A385610 Galileo sequence with ratio k = 2: a(1) = 1, a(2) = k, a(2*n-1) = floor(((k + 1)*a(n) -1)/2), and a(2*n) = floor((k + 1)*a(n)/2) + 1 for n > 2.

1, 2, 2, 4, 2, 4, 5, 7, 2, 4, 5, 7, 7, 8, 10, 11, 2, 4, 5, 7, 7, 8, 10, 11, 10, 11, 11, 13, 14, 16, 16, 17, 2, 4, 5, 7, 7, 8, 10, 11, 10, 11, 11, 13, 14, 16, 16, 17, 14, 16, 16, 17, 16, 17, 19, 20, 20, 22, 23, 25, 23, 25, 25, 26, 2, 4, 5, 7, 7, 8, 10, 11, 10, 11

Offset

1,2

Comments

A Galileo sequence of ratio k > 0 has the property that 1/k = a(1)/a(2) = (a(1) + a(2))/(a(3) + a(4)) = (a(1) + a(2) + a(3))/(a(4) + a(5) + a(6)) = ...

References

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 23.

Links

Stefano Spezia, Table of n, a(n) for n = 1..10000

William Cheah and David Treeby, Structure and Growth of Galileo Sequences, arXiv:2604.20889 [math.GM], 2026.

Example

1/2 = (1 + 2)/(2 + 4) = (1 + 2 + 2)/(4 + 2 + 4) = ...

Mathematica

k=2; a[1]=1; a[2]=k; a[n_]:=a[n]=If[OddQ[n], Floor[((k+1)*a[(n+1)/2]-1)/2], Floor[(k+1)*a[n/2]/2]+1]; Array[a, 75]

Crossrefs

Similar sequences for k=1..5: A037861, this sequence, A005408 [Galileo, 1615], A385587, A385643.

Sequence in context: A060609 A330882 A205138 * A233763 A109526 A309894

Adjacent sequences: A385607 A385608 A385609 * A385611 A385612 A385613

Keyword

nonn,easy,look

Author

Stefano Spezia, Jul 04 2025

Status

approved