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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

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

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

Programs

Magma

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

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 *)

PARI

a(n)=(n+3)^2\8;

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

Extensions

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