A005708 -id:A005708 - cp's OEIS frontend

A373958 Number of compositions of 6*n-3 into parts 1 and 6.

1, 5, 21, 92, 414, 1869, 8427, 37975, 171121, 771119, 3474913, 15659094, 70564951, 317988473, 1432958824, 6457375642, 29099021980, 131129599227, 590912361256, 2662842109828, 11999627299824, 54074199444301, 243675821963849, 1098082020999797, 4948312537227216

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (7,-15,20,-15,6,-1).

Cf. A099242, A371125, A373302, A373959, A373960.

Cf. A005708.

LinearRecurrence[{7,-15,20,-15,6,-1},{1,5,21,92,414,1869},30] (* Harvey P. Dale, Nov 04 2024 *)

a(n) = sum(k=0, n, binomial(n+2+5*k, n-1-k));

a(n) = A005708(6*n-3).
a(n) = Sum_{k=0..n} binomial(n+2+5*k,n-1-k).
a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1-x)^2/((1-x)^6 - x).
a(n) = A373302(n) - A373302(n-1).

A373959 Number of compositions of 6*n-4 into parts 1 and 6.

Original entry on oeis.org

1, 4, 16, 71, 322, 1455, 6558, 29548, 133146, 599998, 2703794, 12184181, 54905857, 247423522, 1114970351, 5024416818, 22641646338, 102030577247, 459782762029, 2071929748572, 9336785189996, 42074572144477, 189601622519548, 854406199035948, 3850230516227419

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (7,-15,20,-15,6,-1).

Cf. A099242, A371125, A373302, A373958, A373960.

Cf. A005708.

a(n) = sum(k=0, n, binomial(n+1+5*k, n-1-k));

a(n) = A005708(6*n-4).
a(n) = Sum_{k=0..n} binomial(n+1+5*k,n-1-k).
a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1-x)^3/((1-x)^6 - x).
a(n) = A373958(n) - A373958(n-1).

A373960 Number of compositions of 6*n-5 into parts 1 and 6.

Original entry on oeis.org

1, 3, 12, 55, 251, 1133, 5103, 22990, 103598, 466852, 2103796, 9480387, 42721676, 192517665, 867546829, 3909446467, 17617229520, 79388930909, 357752184782, 1612146986543, 7264855441424, 32737786954481, 147527050375071, 664804576516400, 2995824317191471

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (7,-15,20,-15,6,-1).

Cf. A099242, A371125, A373302, A373958, A373959.

Cf. A005708.

a(n) = sum(k=0, n, binomial(n+5*k, n-1-k));

a(n) = A005708(6*n-5).
a(n) = Sum_{k=0..n} binomial(n+5*k,n-1-k).
a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1-x)^4/((1-x)^6 - x).
a(n) = A373959(n) - A373959(n-1).

A143284 Number of binary words of length n containing at least one subword 100001 and no subwords 10^{i}1 with i<4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 11, 17, 25, 35, 48, 66, 92, 129, 180, 249, 342, 468, 640, 875, 1195, 1629, 2216, 3009, 4080, 5526, 7477, 10107, 13649, 18415, 24823, 33433, 44995, 60513, 81330, 109241, 146644, 196742, 263813, 353570, 473640, 634201

Offset: 0

Alois P. Heinz, Aug 04 2008

			a(7)=2 because 2 binary words of length 7 have at least one subword 100001 and no subwords 10^{i}1 with i<4: 0100001, 1000010.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,0,-1,0,0,0,-1).

Cf. A003520, A005708, 4th column of A143291.

[n le 6 select 0 else n le 11 select n-6 else 2*Self(n-1)-Self(n-2) +Self(n-5)-Self(n-7)-Self(n-11): n in [1..60]]; // Vincenzo Librandi, Jun 05 2013

a:= n-> coeff(series(x^6/((x^5+x-1)*(x^6+x-1)), x, n+1), x, n):
seq(a(n), n=0..60);

CoefficientList[Series[x^6 / ((x^5 + x - 1) (x^6 + x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 04 2013 *)

G.f.: x^6/((x^5+x-1)*(x^6+x-1)).
a(n) = A003520(n+4) - A005708(n+5).
a(n) = 2*a(n-1)-a(n-2)+a(n-5)-a(n-7)-a(n-11). - Vincenzo Librandi, Jun 05 2013

A143285 Number of binary words of length n containing at least one subword 1000001 and no subwords 10^{i}1 with i<5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 12, 18, 26, 36, 48, 63, 83, 111, 150, 203, 273, 364, 482, 636, 839, 1108, 1464, 1933, 2548, 3352, 4402, 5774, 7568, 9914, 12980, 16983, 22204, 29008, 37870, 49408, 64425, 83963, 109373, 142406, 185331, 241088, 313486

Offset: 0

Alois P. Heinz, Aug 04 2008

			a(8)=2 because 2 binary words of length 8 have at least one subword 1000001 and no subwords 10^{i}1 with i<5: 01000001, 10000010.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,0,-1,0,0,0,0,-1).

Cf. A005708, A005709, 5th column of A143291.

[n le 7 select 0 else n le 13 select n-7 else 2*Self(n-1)-Self(n-2) +Self(n-6)-Self(n-8)-Self(n-13): n in [1..60]]; // Vincenzo Librandi, Jun 05 2013

a:= n-> coeff(series(x^7/((x^6+x-1)*(x^7+x-1)), x, n+1), x, n):
seq(a(n), n=0..60);

CoefficientList[Series[x^7 / ((x^6 + x - 1) (x^7 + x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 04 2013 *)

G.f.: x^7/((x^6+x-1)*(x^7+x-1)).
a(n) = A005708(n+5) - A005709(n+6).
a(n) = 2*a(n-1) -a(n-2) +a(n-6) -a(n-8) -a(n-13). - Vincenzo Librandi, Jun 05 2013

A224812 Number of subsets of {1,2,...,n-10} without differences equal to 2, 4, 6, 8 or 10.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 63, 81, 108, 144, 192, 256, 336, 441, 567, 729, 918, 1156, 1462, 1849, 2365, 3025, 3905, 5041, 6532, 8464, 10948, 14161, 18207, 23409, 29988, 38416, 49196, 63001, 80822, 103684, 133308, 171396, 220662, 284089, 365638, 470596, 605052, 777924, 999306, 1283689, 1648515

Offset: 0

Vladimir Baltic, May 18 2013

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=2, r=10, I={-2,0,10}.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-13
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, 3, -2, 2, -1, 1, 0, 0, -2, 1, -2, 0, 0, -3, 1, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224815.

CoefficientList[Series[-(x + 1)*(x^23 - x^22 + x^21 - x^20 + x^19 - x^13 + x^12 - 3*x^11 + 3*x^10 - 3*x^9 + 2*x^8 - 2*x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x^6 + x - 1)*(x^30 + x^24 - 2*x^20 - 2*x^18 - x^14 - 2*x^12 + x^10 + x^8 + x^6 + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 28 2017 *)

x='x+O('x^50); Vec(-(x + 1)*(x^23 - x^22 + x^21 - x^20 + x^19 - x^13 + x^12 - 3*x^11 + 3*x^10 - 3*x^9 + 2*x^8 - 2*x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x^6 + x - 1)*(x^30 + x^24 - 2*x^20 - 2*x^18 - x^14 - 2*x^12 + x^10 + x^8 + x^6 + 1))) \\ G. C. Greubel, Oct 28 2017

a(n) = a(n-1) +a(n-7) -a(n-8) +a(n-9) -a(n-10) +a(n-11) +3*a(n-12) -2*a(n-13) +2*a(n-14) -a(n-15) +a(n-16) -2*a(n-19) +a(n-20) -2*a(n-21) -3*a(n-24) +a(n-25) -2*a(n-26) +a(n-31) +a(n-36).
G.f.: -(x+1) *(x^23 -x^22 +x^21 -x^20 +x^19 -x^13 +x^12 -3*x^11 +3*x^10 -3*x^9 +2*x^8 -2*x^7 +x^6 -x^5 +x^4 -x^3 +x^2 -x +1)/ ((x^6 +x -1) *(x^30 +x^24 -2*x^20 -2*x^18 -x^14 -2*x^12 +x^10 +x^8 +x^6+1) ).
a(2*k) = (A005708(k))^2, a(2*k+1) = A005708(k) * A005708(k+1).

A329146 Triangle read by rows: T(n,k) is the number of subsets of {1,...,n} such that the difference between any two elements is k or greater, 1 <= k <= n.

Original entry on oeis.org

2, 4, 3, 8, 5, 4, 16, 8, 6, 5, 32, 13, 9, 7, 6, 64, 21, 13, 10, 8, 7, 128, 34, 19, 14, 11, 9, 8, 256, 55, 28, 19, 15, 12, 10, 9, 512, 89, 41, 26, 20, 16, 13, 11, 10, 1024, 144, 60, 36, 26, 21, 17, 14, 12, 11, 2048, 233, 88, 50, 34, 27, 22, 18, 15, 13, 12, 4096, 377, 129

Offset: 1

Gerhard Kirchner, Nov 06 2019

The restriction "the difference between any two elements is k or greater" does not apply to subsets with fewer than two elements.
Therefore T(n,k) = n+1 is valid not only for n=k, but also for n < k. These terms do not occur in the triangular matrix, but they help to simplify formula(3).
T(n,k) is, for 1 <= k <= 16, a subsequence of another sequence:
T(n,1) =  A000079(n)
T(n,2) =  A000045(n+2)
T(n,3) =  A000930(n+2)
T(n,4) =  A003269(n+4)
T(n,5) =  A003520(n+4)
T(n,6) =  A005708(n+5)
T(n,7) =  A005709(n+6)
T(n,8) =  A005710(n+7)
T(n,9) =  A005711(n+7)
T(n,10) = A017904(n+19)
T(n,11) = A017905(n+21)
T(n,12) = A017906(n+23)
T(n,13) = A017907(n+25)
T(n,14) = A017908(n+27)
T(n,15) = A017909(n+29)
T(n,16) = A291149(n+16)
Note the recurrence formula(3) below: T(n,k) = T(n-1,k) + T(n-k,k), n >= 2*k.
As to the corresponding recurrence A..(n) = A..(n-1) + A..(n-k), see definition (1 <= k <= 9) or formula (k=13) or b-files in the remaining cases.

			a(1) = T(1,1) = |{}, {1}| = 2
a(2) = T(2,1) = |{}, {1}, {2}, {1,2}| = 4
a(3) = T(2,2) = |{}, {1}, {2}| = 3
a(4) = T(3,1) = |{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}| = 8
a(5) = T(3,2) = |{}, {1}, {2}, {3}, {1,3}| = 5
etc.
The triangle begins:
   2;
   4,  3;
   8,  5,  4;
  16,  8,  6,  5;
  32, 13,  9,  7,  6;
  ...

Gerhard Kirchner, First 141 rows of T(n,k): Table of n, a(n) for n = 1..10011

Cf. A000079, A000045, A000930, A003269, A003520, A005708, A005709, A005710.

Cf. A005711, A017904, A017905, A017906, A017907, A017908, A017909, A291149.

T(n,k) = sum(j=0, ceil(n/k), binomial(n-(k-1)*(j-1), j)); \\ Andrew Howroyd, Nov 06 2019

Let g(n,k,j) be the number of subsets of {1,...,n} with j elements such that the difference between any two elements is k or greater. Then
(1) T(n,k) = Sum_{j = 0..n} g(n,k,j)
(2) g(n,k,j) = binomial(n-(k-1)*(j-1),j) with the convention binomial(m,j)=0 for j > m
(3) T(n,k) = T(n-1,k) + T(n-k,k), n >= 2*k
or: T(n,k) = n+1 for n <= k and T(n,k) = T(n-1,k) + T(n-k,k) for n > k (see comments).
Formula(1) is evident.
Proof of formula(2):
Let C(n,k,j) be the class of subsets of {1,...,n} with j elements such that the difference between any two elements is k or greater. Let S be one of these subsets and let us write it as a j-tuple (c(1),..,c(j)) with c(i+1)-c(i)>=k, 1<=iIn particular, the number of subsets of C(m,1,j) is binomial(m,j), the basic tuple is (1,...,j) and the generating tuple is (d(1),...,d(j)) with 0 <= d(1) <= ... <= d(j) <= m-j.
With m-j = n-(j-1)*k-1, i.e., m = n-(j-1)*(k-1), the numbers of subsets in C(n,k,j) and C(m,1,j) are equal: g(n,k,j) = binomial(n-(k-1)*(j-1),j) qed
Proof of formula(3):
Using the binomial recurrence binomial(m,j) = binomial(m-1,j) + binomial(m-1,j-1) for m = n-(j-1)*(k-1), we find:
T(n,k) = Sum_{j = 0..n} binomial(n-(k-1)*(j-1),j)
       = Sum_{j = 0..n-1} binomial(n-1-(k-1)*(j-1),j)
          + Sum_{j = 1..n} binomial(n-1-(k-1)*(j-1),j-1)
       = T(n-1,k) + Sum_{j = 0..n-1} binomial(n-1-(k-1)*j,j)
       = T(n-1,k) + Sum_{j = 0..n-k} binomial(n-k-(k-1)*(j-1),j)
       = T(n-1,k) + T(n-k,k) qed
T(n-k,k) must be known in this recurrence, therefore n >= 2*k.
For k <= n < 2*k, formula(1) must be applied.

A003412 From a nim-like game.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 32, 43, 54, 68, 86, 110, 142, 185, 239, 307, 393, 503, 645, 830, 1069, 1376, 1769, 2272, 2917, 3747, 4816, 6192, 7961, 10233, 13150, 16897, 21713, 27905, 35866, 46099, 59249, 76146, 97859, 125764, 161630, 207729, 266978

Offset: 0

N. J. A. Sloane

nonn

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 1).

Cf. A005708.

Join[{1, 2, 3, 4, 6, 8}, LinearRecurrence[{1, 0, 0, 0, 0, 1}, {11, 14, 18, 24, 32, 43}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

Recurrence: a(n) = a(n-1) + a(n-6) for n >= 12.

O.g.f.: -(1+x+x^2+x^3+2*x^4+2*x^5+2*x^6+x^7+x^8+2*x^9+2*x^10+3*x^11) / (-1+x+x^6). - R. J. Mathar, Dec 05 2007

A003413 From a nim-like game.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 31, 40, 52, 67, 86, 110, 141, 181, 233, 300, 386, 496, 637, 818, 1051, 1351, 1737, 2233, 2870, 3688, 4739, 6090, 7827, 10060, 12930, 16618, 21357, 27447, 35274, 45334, 58264, 74882, 96239, 123686, 158960, 204294, 262558

Offset: 0

N. J. A. Sloane

nonn

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 1).

Cf. A005708.

A003413:=-(z**5+z**3+1)*(z**2+z+1)/(z**6+z-1); # Simon Plouffe in his 1992 dissertation

Join[{1, 2}, LinearRecurrence[{1, 0, 0, 0, 0, 1}, {3, 4, 5, 7, 9, 12}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

Recurrence: a(n) = a(n-1) + a(n-6) for n >= 8.

O.g.f.: -(x^2+x+1)*(x^5+x^3+1)/(-1+x+x^6) = -x-1+(-2-x-x^3-x^4-2*x^5)/(-1+x+x^6). - R. J. Mathar, Dec 05 2007

A193518 T(n,k) = number of ways to place any number of 6X1 tiles of k distinguishable colors into an nX1 grid.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 1, 5, 7, 7, 5, 1, 1, 1, 1, 1, 6, 9, 10, 9, 6, 1, 1, 1, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 1, 1, 1, 8, 13, 16, 17, 16, 13, 9, 1, 1, 1, 1, 1, 9, 15, 19, 21, 21, 19, 19, 12, 1, 1, 1, 1, 1, 10, 17

Offset: 1

R. H. Hardin, with proof and formula from Robert Israel in the Sequence Fans Mailing List, Jul 29 2011

Table starts:
..1..1...1...1...1...1...1...1...1....1....1....1....1....1....1....1....1....1
..1..1...1...1...1...1...1...1...1....1....1....1....1....1....1....1....1....1
..1..1...1...1...1...1...1...1...1....1....1....1....1....1....1....1....1....1
..1..1...1...1...1...1...1...1...1....1....1....1....1....1....1....1....1....1
..1..1...1...1...1...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...15...16...17...18...19
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
..4..7..10..13..16..19..22..25..28...31...34...37...40...43...46...49...52...55
..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73
..6.11..16..21..26..31..36..41..46...51...56...61...66...71...76...81...86...91
..7.13..19..25..31..37..43..49..55...61...67...73...79...85...91...97..103..109
..9.19..31..45..61..79..99.121.145..171..199..229..261..295..331..369..409..451
.12.29..52..81.116.157.204.257.316..381..452..529..612..701..796..897.1004.1117
.16.43..82.133.196.271.358.457.568..691..826..973.1132.1303.1486.1681.1888.2107
.21.61.121.201.301.421.561.721.901.1101.1321.1561.1821.2101.2401.2721.3061.3421

			Some solutions for n=13 k=3; colors=1, 2, 3; empty=0
..0....0....0....0....0....3....0....0....0....0....2....0....0....2....2....1
..3....0....1....2....1....3....0....0....0....2....2....0....0....2....2....1
..3....0....1....2....1....3....0....0....0....2....2....0....0....2....2....1
..3....0....1....2....1....3....0....0....0....2....2....2....0....2....2....1
..3....0....1....2....1....3....0....0....3....2....2....2....0....2....2....1
..3....1....1....2....1....3....0....0....3....2....2....2....0....2....2....1
..3....1....1....2....1....0....3....0....3....2....3....2....0....0....0....2
..1....1....1....0....0....1....3....3....3....3....3....2....2....2....0....2
..1....1....1....0....0....1....3....3....3....3....3....2....2....2....0....2
..1....1....1....0....0....1....3....3....3....3....3....0....2....2....0....2
..1....1....1....0....0....1....3....3....0....3....3....0....2....2....0....2
..1....0....1....0....0....1....3....3....0....3....3....0....2....2....0....2
..1....0....1....0....0....1....0....3....0....3....0....0....2....2....0....0

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A005708,
Column 2 is A143448(n-5),
Column 3 is A143456(n-5),
Row 12 is A190576(n+1),
Row 15 is A069133(n+1).

T:= proc(n, k) option remember;
      `if`(n<0, 0,
      `if`(n<6 or k=0, 1, k*T(n-6, k) +T(n-1, k)))
    end:
seq(seq(T(n, d+1-n), n=1..d), d=1..13); # Alois P. Heinz, Jul 29 2011

T[n_, k_] := T[n, k] = If[n<0, 0, If[n < 6 || k == 0, 1, k*T[n-6, k]+T[n-1, k]]]; Table[Table[T[n, d+1-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Mar 04 2014, after Alois P. Heinz *)

With z X 1 tiles of k colors on an n X 1 grid (with n >= z), either there is a tile (of any of the k colors) on the first spot, followed by any configuration on the remaining (n-z) X 1 grid, or the first spot is vacant, followed by any configuration on the remaining (n-1) X 1. So T(n,k) = T(n-1,k) + k*T(n-z,k), with T(n,k) = 1 for n=0,1,...,z-1. The solution is T(n,k) = sum_r r^(-n-1)/(1 + z k r^(z-1)) where the sum is over the roots of the polynomial k x^z + x - 1.
T(n,k) = sum {s=0..[n/6]} (binomial(n-5*s,s)*k^s).
For z X 1 tiles, T(n,k,z) = sum{s=0..[n/z]} (binomial(n-(z-1)*s,s)*k^s). - R. H. Hardin, Jul 31 2011

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A373958 Number of compositions of 6*n-3 into parts 1 and 6.

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A373959 Number of compositions of 6*n-4 into parts 1 and 6.

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A373960 Number of compositions of 6*n-5 into parts 1 and 6.

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A143284 Number of binary words of length n containing at least one subword 100001 and no subwords 10^{i}1 with i<4.

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A143285 Number of binary words of length n containing at least one subword 1000001 and no subwords 10^{i}1 with i<5.

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A224812 Number of subsets of {1,2,...,n-10} without differences equal to 2, 4, 6, 8 or 10.

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A329146 Triangle read by rows: T(n,k) is the number of subsets of {1,...,n} such that the difference between any two elements is k or greater, 1 <= k <= n.

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A003412 From a nim-like game.

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