) = 1 for n = 1..6, a(7) = 8. generating function = (x+7*x^7)/(1-x-x^7).

A058367 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 6 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 8, 9, 10, 11, 12, 19, 27, 36, 46, 57, 69, 88, 115, 151, 197, 254, 323, 411, 526, 677, 874, 1128, 1451, 1862, 2388, 3065, 3939, 5067, 6518, 8380, 10768, 13833, 17772, 22839, 29357, 37737, 48505, 62338, 80110, 102949, 132306, 170043, 218548

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(6) = 7 because there is one way to put zero molecule to the necklace and 6 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-5*i, i-1)/i, i=1..n/6). a(n) = a(n-1) + a(n-6), a(n) = 1 for n = 1..5, a(6) = 7. generating function = (x+6*x^6)/(1-x-x^6).

cp's OEIS Frontend

A017899 Expansion of 1/(1 -x^5 -x^6 -x^7 - ...).

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A264878 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,1 0,-1 -2,0 or 1,1.

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A278657 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape and monominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A014097 a(n) = a(n-1)+a(n-4).

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A079675 a(1)=1; a(n)=sum(u=1,n-1,sum(v=1,u,sum(w=1,v,sum(x=1, w,sum(y=1,x,a(y)))))).

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A058368 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 5 sites wide.

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A058364 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 9 sites wide.

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A058365 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 8 sites wide.

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