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A033190
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a(n) = Sum_{k=0..n} binomial(n,k) * binomial(Fibonacci(k)+1,2).
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3
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0, 1, 3, 9, 28, 90, 297, 1001, 3431, 11917, 41820, 147918, 526309, 1881009, 6744843, 24244145, 87300092, 314765506, 1135980801, 4102551897, 14823628015, 53581222773, 193724727804, 700551945014, 2533702591613, 9164618329825, 33151607475987, 119927166988761
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OFFSET
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0,3
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COMMENTS
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Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 3. - Herbert Kociemba, Jun 14 2004
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LINKS
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FORMULA
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G.f.: (-x^4+6x^3-5x^2+x)/((1-3x+x^2)*(1-5x+5x^2)).
a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)*sin(3*r*Pi/10)*(2*cos(r*Pi/10))^(2*n), n >= 1.
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), n >= 5. (End)
a(2n) = (5*Fibonacci(4*n) + (5^n)*Lucas(2*n))/10 for n > 0.
a(2n+1) = (Fibonacci(4*n+2) + (5^n)*Fibonacci(2*n+1))/2 for n >= 0.
(End)
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MAPLE
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add(binomial(n, k)*binomial(combinat[fibonacci](k)+1, 2), k=0..n) ;
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MATHEMATICA
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LinearRecurrence[{8, -21, 20, -5}, {0, 1, 3, 9, 28}, 30] (* Harvey P. Dale, Jan 24 2019 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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