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A105045 a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=12; for n > 4, a(n) = 8*a(n-2) - a(n-4) - 3.

%I A105045 #41 Sep 02 2023 19:28:13
%S A105045 0,1,2,3,12,20,91,154,713,1209,5610,9515,44164,74908,347699,589746,
%T A105045 2737425,4643057,21551698,36554707,169676156,287794596,1335857547,
%U A105045 2265802058,10517184217,17838621865,82801616186,140443172859,651895745268,1105706761004,5132364345955
%N A105045 a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=12; for n > 4, a(n) = 8*a(n-2) - a(n-4) - 3.
%C A105045 It appears that this sequence gives all nonnegative m such that 60*m^2 - 60*m + 1 is a square and that for n > 3, a(n+1) = A103200(n) + 1.
%C A105045 From _Paul Weisenhorn_, Jun 30 2010: (Start)
%C A105045 Place b(n) red and a(n) blue balls in an urn, then draw 6 balls without replacement.
%C A105045 This gives binomial(b(n), 6) = binomial(b(n), 4) * binomial(a(n), 2), where b(n) = A179123(n). (End)
%H A105045 Muniru A Asiru, <a href="/A105045/b105045.txt">Table of n, a(n) for n = 0..2000</a>
%H A105045 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,8,-8,-1,1).
%F A105045 a(n) = 8*a(n-2) - a(n-4) - 3, for n > 4.
%F A105045 From _Paul Weisenhorn_, Jun 30 2010: (Start)
%F A105045 Let r=sqrt(15), then
%F A105045 a(n) = ((15+r)*(4+r)^((n-1)/2) + (15-r)*(4-r)^((n-1)/2) + 30)/60 when n is odd, and
%F A105045 a(n) = ((45+11*r)*(4+r)^((n-2)/2) + (45-11*r)*(4-r)^((n-2)/2) + 30)/60 when n is even. (End)
%F A105045 a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) - a(n-4) + a(n-5), a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=12, a(5)=20. - _Harvey P. Dale_, Nov 10 2011
%F A105045 G.f.: x*(1 +x -7*x^2 +x^3 +x^4)/((1-x)*(1-8*x^2+x^4)). - _Colin Barker_, Jan 01 2013
%e A105045 For n=3, a(3)=3; b(3)=14; binomial(14,6)=3003; binomial(14,4)*binomial(3,2) = 1001*3 = 3003. - _Paul Weisenhorn_, Jun 30 2010
%p A105045 n:=1: for m from 1 to 2000 do w:=sqrt(1+60*m*(m-1)):
%p A105045 if (w=floor(w)) then a(n)=m: b(n)=(9+w)/2: inc(n): end if: end do # _Paul Weisenhorn_, Jun 30 2010
%t A105045 Join[{0},RecurrenceTable[{a[1]==1,a[2]==2,a[3]==3,a[4]==12,a[n] == 8a[n-2]-a[n-4]-3},a,{n,30}]] (* or *) LinearRecurrence[{1,8,-8,-1, 1}, {0,1,2,3,12,20}, 30] (* _Harvey P. Dale_, Nov 10 2011 *)
%o A105045 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x-7*x^2+x^3+x^4)/((1-x)*(1-8*x^2+x^4)) )); // _G. C. Greubel_, Mar 14 2023
%o A105045 (SageMath)
%o A105045 @CachedFunction
%o A105045 def a(n): # a = A105045
%o A105045     if (n<6): return (0,1,2,3,12,20)[n]
%o A105045     else: return a(n-1) +8*a(n-2) -8*a(n-3) -a(n-4) +a(n-5)
%o A105045 [a(n) for n in range(41)] # _G. C. Greubel_, Mar 14 2023
%Y A105045 Cf. A001090, A103200, A179123.
%K A105045 nonn,easy
%O A105045 0,3
%A A105045 _Gerald McGarvey_, Apr 03 2005
%E A105045 More terms from _Colin Barker_, Jan 01 2013