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