A004120 Expansion of (1 + x - x^5) / (1 - x)^3. (Formerly M3354)

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, 1255, 1306, 1358, 1411, 1465, 1520, 1576, 1633

Offset

0,2

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), 1996.

Murray S. 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).

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

Sum_{n>=0} 1/a(n) = 625931/813960 + 2*Pi*tan(sqrt(161)*Pi/2)/sqrt(161). - Amiram Eldar, Feb 03 2026

E.g.f.: (12 + 6*x + x^2 - exp(x)*(10 - 12*x - x^2))/2. - Stefano Spezia, Feb 04 2026

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 *)

Prog

(Magma) [1, 4, 9] cat [n*(n+11)/2-5: n in [3..30]]; // Vincenzo Librandi, Oct 08 2011
(PARI) a(n)=if(n>2, (n^2+11*n)/2-5, (n+1)^2) \\ Charles R Greathouse IV, Sep 30 2015

Crossrefs

Cf. A302537.

Sequence in context: A010457 A244833 A331220 * A052118 A070470 A070469

Adjacent sequences: A004117 A004118 A004119 * A004121 A004122 A004123

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved