A099134 -id:A099134 - cp's OEIS frontend

A099133 4^(n-1)*Fibonacci(n).

0, 1, 4, 32, 192, 1280, 8192, 53248, 344064, 2228224, 14417920, 93323264, 603979776, 3909091328, 25300041728, 163745628160, 1059783180288, 6859062771712, 44392781971456, 287316132233216, 1859549040476160, 12035254277636096, 77893801758162944

Offset: 0

Paul Barry, Sep 29 2004

Binomial transform of A099134.
Second binomial transform of x/(1-20x^2), or (0,1,0,20,0,400,0,8000,....).
In general k^(n-1)*Fibonacci(n) has g.f. x/(1-kx-k^2x^2).
The ratio a(n+1)/a(n) converges to 4 times the golden ratio as n approaches infinity. In general, the ratio a(n+1)/a(n) of the sequence which is the solution to the linear recurrence relation a(n) = m*a(n-1)+m^2*a(n-2) with a(0)=0 and a(1) = 1 converges to m times the golden ratio as n approaches infinity where m is a positive integer. - Felix P. Muga II, Mar 10 2014

			G.f. = x + 4*x^2 + 32*x^3 + 192*x^4 + 1280*x^5 + 8192*x^6 + 53248*x^7 + ...

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivré Formula, March 2014; Preprint on ResearchGate.

Index entries for linear recurrences with constant coefficients, signature (4,16).

Cf. A000045, A099012, A085449. Fourth row of A234357.

Join[{a=0,b=1},Table[c=4*b+16*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011*)
Table[4^(n-1) Fibonacci[n],{n,0,20}] (* Harvey P. Dale, Aug 22 2012 *)
LinearRecurrence[{4,16},{0,1},30] (* Harvey P. Dale, Aug 22 2012 *)

a(n) = 4^(n-1)*fibonacci(n); \\ Michel Marcus, Jan 10 2014

G.f.: x/(1-4*x-16*x^2).
a(n) = 4*a(n-1) + 16*a(n-2).
a(n) = (2+2*sqrt(5))^n/(4*sqrt(5))-(2-sqrt(5))^n/(4*sqrt(5)).
a(-n) = -(-1)^n * a(n) / 16^n for all n in Z. - Michael Somos, Mar 18 2014

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).

Original entry on oeis.org

5, 7, 101, 11, 13, 269, 17, 19, 509, 23, 709, 821, 29, 31, 46957, 55399, 37, 168846239, 41, 43, 9177868096974864412935432937651459122761, 47, 485329129, 2789, 53, 3229, 3461, 59, 61, 1563353111, 139237612541, 67, 5021, 71, 73, 484639, 6221, 79, 6869, 83, 7549

Offset: 1

XU Pingya, Sep 24 2017

nonn

			For k = {1, 2, 3, 4, 5}, ((1 + sqrt(6))^k - (1 - sqrt(6))^k)/(2*sqrt(6)) = {1, 2, 9, 28, 101}. 101 is odd prime, so a(3) = 101.

Cf. A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914, A079773, A161007, A099134.

g[n_, k_] := ((1 + Sqrt[n])^k - (1 - Sqrt[n])^k)/(2Sqrt[n]);
Table[k = 3; While[! PrimeQ[Expand@g[2n, k]], k++]; Expand@g[2n, k], {n, 41}]

g(n,k) = ([0,1;2*n-1,2]^k*[0;1])[1,1]
a(n) = for(k=3,oo,if(ispseudoprime(g(n,k)),return(g(n,k)))) \\ Jason Yuen, Apr 12 2025

When 2*n + 3 = p is prime, a(n) = p.

cp's OEIS Frontend

A099133 4^(n-1)*Fibonacci(n).

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A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).

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cp's OEIS Frontend

A099133 4^(n-1)*Fibonacci(n).

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A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2*n))^k - (1 - sqrt(2*n))^k)/(2*sqrt(2*n)).

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A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).