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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. 4

%I

%S 0,1,2,3,12,20,91,154,713,1209,5610,9515,44164,74908,347699,589746,

%T 2737425,4643057,21551698,36554707,169676156,287794596,1335857547,

%U 2265802058,10517184217,17838621865,82801616186,140443172859,651895745268,1105706761004,5132364345955

%N 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 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 From _Paul Weisenhorn_, Jun 30 2010: (Start)

%C b(n) red and a(n) blue balls in an urn; draw 6 balls without replacement;

%C binomial(b(n),6) = binomial(b(n),4)*binomial(a(n),2), where b(n)=A179123(n).

%C (End)

%H Muniru A Asiru, <a href="/A105045/b105045.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,8,-8,-1,1).

%F From _Paul Weisenhorn_, Jun 30 2010: (Start)

%F r=sqrt(15);

%F a(n)=((15+r)*(4+r)^((n-1)/2)+(15-r)*(4-r)^((n-1)/2)+30)/60; n odd

%F a(n)=((45+11*r)*(4+r)^((n-2)/2)+(45-11*r)*(4-r)^((n-2)/2)+30)/60; n even

%F (End)

%F 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 G.f.: -x*(x^4 + x^3 - 7*x^2 + x + 1) / ((x-1)*(x^4-8*x^2+1)). - _Colin Barker_, Jan 01 2013

%e 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 n:=1: for m from 1 to 2000 do w:=sqrt(1+60*m*(m-1)):

%p 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 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 *) Join[{0},LinearRecurrence[ {1,8,-8,-1,1},{1,2,3,12,20},30]] (* _Harvey P. Dale_, Nov 10 2011 *)

%Y Cf. A103200, A001090.

%K nonn,easy

%O 0,3

%A _Gerald McGarvey_, Apr 03 2005

%E More terms from _Colin Barker_, Jan 01 2013

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Last modified October 22 07:48 EDT 2019. Contains 328315 sequences. (Running on oeis4.)