A385643 Galileo sequence with ratio k = 5: 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, 5, 14, 16, 41, 43, 47, 49, 122, 124, 128, 130, 140, 142, 146, 148, 365, 367, 371, 373, 383, 385, 389, 391, 419, 421, 425, 427, 437, 439, 443, 445, 1094, 1096, 1100, 1102, 1112, 1114, 1118, 1120, 1148, 1150, 1154, 1156, 1166, 1168, 1172, 1174, 1256, 1258, 1262

Offset

1,2

Comments

Solution to Exercise 1.2.3 on page 35 in Tattersall.

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, pages 23, 35.

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

1/5 = (1 + 5)/(14 + 16) = (1 + 5 + 14)/(16 + 41 + 43) = ...

Mathematica

k=5; 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, 51]

Crossrefs

Similar sequences for k=1..5: A037861, A385610, A005408 [Galileo, 1615], A385587, this sequence.

Sequence in context: A222561 A245015 A196366 * A306772 A230058 A230091

Adjacent sequences: A385640 A385641 A385642 * A385644 A385645 A385646

Keyword

nonn,easy,look

Author

Stefano Spezia, Jul 06 2025

Status

approved