A370175 a(n) = floor(x*a(n-1)) for n > 0 where x = (5+3*sqrt(5))/2, a(0) = 1.

1, 5, 29, 169, 989, 5789, 33889, 198389, 1161389, 6798889, 39801389, 233001389, 1364013889, 7985076389, 46745451389, 273652638889, 1601990451389, 9378215451389, 54901029513889, 321396224826389, 1881486271701389, 11014412482638889, 64479493771701389

Offset: 0

Author

Philippe Deléham, Mar 18 2024

Keywords

• nonn
• easy

Comments

x = A090550 = 1 + 3*phi = 5.854101966..., where phi is the golden ratio.

Examples

			a(0) = 1, a(1) = floor(x) = 5 where x = (5+3*sqrt(5))/2.
a(2) = floor(5*x) = 29, a(3) = floor(29*x) = 169.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,0,-5).

Crossrefs

Cf. A057088, A090550, A370174.

Programs

Mathematica

NestList[Floor[#*(5 + 3*Sqrt[5])/2] &, 1, 30] (* or *)
LinearRecurrence[{6, 0, -5}, {1, 5, 29}, 30] (* Paolo Xausa, May 25 2024 *)

Formula

a(n) = 6*a(n-1) - 5*a(n-3), a(0) = 1, a(1) = 5, a(2) = 29.

a(n) = 5*a(n-1) + 5*a(n-2) - 1.

a(n) = (4*(5-2*sqrt(5))*((5-3*sqrt(5))/2)^n + 4*(5+2*sqrt(5))*((5+3*sqrt(5))/2)^n + 5)/45.

G.f.: (1 - x - x^2)/(1 - 6*x + 5*x^3).

a(n) = Sum_{k = 0..n} A370174(n,k)*4^k.

a(n) = (8*A057088(n) + 4*A057088(n-1) + 1)/9.