A004120 Expansion of (1 + x - x^5) / (1 - x)^3.

1, 4, 9, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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
• D. R. Breach, Solution to Problem 68-16, SIAM Rev. 12 (1970), 294-297.
• D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
• Philippe Flajolet, Balls and urns, etc. A problem in submarine detection (solution to 68-16)
• M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 109-111.
• 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
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

Programs

Magma

[1,4,9] cat [n*(n+11)/2-5: n in [3..30]]; // Vincenzo Librandi, Oct 08 2011

Maple

A004120:=(-1-z+z**5)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Mathematica

i=7;s=1;lst={s};Do[s+=n+i;If[s>=0, AppendTo[lst, s]], {n, 0, 6!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 30 2008 *)
CoefficientList[Series[(1+x-x^5)/(1-x)^3,{x,0,50}],x] (* or *) Join[ {1,4,9}, LinearRecurrence[{3,-3,1},{16,25,35},50]] (* Harvey P. Dale, Oct 11 2011 *)

PARI

a(n)=if(n>2,(n^2+11*n)/2-5,(n+1)^2) \\ Charles R Greathouse IV, Sep 30 2015

Formula

a(n) = n*(n + 11)/2 - 5, n>=3. - R. J. Mathar, Mar 15 2011

a(n) = A302537(n-1), n>=3. - R. J. Mathar, Apr 24 2024