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

%I A001972 M0551 N0199 #86 Jan 18 2024 01:24:15
%S A001972 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,
%T A001972 98,105,112,120,128,136,144,153,162,171,180,190,200,210,220,231,242,
%U A001972 253,264,276,288,300,312,325,338,351,364,378,392,406,420,435,450,465
%N A001972 Expansion of 1/((1-x)^2*(1-x^4)) = 1/( (1+x)*(1+x^2)*(1-x)^3 ).
%C A001972 First differences are A008621. - _Amarnath Murthy_, Apr 26 2004
%C A001972 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
%C A001972 From _Jon Perry_, Nov 16 2010: (Start)
%C A001972 Column sums of the following array:
%C A001972   1 2 3 4 5 6 7  8  9...
%C A001972           1 2 3  4  5...
%C A001972                     1...
%C A001972   --------------------
%C A001972   1 2 3 4 6 8 10 12 15 (End)
%C A001972 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
%C A001972 Number of partitions of n into parts 1 (of two sorts) and 4 (of one sort). - _Joerg Arndt_, Aug 08 2013
%C A001972 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
%D A001972 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.
%D A001972 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001972 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001972 Vincenzo Librandi, <a href="/A001972/b001972.txt">Table of n, a(n) for n = 0..10000</a>
%H A001972 A. Cayley, <a href="/A001971/a001971.pdf">Numerical tables supplementary to second memoir on quantics</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
%H A001972 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=208">Encyclopedia of Combinatorial Structures 208</a>
%H A001972 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H A001972 Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=4]
%H A001972 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001972 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001972 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F A001972 From _Michael Somos_, Apr 21 2000: (Start)
%F A001972 a(n) = a(n-1) + a(n-4) - a(n-5) + 1.
%F A001972 a(n) = floor((n+3)^2/8). (End)
%F A001972 a(n) = Sum_{k=0..n} floor((k+4)/4) = n + 1 + Sum_{k=0..n} floor(k/4). - _Paul Barry_, Aug 19 2003
%F A001972 a(n) = a(n-4) + n + 1. - _Paul Barry_, Jul 14 2004
%F A001972 From _Mitch Harris_, Sep 08 2008: (Start)
%F A001972 a(n) = Sum_{j=0..n+4} floor(j/4);
%F A001972 a(n-4) = (1/2)*floor(n/4)*(2*n - 2 - 4*floor(n/4)). (End)
%F A001972 A002620(n+1) = a(2*n-1)/2.
%F A001972 A000217(n+1) = a(2*n).
%F A001972 a(n)+a(n+1)+a(n+2)+a(n+3) = (n+4)*(n+5)/2. - _Amarnath Murthy_, Apr 26 2004
%F A001972 a(n) = n^2/8 + 3*n/4 + 15/16 + (-1)^n/16 + A056594(n+3)/4. - _Amarnath Murthy_, Apr 26 2004
%F A001972 a(n) = A130519(n+4). - _Franklin T. Adams-Watters_, Jul 10 2009
%F A001972 a(n) = floor((n+1)/(1-e^(-8/(n+1)))). - _Richard R. Forberg_, Aug 07 2013
%F A001972 a(n) = a(-6-n) for all n in Z. - _Michael Somos_, Mar 13 2020
%F A001972 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
%p A001972 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
%t A001972 CoefficientList[Series[1/((1-x)^2(1-x^4)),{x,0,80}],x]  (* _Harvey P. Dale_, Mar 27 2011 *)
%o A001972 (PARI) a(n)=(n+3)^2\8;
%o A001972 (Magma) [Floor((n+3)^2/8): n in [0..60]]; // _Vincenzo Librandi_, Aug 15 2011
%Y A001972 Bisections are A000217 and A007590. - _Amarnath Murthy_, Apr 26 2004
%Y A001972 Cf. A001972, A002620, A008621, A056594, A130519, A333260.
%K A001972 nonn,easy
%O A001972 0,2
%A A001972 _N. J. A. Sloane_
%E A001972 Partially edited by _R. J. Mathar_, Jul 11 2009