A081168 Differences of Beatty sequence for square root of 10.

3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4

Offset

0,1

Comments

Let S(0) = 3; obtain S(k) from S(k-1) by applying the morphism 3 -> 333334, 4 -> 3333334; sequence is S(0), S(1), S(2), ...

More generally, for a(n,m) = floor((n+1)*sqrt(m^2+ 1)) - floor(n*sqrt(m^2+1)) start with m and apply the morphism: m -> m^(2m-1), m+1; m+1 -> m^(2m), m+1.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor((n+1)*sqrt(10)) - floor(n*sqrt(10)).

Mathematica

Differences[Floor[Sqrt[10]*Range[0, 120]]] (* G. C. Greubel, Jan 15 2024 *)

Prog

(PARI) a(n)=floor((n+1)*sqrt(10))-floor(n*sqrt(10))
(Magma)
A081168:= func< n | Floor((n+1)*Sqrt(10)) - Floor(n*Sqrt(10)) >;
[A081168(n): n in [0..120]]; // G. C. Greubel, Jan 15 2024
(SageMath)
def A081168(n): return floor((n+1)*sqrt(10)) - floor(n*sqrt(10))
[A081168(n) for n in range(121)] # G. C. Greubel, Jan 15 2024

Crossrefs

Cf. A010467, A006337, A177102.

Sequence in context: A247970 A063438 A276870 * A301415 A210509 A362049

Adjacent sequences: A081165 A081166 A081167 * A081169 A081170 A081171

Keyword

nonn

Author

Benoit Cloitre, Apr 16 2003

Status

approved