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A105045
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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.
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2
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0, 1, 2, 3, 12, 20, 91, 154, 713, 1209, 5610, 9515, 44164, 74908, 347699, 589746, 2737425, 4643057, 21551698, 36554707, 169676156, 287794596, 1335857547, 2265802058, 10517184217, 17838621865, 82801616186, 140443172859
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| It appears that this sequences 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.
Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jun 30 2010: (Start)
b(n) red and a(n) blue balls in an urn; draw 6 balls without return;
binomial(b(n),6)=binomial(b(n),4)*binomial(a(n),2)
b(n)=A179123(n);
(End)
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FORMULA
| a(n)=1/2 + 1/2*(4-sqrt(15))^(1/2*(n-1))*(4-sqrt(15))^(1/4*(-1)^(n-1))*(4-sqrt(15))^(-1/4)-1/4*( -1)^(n-1)*(4-sqrt(15))^(1/2*(n-1))*(4-sqrt(15))^(1/4*(-1)^(n-1))*(4-sqrt(15))^(-1/4) + 1/10*(4 + sqrt(15))^(-1/4)*sqrt(15)*(4 + sqrt(15))^(1/2*(n-1))*(4 + sqrt(15))^(1/4*(-1)^(n-1))-1/12 *(4 + sqrt(15))^(-1/4)*(-1)^(n-1)*sqrt(15)*(4 + sqrt(15))^(1/2*(n-1))*(4 + sqrt(15))^(1/4*( -1)^(n-1)) + 1/2*(4 + sqrt(15))^(-1/4)*(4 + sqrt(15))^(1/2*(n-1))*(4 + sqrt(15))^(1/4*(-1)^(n-1)) + 1/12*(-1)^(n-1)*sqrt(15)*(4-sqrt(15))^(1/2*(n-1))*(4-sqrt(15))^(1/4*(-1)^(n-1))*(4 -sqrt(15))^(-1/4)-1/4*(4 + sqrt(15))^(-1/4)*(-1)^(n-1)*(4 + sqrt(15))^(1/2*(n-1))*(4 + sqrt(15))^(1/4*(-1)^(n-1))-1/10*sqrt(15)*(4-sqrt(15))^(1/2*(n-1))*(4-sqrt(15))^(1/4*( -1)^(n-1))*(4-sqrt(15))^(-1/4)-[C(2*n,n) mod 2], with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Aug 28 2009]
Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jun 30 2010: (Start)
r=sqrt(15);
a(n)=((15+r)*(4+r)^((n-1)/2)+(15-r)*(4-r)^((n-1)/2)+30)/60; n odd
a(n)=((45+11*r)*(4+r)^((n-2)/2)+(45-11*r)*(4-r)^((n-2)/2)+30)/60; n even
(End)
a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=12, a(5)=20, a(n)=a(n-1)+ 8*a(n-2)- 8*a(n-3)-a(n-4)+a (n-5) [From Harvey P. Dale, Nov 10 2011]
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EXAMPLE
| Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jun 30 2010: (Start)
for n=3 a(3)=3; b(3)=14; binomial(14,6)=3003;
binomial(14,4)*binomial(3,2)=1001*3=3003;
(End)
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MAPLE
| Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jun 30 2010: (Start)
n:=1: for m from 1 to 2000 do w:=sqrt(1+60*m*(m-1)):
if (w=floor(w)) then a(n)=m: b(n)=(9+w)/2: inc(n): end if: end do
(End)
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MATHEMATICA
| 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]] (* From Harvey P. Dale, Nov 10 2011 *)
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CROSSREFS
| Cf. A103200, A001090.
Sequence in context: A083265 A067391 A096361 * A205825 A076000 A096632
Adjacent sequences: A105042 A105043 A105044 * A105046 A105047 A105048
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KEYWORD
| nonn
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AUTHOR
| Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Apr 03 2005
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