A005709 -id:A005709 - cp's OEIS frontend

A373928 Number of compositions of 7*n-2 into parts 1 and 7.

1, 7, 35, 168, 819, 4025, 19796, 97315, 478304, 2350860, 11554621, 56791883, 279136551, 1371977475, 6743373646, 33144194898, 162906243014, 800696596250, 3935484773527, 19343207491818, 95073338508548, 467292702057555, 2296779231936167, 11288844908179562

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373929, A373930, A373931, A373932.

Cf. A005709, A369805, A369837.

a[n_]:= n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*HypergeometricPFQ[{1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6}, {6/7, 8/7, 9/7, 10/7, 11/7, 12/7}, -6^6/7^7]/120; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
LinearRecurrence[{8,-21,35,-35,21,-7,1},{1,7,35,168,819,4025,19796},40] (* Harvey P. Dale, Jul 28 2024 *)

a(n) = sum(k=0, n, binomial(n+4+6*k, n-1-k));

a(n) = A005709(7*n-2).
a(n) = Sum_{k=0..n} binomial(n+4+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6], [6/7, 8/7, 9/7, 10/7, 11/7, 12/7], -6^6/7^7)/120. - Stefano Spezia, Jun 23 2024

A373929 Number of compositions of 7*n-3 into parts 1 and 7.

Original entry on oeis.org

1, 6, 28, 133, 651, 3206, 15771, 77519, 380989, 1872556, 9203761, 45237262, 222344668, 1092840924, 5371396171, 26400821252, 129762048116, 637790353236, 3134788177277, 15407722718291, 75730131016730, 372219363549007, 1829486529878612, 8992065676243395

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373928, A373930, A373931, A373932.

Cf. A005709, A369806.

a[n_]:=n*(1 + n)*(2 + n)*(3 + n)*HypergeometricPFQ[{1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6}, {5/7, 6/7, 8/7, 9/7, 10/7, 11/7}, -6^6/7^7]/24; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)

a(n) = sum(k=0, n, binomial(n+3+6*k, n-1-k));

a(n) = A005709(7*n-3).
a(n) = Sum_{k=0..n} binomial(n+3+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^2/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*(3 + n)*hypergeom([1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6], [5/7, 6/7, 8/7, 9/7, 10/7, 11/7], -6^6/7^7)/24. - Stefano Spezia, Jun 23 2024

A373930 Number of compositions of 7*n-4 into parts 1 and 7.

Original entry on oeis.org

1, 5, 22, 105, 518, 2555, 12565, 61748, 303470, 1491567, 7331205, 36033501, 177107406, 870496256, 4278555247, 21029425081, 103361226864, 508028305120, 2496997824041, 12272934541014, 60322408298439, 296489232532277, 1457267166329605, 7162579146364783

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373928, A373929, A373931, A373932.

Cf. A005709, A369807.

a[n_]:=n*(1 + n)*(2 + n)*HypergeometricPFQ[{1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6}, {4/7, 5/7, 6/7, 8/7, 9/7, 10/7}, -6^6/7^7]/6; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)

a(n) = sum(k=0, n, binomial(n+2+6*k, n-1-k));

a(n) = A005709(7*n-4).
a(n) = Sum_{k=0..n} binomial(n+2+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^3/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*hypergeom([1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6], [4/7, 5/7, 6/7, 8/7, 9/7, 10/7], -6^6/7^7)/6. - Stefano Spezia, Jun 23 2024

A373931 Number of compositions of 7*n-5 into parts 1 and 7.

Original entry on oeis.org

1, 4, 17, 83, 413, 2037, 10010, 49183, 241722, 1188097, 5839638, 28702296, 141073905, 693388850, 3408058991, 16750869834, 82331801783, 404667078256, 1988969518921, 9775936716973, 48049473757425, 236166824233838, 1160777933797328, 5705311980035178

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373928, A373929, A373930, A373932.

Cf. A005709, A369808.

a[n_]:=n*(1 + n)*HypergeometricPFQ[{1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6}, {3/7, 4/7, 5/7, 6/7, 8/7, 9/7}, -6^6/7^7]/2; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)

a(n) = sum(k=0, n, binomial(n+1+6*k, n-1-k));

a(n) = A005709(7*n-5).
a(n) = Sum_{k=0..n} binomial(n+1+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^4/((1-x)^7 - x).
a(n) = n*(1 + n)*hypergeom([1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6], [3/7, 4/7, 5/7, 6/7, 8/7, 9/7], -6^6/7^7)/2. - Stefano Spezia, Jun 23 2024

A373932 Number of compositions of 7*n-6 into parts 1 and 7.

Original entry on oeis.org

1, 3, 13, 66, 330, 1624, 7973, 39173, 192539, 946375, 4651541, 22862658, 112371609, 552314945, 2714670141, 13342810843, 65580931949, 322335276473, 1584302440665, 7786967198052, 38273537040452, 188117350476413, 924611109563490, 4544534046237850

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373928, A373929, A373930, A373931.

Cf. A005709.

a[n_]:=n*HypergeometricPFQ[{1-n,(1+n)/6,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6}, {2/7, 3/7, 4/7, 5/7, 6/7, 8/7}, -6^6/7^7]; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)

a(n) = sum(k=0, n, binomial(n+6*k, n-1-k));

a(n) = A005709(7*n-6).
a(n) = Sum_{k=0..n} binomial(n+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^5/((1-x)^7 - x).
a(n) = n*hypergeom([1-n,(1+n)/6,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6], [2/7, 3/7, 4/7, 5/7, 6/7, 8/7], -6^6/7^7). - Stefano Spezia, Jun 23 2024

A017901 Expansion of 1/(1 - x^7 - x^8 - ...).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 53, 66, 83, 105, 133, 168, 211, 264, 330, 413, 518, 651, 819, 1030, 1294, 1624, 2037, 2555, 3206, 4025, 5055, 6349, 7973, 10010, 12565, 15771, 19796, 24851, 31200, 39173

Offset: 0

N. J. A. Sloane

A Lamé sequence of higher order.
a(n) = number of compositions of n in which each part is >= 7. - Milan Janjic, Jun 28 2010
a(n+7) equals the number of n-length binary words such that 0 appears only in a run length that is a multiple of 7. - Milan Janjic, Feb 17 2015
A017847(n) = |a(-n)| for n>=0. - Michael Somos, Oct 28 2018

			G.f. = 1 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + 2*x^14 + ... - _Michael Somos_, Oct 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
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.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1).

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898, A017899, A017900, A017901, A017902, A017903, A017904.

Cf. A005709, A017847.

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(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$5, 1][i] else 0 fi)^n)[7,7]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 04 2008

LinearRecurrence[{1,0,0,0,0,0,1}, {1,0,0,0,0,0,0}, 60] (* Jean-François Alcover, Mar 28 2017 *)

Vec((x-1)/(x-1+x^7)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = if( n < 0, polcoeff( 1 / (1 + x^6 - x^7) + x * O(x^-n), -n), polcoeff( (1 - x) / (1 - x - x^7) + x * O(x^n), n))}; /* Michael Somos, Oct 28 2018 */

G.f.: (x-1)/(x-1+x^7). - Alois P. Heinz, Aug 04 2008
For positive integers n and k such that k <= n <= 7*k, and 6 divides n-k, define c(n,k) = binomial(k,(n-k)/6), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+7) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
a(n) = A005709(n) - A005709(n-1). - R. J. Mathar, Sep 07 2016
0 == a(n) + a(n+6) - a(n+7) for all n in Z. - Michael Somos, Oct 28 2018

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 *)

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

A143286 Number of binary words of length n containing at least one subword 10^{6}1 and no subwords 10^{i}1 with i<6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9, 13, 19, 27, 37, 49, 63, 80, 102, 132, 173, 228, 300, 392, 508, 654, 839, 1076, 1382, 1778, 2289, 2945, 3783, 4850, 6207, 7934, 10135, 12943, 16526, 21095, 26915, 34320, 43733, 55692, 70882, 90174, 114673, 145778

Offset: 0

Alois P. Heinz, Aug 04 2008

			a(9)=2 because 2 binary words of length 9 have at least one subword 10^{6}1 and no subwords 10^{i}1 with i<6: 010000001, 100000010.

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

Cf. A005709, A005710, 6th column of A143291.

[n le 8 select 0 else n le 15 select n-8 else 2*Self(n-1)-Self(n-2) +Self(n-7)-Self(n-9)-Self(n-15): n in [1..60]]; // Vincenzo Librandi, Jun 05 2013

a:= n-> coeff(series(x^8/((x^7+x-1)*(x^8+x-1)), x, n+1), x, n):
seq(a(n), n=0..60);

CoefficientList[Series[x^8 / ((x^7 + x - 1) (x^8 + x - 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 04 2013 *)

G.f.: x^8/((x^7+x-1)*(x^8+x-1)).
a(n) = A005709(n+6)-A005710(n+7).
a(n) = 2*a(n-1) - a(n-2) + a(n-7) - a(n-9) - a(n-15). - Vincenzo Librandi, Jun 05 2013

A224813 Number of subsets of {1,2,...,n-12} without differences equal to 2, 4, 6, 8, 10 or 12.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 80, 100, 130, 169, 221, 289, 374, 484, 616, 784, 980, 1225, 1505, 1849, 2279, 2809, 3498, 4356, 5478, 6889, 8715, 11025, 13965, 17689, 22344, 28224, 35448, 44521, 55704, 69696, 87120, 108900, 136290, 170569, 213934, 268324, 337218, 423801, 533169, 670761, 843570

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=12, I={-2,0,12}.

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, -1, 1, 3, -2, 2, -1, 1, 0, 0, -3, 2, -4, 2, -3, 0, 0, -3, 1, -2, 0, 0, 0, 0, 3, -1, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1).

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

CoefficientList[Series[-(-1 + x^7 + x^9 + x^11 + 2*x^14 + x^16 - 2*x^21 - 2*x^23 - x^28 + x^35)/((x^7 + x - 1)*(x^42 - x^36 - 2*x^30 - 3*x^28 + 2*x^24 + 2*x^22 + x^18 + 2*x^16 + 3*x^14 - x^12 - x^10 - x^8 - 1)), {x, 0, 1000}], x] (* G. C. Greubel, Oct 28 2017 *)

x='x+O('x^50); Vec(-(-1 + x^7 + x^9 + x^11 + 2*x^14 + x^16 - 2*x^21 - 2*x^23 - x^28 + x^35)/((x^7 + x - 1)*(x^42 - x^36 - 2*x^30 - 3*x^28 + 2*x^24 + 2*x^22 + x^18 + 2*x^16 + 3*x^14 - x^12 - x^10 - x^8 - 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) -a(n-12) +a(n-13) +3*a(n-14) -2*a(n-15) +2*a(n-16) -a(n-17) +a(n-18) -3*a(n-21) +2*a(n-22) -4*a(n-23) +2*a(n-24) -3*a(n-25) -3*a(n-28) +a(n-29) -2*a(n-30) +3*a(n-35) -a(n-36) +3*a(n-37) +a(n-42) -a(n-49).
G.f.: -(-1 +x^7 +x^9 +x^11 +2*x^14 +x^16 -2*x^21 -2*x^23 -x^28 +x^35)/( (x^7+x-1) *(x^42 -x^36 -2*x^30 -3*x^28 +2*x^24 +2*x^22 +x^18 +2*x^16 +3*x^14 -x^12 -x^10 -x^8 -1) ).
a(2*k) = (A005709(k))^2, a(2*k+1) = A005709(k) * A005709(k+1).

cp's OEIS Frontend

A373928 Number of compositions of 7*n-2 into parts 1 and 7.

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A373929 Number of compositions of 7*n-3 into parts 1 and 7.

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A373930 Number of compositions of 7*n-4 into parts 1 and 7.

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A373931 Number of compositions of 7*n-5 into parts 1 and 7.

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A373932 Number of compositions of 7*n-6 into parts 1 and 7.

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A017901 Expansion of 1/(1 - x^7 - x^8 - ...).

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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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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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