A000115 Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)). (Formerly M0279 N0098)

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 18, 20, 22, 24, 26, 29, 31, 34, 36, 39, 42, 45, 48, 51, 54, 58, 61, 65, 68, 72, 76, 80, 84, 88, 92, 97, 101, 106, 110, 115, 120, 125, 130, 135, 140, 146, 151, 157, 162, 168, 174, 180, 186, 192, 198, 205, 211, 218, 224, 231, 238

Offset

0,3

Comments

Number of partitions of n into parts 1, 2, or 5.

First differences are in A008616. First differences of A001304. Pairwise sums of A008720.

References

Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, pp. 139-140.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,2,5).

M. Jeger, Ein partitions problem ..., Elemente de Math., 13 (1958), 97-120.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1,-1,-1,1)

Formula

a(n) = round((n+4)^2/20).

a(n) = a(-8 - n) for all n in Z. - Michael Somos, May 28 2014

Example

G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + ...

Maple

1/((1-x)*(1-x^2)*(1-x^5)): seq(coeff(series(%, x, n+1), x, n), n=0..65);
# Alternative:
s:=proc(n) if n mod 5 = 0 then RETURN(1); fi; if n mod 5 = 1 then RETURN(0); fi; if n mod 5 = 2 then RETURN(1); fi; if n mod 5 = 3 then RETURN(-1); fi; if n mod 5 = 4 then RETURN(-1); fi; end: f:=n->(2*n^2+16*n+27+5*(-1)^n+8*s(n))/40: seq(f(n), n=0..65);  # from Jeger's paper

Mathematica

nn=50; CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^5), {x, 0, nn}], x]  (* Geoffrey Critzer, Jan 20 2013 *)
LinearRecurrence[{1, 1, -1, 0, 1, -1, -1, 1}, {1, 1, 2, 2, 3, 4, 5, 6}, 70] (* Harvey P. Dale, Sep 27 2019 *)

Prog

(Magma) [Round((n+4)^2/20): n in [0..70]]; // Vincenzo Librandi, Jun 23 2011
(PARI) a(n)=(n^2+8*n+26)\20 \\ Charles R Greathouse IV, Jun 23 2011
(SageMath) (1/((1-x)*(1-x^2)*(1-x^5))).series(x, 66).coefficients(x, sparse=False) # Stefano Spezia, Mar 26 2026

Crossrefs

Cf. A001304, A008616, A008720.

Sequence in context: A274165 A388059 A011874 * A367253 A033552 A062420

Adjacent sequences: A000112 A000113 A000114 * A000116 A000117 A000118

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved