A000115 Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).
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
Examples
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 + ...
References
- 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)
Programs
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Magma
[Round((n+4)^2/20): n in [0..70]]; // Vincenzo Librandi, Jun 23 2011
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Maple
1/((1-x)*(1-x^2)*(1-x^5)): seq(coeff(series(%, x, n+1), x, n), n=0..65); # next Maple program: 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
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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 *) -
PARI
a(n)=(n^2+8*n+26)\20 \\ Charles R Greathouse IV, Jun 23 2011
Formula
a(n) = round((n+4)^2/20).
a(n) = a(-8 - n) for all n in Z. - Michael Somos, May 28 2014
Comments