A017900 -id:A017900 - cp's OEIS frontend

A339104 Number of compositions (ordered partitions) of n into distinct parts >= 6.

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 17, 17, 25, 31, 39, 45, 59, 65, 79, 115, 129, 165, 209, 269, 313, 403, 471, 585, 683, 941, 1063, 1375, 1641, 2097, 2537, 3161, 3745, 4663, 5535, 6741, 8627, 10241, 12535, 15307, 18849, 22869, 28409

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(13) = 3 because we have [13], [7, 6] and [6, 7].

Index entries for sequences related to compositions

Cf. A017900, A025151, A032020, A032022, A185326, A339101, A339102, A339103.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-5)*(i+6)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020

nmax = 57; CoefficientList[Series[Sum[k! x^(k (k + 11)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(k + 11)/2) / Product_{j=1..k} (1 - x^j).

A141539 Square array A(n,k) of numbers of length n binary words with at least k "0" between any two "1" digits (n,k >= 0), read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 8, 1, 2, 3, 5, 16, 1, 2, 3, 4, 8, 32, 1, 2, 3, 4, 6, 13, 64, 1, 2, 3, 4, 5, 9, 21, 128, 1, 2, 3, 4, 5, 7, 13, 34, 256, 1, 2, 3, 4, 5, 6, 10, 19, 55, 512, 1, 2, 3, 4, 5, 6, 8, 14, 28, 89, 1024, 1, 2, 3, 4, 5, 6, 7, 11, 19, 41, 144, 2048, 1, 2, 3, 4, 5, 6, 7, 9, 15, 26, 60, 233, 4096

Offset: 0

Alois P. Heinz, Aug 15 2008

A(n,k+1) = A(n,k) - A143291(n,k).
From Gary W. Adamson, Dec 19 2009: (Start)
Alternative method generated from variants of an infinite lower triangle T(n) = A000012 = (1; 1,1; 1,1,1; ...) such that T(n) has the leftmost column shifted up n times. Then take lim_{k->infinity} T(n)^k, obtaining a left-shifted vector considered as rows of an array (deleting the first 1) as follows:
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... = powers of 2
  1, 1, 2, 3,  5,  8, 13,  21,  34, ... = Fibonacci numbers
  1, 1, 1, 2,  3,  4,  6,   9,  13, ... = A000930
  1, 1, 1, 1,  2,  3,  4,   5,   7, ... = A003269
  ... with the next rows A003520, A005708, A005709, ... such that beginning with the Fibonacci row, the succession of rows are recursive sequences generated from a(n) = a(n-1) + a(n-2); a(n) = a(n-1) + a(n-3), ... a(n) = a(n-1) + a(n-k); k = 2,3,4,... Last, columns going up from the topmost 1 become rows of triangle A141539. (End)

			A(4,2) = 6, because 6 binary words of length 4 have at least 2 "0" between any two "1" digits: 0000, 0001, 0010, 0100, 1000, 1001.
Square array A(n,k) begins:
    1,  1,  1,  1,  1,  1,  1,  1, ...
    2,  2,  2,  2,  2,  2,  2,  2, ...
    4,  3,  3,  3,  3,  3,  3,  3, ...
    8,  5,  4,  4,  4,  4,  4,  4, ...
   16,  8,  6,  5,  5,  5,  5,  5, ...
   32, 13,  9,  7,  6,  6,  6,  6, ...
   64, 21, 13, 10,  8,  7,  7,  7, ...
  128, 34, 19, 14, 11,  9,  8,  8, ...

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

Cf. column k=0: A000079, k=1: A000045(n+2), k=2: A000930(n+2), A068921, A078012(n+5), k=3: A003269(n+4), A017898(n+7), k=4: A003520(n+4), A017899(n+9), k=5: A005708(n+5), A017900(n+11), k=6: A005709(n+6), A017901(n+13), k=7: A005710(n+7), A017902(n+15), k=8: A005711(n+7), A017903(n+17), k=9: A017904(n+19), k=10: A017905(n+21), k=11: A017906(n+23), k=12: A017907(n+25), k=13: A017908(n+27), k=14: A017909(n+29).
Main diagonal gives A000027(n+1).
A(2n,n) gives A000217(n+1)
A(3n,n) gives A008778.
A(3n,2n) gives A034856(n+1).
A(2n,3n) gives A005408.
A(2^n-1,n) gives A376697.
See also A143291.

A:= proc(n, k) option remember;
      if k=0 then 2^n
    elif n<=k and n>=0 then n+1
    elif n>0 then A(n-1, k) +A(n-k-1, k)
    else          A(n+1+k, k) -A(n+k, k)
      fi
    end:
seq(seq(A(n, d-n), n=0..d), d=0..15);

a[n_, k_] := a[n, k] = Which[k == 0, 2^n, n <= k && n >= 0, n+1, n > 0, a[n-1, k] + a[n-k-1, k], True, a[n+1+k, k] - a[n+k, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

G.f. of column k: x^(-k)/(1-x-x^(k+1)).

A(n,k) = 2^n if k=0, otherwise A(n,k) = n+1 if n<=k, otherwise A(n,k) = A(n-1,k) + A(n-k-1,k).

A371213 Number of ordered factorizations of n into factors > 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 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, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 3, 3, 1, 3, 2, 1, 1, 5, 1, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 5, 1, 3, 3, 2, 1, 3, 1, 3, 3, 1, 1, 5

Offset: 1

Ilya Gutkovskiy, Mar 15 2024

nonn

			a(42) = 3: 42 = 6*7 = 7*6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002033, A017900, A074206, A185326, A371209, A371211, A371212.

a[n_] := a[n] = If[n == 1, n, Sum[If[n/d > 5, a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 96}]

memoA371213 = Map();
A371213(n) = if(1==n,1,my(v); if(mapisdefined(memoA371213,n,&v), v, v = sumdiv(n,d,if((n/d)<=5, 0, A371213(d))); mapput(memoA371213,n,v); (v))); \\ Antti Karttunen, Jan 13 2025

a(1) = 1; a(n) = Sum_{d|n, n/d > 5} a(d).

More terms from Antti Karttunen, Jan 13 2025

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

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A339104 Number of compositions (ordered partitions) of n into distinct parts >= 6.

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A141539 Square array A(n,k) of numbers of length n binary words with at least k "0" between any two "1" digits (n,k >= 0), read by antidiagonals.

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A371213 Number of ordered factorizations of n into factors > 5.

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

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