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A008732 Molien series for 3-dimensional group [2,n] = *22n.

%I A008732 #50 Sep 08 2022 08:44:36
%S A008732 1,2,3,4,5,7,9,11,13,15,18,21,24,27,30,34,38,42,46,50,55,60,65,70,75,
%T A008732 81,87,93,99,105,112,119,126,133,140,148,156,164,172,180,189,198,207,
%U A008732 216,225,235,245,255,265
%N A008732 Molien series for 3-dimensional group [2,n] = *22n.
%H A008732 Vincenzo Librandi, <a href="/A008732/b008732.txt">Table of n, a(n) for n = 0..10000</a>
%H A008732 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=188">Encyclopedia of Combinatorial Structures 188</a>
%H A008732 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=5]
%H A008732 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008732 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).
%F A008732 a(n) = floor( (n+3)*(n+4)/10 ) = (n+2)*(n+5)/10 + b(n)/5 where b(n) = A010891(n-2) + 2*A092202(n-1) = 0, 1, 1, 0, -2, ... with period length 5.
%F A008732 G.f.: 1/((1-x)^2*(1-x^5)).
%F A008732 a(n) = a(n-5) + n + 1. - _Paul Barry_, Jul 14 2004
%F A008732 From _Mitch Harris_, Sep 08 2008: (Start)
%F A008732 a(n) = Sum_{j=0..n+5} floor(j/5).
%F A008732 a(n-5) = (1/2)floor(n/5)*(2*n - 3 - 5*floor(n/5)). (End)
%F A008732 a(n) = A130520(n+5). - _Philippe Deléham_, Apr 05 2013
%F A008732 a(5n) = A000566(n+1), a(5n+1) = A005476(n+1), a(5n+2) = A005475(n+1), a(5n+3) = A147875(n+2), a(5n+4) = A028895(n+1); these formulas correspond to the 5 columns of the array shown in example. - _Philippe Deléham_, Apr 05 2013
%e A008732 From _Philippe Deléham_, Apr 05 2013: (Start)
%e A008732 Stored in five columns:
%e A008732     1   2   3   4   5
%e A008732     7   9  11  13  15
%e A008732    18  21  24  27  30
%e A008732    34  38  42  46  50
%e A008732    55  60  65  70  75
%e A008732    81  87  93  99 105
%e A008732   112 119 126 133 140
%e A008732 (End)
%p A008732 A092202 := proc(n) op(1+(n mod 5),[0,1,0,-1,0]) ; end proc:
%p A008732 A010891 := proc(n) op(1+(n mod 5),[1,-1,0,0,0]) ; end proc:
%p A008732 A008732 := proc(n) (n+2)*(n+5)/10+(A010891(n-1)+2*A092202(n-1))/5 ; end proc:
%t A008732 LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {1, 2, 3, 4, 5, 7, 9}, 50] (* _Jean-François Alcover_, Jan 18 2018 *)
%o A008732 (Magma) [Floor((n+3)*(n+4)/10): n in [0..50] ]; // _Vincenzo Librandi_, Aug 21 2011
%o A008732 (PARI) a(n)=(n+3)*(n+4)\10 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A008732 (Sage) [floor((n+3)*(n+4)/10) for n in (0..50)] # _G. C. Greubel_, Jul 30 2019
%o A008732 (GAP) List([0..50], n-> Int((n+3)*(n+4)/10)); # _G. C. Greubel_, Jul 30 2019
%Y A008732 Cf. A130520.
%K A008732 nonn,easy,tabf
%O A008732 0,2
%A A008732 _N. J. A. Sloane_