A370176 a(n) = floor(x*a(n-1)) for n > 0 where x = 3+sqrt(15), a(0) = 1.

1, 6, 41, 281, 1931, 13271, 91211, 626891, 4308611, 29613011, 203529731, 1398856451, 9614317091, 66079041251, 454160150051, 3121435147811, 21453571787171, 147450041609891, 1013421680382371, 6965230331953571, 47871912074015651, 329022854435815331, 2261368599058985891

Offset: 0

Author

Philippe Deléham, Mar 19 2024

Keywords

• nonn
• easy

Comments

x = A092294 = 3+sqrt(15) = 6.872983346...

Examples

			a(0) = 1;
a(1) = floor(x) = 6 where x = 3+sqrt(15);
a(2) = floor(6*x) = 41;
a(3) = floor(41*x) = 281.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,0,-6).

Crossrefs

Cf. A057089, A092294, A370174.

Programs

Mathematica

NestList[Floor[(Sqrt[15]+3)*#] &, 1, 25] (* or *)
LinearRecurrence[{7, 0, -6}, {1, 6, 41}, 25] (* Paolo Xausa, Mar 31 2024 *)

Formula

a(n) = 7*a(n-1) - 6*a(n-3), a(0) = 1, a(1) = 6, a(2) = 41.

a(n) = 6*a(n-1) + 6*a(n-2) - 1.

a(n) = ((30-7*sqrt(15))*(3-sqrt(15))^n + (30+7*sqrt(15))*(3+sqrt(15))^n + 6)/66.

G.f.: (1-x-x^2)/(1-7*x+6*x^3).

a(n) = Sum_{k = 0..n} A370174(n,k)*5^k.

a(n) = (10*A057089(n) + 5*A057089(n-1) + 1)/11.