A320519
a(n) = 2*n^n*cos(n*arcsin(sqrt(4*n^2-1)/(2*n))) for n > 0 and a(0) = 2.
Original entry on oeis.org
2, 1, -7, -26, 449, 3001, -81863, -790271, 29441537, 371653678, -17549650999, -273657699479, 15647990550337, 290477695697281, -19502273553366631, -419946226750915874, 32375641678132277249, 793304922112119056161, -69057016330377072921047, -1897210272812469982747199
Offset: 0
A320570
a(n) = L_n(n), where L_n(x) is the Lucas polynomial.
Original entry on oeis.org
2, 1, 6, 36, 322, 3775, 54758, 946043, 18957314, 432083484, 11035502502, 312119004989, 9682664443202, 326872340718053, 11928306344169798, 467875943531657100, 19629328849962024962, 877095358067166709187, 41583555684469161804998, 2084882704791413248133431
Offset: 0
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[2] cat [(&+[(n/(n-j))*(Factorial(n-j)*n^(n-2*j)/(Factorial(j)*Factorial(n-2*j))): j in [0..Floor(n/2)]]): n in [1..20]]; // G. C. Greubel, Oct 15 2018
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Table[LucasL[n, n], {n, 0, 19}] (* or *)
Table[Round[((n + Sqrt[n^2 + 4])^n + (n - Sqrt[n^2 + 4])^n)/2^n], {n, 0, 19}] (* Round is equivalent to FullSimplify here *)
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for(n=0,20, print1(if(n==0,2, sum(j=0,floor(n/2), (n/(n-j))*((n-j)!*n^(n-2*j)/(j!*(n-2*j)!)))), ", ")) \\ G. C. Greubel, Oct 15 2018
A320565
a(n) = ((1 + sqrt(4*n^2 + 1))^n - (1 - sqrt(4*n^2 + 1))^n)/(2^n * sqrt(4*n^2 + 1)).
Original entry on oeis.org
0, 1, 1, 10, 33, 701, 4033, 132301, 1089921, 48460114, 520210801, 29215223489, 386721507745, 26250621340841, 413242502386337, 32899021525375426, 600383148312628737, 54846079150716441949, 1138470675779123657425, 117372939125452004885621
Offset: 0
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[2^(1-n)*(&+[Binomial(n,2*k+1)*(4*n^2+1)^k: k in [0..Floor(n/2)]]): n in [0..20]]; // G. C. Greubel, Oct 15 2018
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a[0] = Limit[n^(n - 1) Fibonacci[n, 1/n], n -> 0]; (* a[0] = 0 *)
a[n_] := a[n] = n^(n - 1) Fibonacci[n, 1/n];
Table[a[n], {n, 0, 19}]
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for(n=0,20, print1( 2^(1-n)*sum(k=0,floor(n/2), binomial(n,2*k+1)*(4*n^2+1)^k) , ", ")) \\ G. C. Greubel, Oct 15 2018
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