A017899 -id:A017899 - cp's OEIS frontend

A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.

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, 367411, 486716, 644761

Offset: 0

N. J. A. Sloane

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 = 0..m-1. The generating function is 1/(1-x-x^m). Also a(n) = Sum_{i=0..n/m} binomial(n-(m-1)*i, i). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
Also counts ordered partitions such that no part is less than 5. For example, a(12) = a(11) + a(7) where a(7) counts 11,6+5 and 5+6 and a(11) counts 15,10+5, 9+6,8+7,7+8,6+9,5+10 and 5+5+5. Thus a(12) = 3 + 8 = 11. a(12) counts 16,11+5,10+6,9+7,8+8,7+9,6+10 and 6+5+5 but also 5+11,5+6+5 and 5+5+6. Similar results hold for the other sequences formed by a(n) = a(n-1) + a(n-k). - Alford Arnold, Aug 06 2003
Number of compositions of n into parts 1 and 5. - Joerg Arndt, Jun 25 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=5, 2*a(n-5) equals the number of 2-colored compositions of n with all parts >= 5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+4) equals the number of binary words of length n having at least 4 zeros between every two successive ones. - Milan Janjic, Feb 07 2015
Number of tilings of a 5 X n rectangle with 5 X 1 pentominoes. - M. Poyraz Torcuk, Mar 26 2022

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 119.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..500
Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See pp. 18, 22.
Roland Bacher, On the number of perfect lattices, arXiv:1704.02234 [math.NT], 2017. See Section 6.
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 9.
Bruce M. Boman, Geometric Capitulum Patterns Based on Fibonacci p-Proportions, Fibonacci Quart. 58 (2020), no. 5, 91-102.
Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,4,1).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 378
S. Kitaev, Independent sets on path-schemes, JIS 9 (2006) # 06.2.2 G(x) for M={1,2,3,4} gives seq. shifted 4 places left
T. G. Lewis, B. J. Smith and M. Z. Smith, Fibonacci sequences and money management, Fib. Quart., 14 (1976), 37-41.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 33.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO] (2016), Section 4.4.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
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
E. Wilson, The Scales of Mt. Meru
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1)

Apart from initial terms, same as A017899.

Cf. A000045, A000079, A000930, A003269, A005708, A005709, A005710, A005711.

a[0]:=1:a[1]:=1:a[2]:=1:a[3]:=1:a[4]:=1:for n from 5 to 60 do a[n]:=a[n-1]+a[n-5] od:seq(a[n],n=0..60);
with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 4)}, unlabeled]: seq(count(SeqSetU, size=j), j=5..55); # Zerinvary Lajos, Oct 10 2006
A003520:=-1/(z**3+z**2-1)/(z**2-z+1); # Simon Plouffe in his 1992 dissertation
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 4)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=4..54); # Zerinvary Lajos, Mar 26 2008
M := Matrix(5, (i,j)-> if j=1 then [1, 0, 0, 0, 1][i] elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1,1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 27 2008

a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = a[n - 1] + a[n - 5]; Table[ a[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 09 2004 *)
CoefficientList[Series[1/(1 - x - x^5), {x, 0, 51}], x] (* Zerinvary Lajos, Mar 29 2007 *)
LinearRecurrence[{1, 0, 0, 0, 1}, {1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,e+a}; NestList[nxt,{1,1,1,1,1},50][[;;,1]] (* Harvey P. Dale, Sep 27 2023 *)

a(n):=sum(binomial(n-1+(-4)*j,j),j,0,(n-1)/4); /* Vladimir Kruchinin, May 23 2011 */

my(x='x+O('x^66)); Vec(x/(1-(x+x^5))) /* Joerg Arndt, Jun 25 2011 */

G.f.: 1/(1-x-x^5) = 1/((1-x+x^2)(1-x^2-x^3)).
a(n) = Sum_{j=0..(n-1)/4} binomial(n-1+(-4)*j,j).
For n>5, a(n) = floor( d*c^n + 1/2) where c is the positive real root of x^5-x^4-1 and d is the positive real root of 161*x^3-23*x^2-12*x-1 ( c=1.32471795724474602... and d=0.3811571478326847...) - Benoit Cloitre, Nov 30 2002
a(n) = term (1,1) in the 5 X 5 matrix [1,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,0,0,0]^n. - Alois P. Heinz, Jul 27 2008
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) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([-1/5*n, 1/5-1/5*n, 2/5-1/5*n, 3/5-1/5*n, 4/5-1/5*n], [-1/4*n, 1/4-1/4*n, 1/2-1/4*n, 3/4-1/4*n], -5^5/4^4) for n>=16. - Peter Luschny, Sep 18 2014
7*a(n) = A117373(n+4) +5*b(n) +4*b(n-1) +b(n-2) where b(n) = A182097(n). - R. J. Mathar, Aug 07 2017

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

A017898 Expansion of (1-x)/(1-x-x^4).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 345, 476, 657, 907, 1252, 1728, 2385, 3292, 4544, 6272, 8657, 11949, 16493, 22765, 31422, 43371, 59864, 82629, 114051, 157422, 217286, 299915, 413966, 571388, 788674

Offset: 0

N. J. A. Sloane

A Lamé sequence of higher order.
Essentially the same as A003269, which has much more information.
Number of compositions of n into parts >= 4. - Joerg Arndt, Aug 13 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Christian Ballot, On Functions Expressible as Words on a Pair of Beatty Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.2.
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.
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
T. Mansour, M. Shattuck, A monotonicity property for generalized Fibonacci sequences, arXiv:1410.6943 [math.CO], 2014.
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
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

For Lamé sequences of orders 1 through 9 see A000045, A000930, this one, and A017899-A017904.

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(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$2, 1][i] else 0 fi)^n)[4,4]: seq(a(n), n=0..42); # Alois P. Heinz, Aug 04 2008

LinearRecurrence[{1, 0, 0, 1}, {1, 0, 0, 0}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
CoefficientList[Series[(1-x)/(1-x-x^4),{x,0,50}],x] (* Harvey P. Dale, Sep 12 2019 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,1]^n*[1;0;0;0])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = a(n-1) + a(n-4). - R. J. Mathar, Mar 06 2008
G.f.: 1/(1-sum(k>=4, x^k)). - Joerg Arndt, Aug 13 2012
Apparently a(n) = hypergeometric([1-1/4*n, 5/4-1/4*n, 3/2-1/4*n, 7/4-1/4*n],[4/3-1/3*n, 5/3-1/3*n, 2-1/3*n], -4^4/3^3) for n>=13. - Peter Luschny, Sep 18 2014
a(n) = A003269(n+1)-A003269(n). - R. J. Mathar, Jun 10 2018

A339103 Number of compositions (ordered partitions) of n into distinct parts >= 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 15, 17, 23, 31, 37, 45, 57, 65, 101, 115, 151, 189, 255, 293, 383, 451, 565, 777, 921, 1157, 1469, 1855, 2311, 2865, 3495, 4313, 5231, 7063, 8269, 10509, 12849, 16217, 19829, 25171, 30031, 37485, 45183

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(11) = 3 because we have [11], [6, 5] and [5, 6].

Index entries for sequences related to compositions

Cf. A017899, A025150, A032020, A032022, A185325, A339101, A339102, A339104.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-4)*(i+5)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020

nmax = 54; CoefficientList[Series[Sum[k! x^(k (k + 9)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(k + 9)/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).

A371212 Number of ordered factorizations of n into factors > 4.

Original entry on oeis.org

1, 0, 0, 0, 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, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 3, 3, 3, 1, 1, 1, 5, 1, 1, 3, 2, 3, 3, 1, 1, 1, 5, 1, 5, 1, 1, 3, 1, 3, 3, 1, 5, 2, 1, 1, 5, 3, 1, 1, 3, 1, 7, 3, 1, 1, 1, 3, 5, 1, 3, 3, 4, 1, 3, 1, 3, 5, 1, 1, 5

Offset: 1

Ilya Gutkovskiy, Mar 15 2024

nonn

			a(30) = 3: 30 = 5*6 = 6*5.

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

Cf. A002033, A017899, A074206, A185325, A371209, A371211, A371213.

a[n_] := a[n] = If[n == 1, n, Sum[If[n/d > 4, a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 92}]

memoA371212 = Map();
A371212(n) = if(1==n,1,my(v); if(mapisdefined(memoA371212,n,&v), v, v = sumdiv(n,d,if((n/d)<=4, 0, A371212(d))); mapput(memoA371212,n,v); (v))); \\ Antti Karttunen, Jan 16 2025

a(1) = 1; a(n) = Sum_{d|n, n/d > 4} a(d).

More terms from Antti Karttunen, Jan 16 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

A371245 Number of factorizations of n into factors > 4.

Original entry on oeis.org

1, 0, 0, 0, 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, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 3, 1, 3, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 1, 2, 1, 4, 2, 1

Offset: 1

Ilya Gutkovskiy, Mar 16 2024

nonn

			a(60) = 3: 60 = 5*12 = 6*10.

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

Cf. A001055, A017899, A185325, A371212, A371243, A371244.

A371245(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>4)&&(d<=m), s += A371245(n/d, d))); (s)); \\ Antti Karttunen, Jan 17 2025

Dirichlet g.f.: Product_{k>=5} 1 / (1 - k^(-s)).

A174469 Number of permutations p of {1,...,n} satisfying p(1)=1 and, if n>1, |p(i)-p((i mod n)+1)| is in {2,3} for i from 1 to n.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2, 4, 6, 8, 10, 12, 16, 22, 30, 40, 52, 68, 90, 120, 160, 212, 280, 370, 490, 650, 862, 1142, 1512, 2002, 2652, 3514, 4656, 6168, 8170, 10822, 14336, 18992, 25160, 33330, 44152, 58488, 77480, 102640, 135970

Offset: 1

Alois P. Heinz, Nov 28 2010

Also the number of directed Hamiltonian cycles in the graph on n vertices {1,...,n}, with i adjacent to j iff 2 <= |i-j| <= 3.

			For n = 10 the a(10) = 2 permutations are (1,3,6,9,7,10,8,5,2,4), (1,4,2,5,8,10,7,9,6,3).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1).

Cf. A017899.

a:= n-> `if`(n<2, n, (Matrix (5, (i,j)-> `if`(j-i=1 or i=5 and j in {1,5}, 1, 0))^n. <<2, -2, (0$3)>>)[1,1]): seq(a(n), n=1..60);

Join[{1}, LinearRecurrence[{1, 0, 0, 0, 1}, {0, 0, 0, 2, 0}, 60]] (* Jean-François Alcover, Oct 28 2021 *)

G.f.: (3*x^5-2*x^4+x-1)*x / (x^5+x-1).

a(n) = 2*A017899(n-5) for n>=5.

cp's OEIS Frontend

A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.

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A017898 Expansion of (1-x)/(1-x-x^4).

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A339103 Number of compositions (ordered partitions) of n into distinct parts >= 5.

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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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A371212 Number of ordered factorizations of n into factors > 4.

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

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A371245 Number of factorizations of n into factors > 4.

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