A001972 Expansion of 1/((1-x)^2*(1-x^4)) = 1/( (1+x)*(1+x^2)*(1-x)^3 ). (Formerly M0551 N0199)

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450, 465

Offset

0,2

Comments

First differences are A008621. - Amarnath Murthy, Apr 26 2004

a(n) = least k > a(n-1) such that k + a(n-1) + a(n-2) + a(n-3) is triangular. - Amarnath Murthy, Apr 26 2004

From Jon Perry, Nov 16 2010: (Start)

Column sums of the following array:

1 2 3 4 5 6 7 8 9...

1 2 3 4 5...

1...

--------------------

1 2 3 4 6 8 10 12 15 (End)

A001972(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2=4x+y. - Clark Kimberling, Jun 04 2012

Number of partitions of n into parts 1 (of two sorts) and 4 (of one sort). - Joerg Arndt, Aug 08 2013

In the polynomial sequence s(n) = (x*s(n-1)*s(n-4) + y*s(n-2)*s(n-3))/s(n-5), with s(k) = 1 for k = 0..4, the leading term of s(n+5) is x^a(n). See A333260. - Michael Somos, Mar 13 2020

References

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.

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

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 208

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=4]

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 (2,-1,0,1,-2,1).

Formula

From Michael Somos, Apr 21 2000: (Start)

a(n) = a(n-1) + a(n-4) - a(n-5) + 1.

a(n) = floor((n+3)^2/8). (End)

a(n) = Sum_{k=0..n} floor((k+4)/4) = n + 1 + Sum_{k=0..n} floor(k/4). - Paul Barry, Aug 19 2003

a(n) = a(n-4) + n + 1. - Paul Barry, Jul 14 2004

From Mitch Harris, Sep 08 2008: (Start)

a(n) = Sum_{j=0..n+4} floor(j/4);

a(n-4) = (1/2)*floor(n/4)*(2*n - 2 - 4*floor(n/4)). (End)

A002620(n+1) = a(2*n-1)/2.

A000217(n+1) = a(2*n).

a(n)+a(n+1)+a(n+2)+a(n+3) = (n+4)*(n+5)/2. - Amarnath Murthy, Apr 26 2004

a(n) = n^2/8 + 3*n/4 + 15/16 + (-1)^n/16 + A056594(n+3)/4. - Amarnath Murthy, Apr 26 2004

a(n) = A130519(n+4). - Franklin T. Adams-Watters, Jul 10 2009

a(n) = floor((n+1)/(1-e^(-8/(n+1)))). - Richard R. Forberg, Aug 07 2013

a(n) = a(-6-n) for all n in Z. - Michael Somos, Mar 13 2020

E.g.f.: ((8 + 7*x + x^2)*cosh(x) + 2*sin(x) + (7 + 7*x + x^2)*sinh(x))/8. - Stefano Spezia, May 09 2023

Maple

A001972:=-(2-z+z**3-2*z**4+z**5)/(z+1)/(z**2+1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 1

Mathematica

CoefficientList[Series[1/((1-x)^2(1-x^4)), {x, 0, 80}], x]  (* Harvey P. Dale, Mar 27 2011 *)
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {1, 2, 3, 4, 6, 8}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

Prog

(PARI) a(n)=(n+3)^2\8;
(Magma) [Floor((n+3)^2/8): n in [0..60]]; // Vincenzo Librandi, Aug 15 2011

Crossrefs

Bisections are A000217 and A007590. - Amarnath Murthy, Apr 26 2004

Cf. A001972, A002620, A008621, A056594, A130519, A333260.

Sequence in context: A054041 A019293 A130519 * A328325 A005705 A139542

Adjacent sequences: A001969 A001970 A001971 * A001973 A001974 A001975

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Partially edited by R. J. Mathar, Jul 11 2009

Status

approved