A385587 Galileo sequence with ratio k = 4: 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, 4, 9, 11, 22, 23, 27, 28, 54, 56, 57, 58, 67, 68, 69, 71, 134, 136, 139, 141, 142, 143, 144, 146, 167, 168, 169, 171, 172, 173, 177, 178, 334, 336, 339, 341, 347, 348, 352, 353, 354, 356, 357, 358, 359, 361, 364, 366, 417, 418, 419, 421, 422, 423, 427, 428, 429

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)) = ...

In Tattersall reference the terms a(7) = 27 and a(8) = 28 miss.

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

1/4 = (1 + 4)/(9 + 11) = (1 + 4 + 9)/(11 + 22 + 23) = ...

Mathematica

k=4; 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, 57]

Crossrefs

Similar sequences for k=1..5: A037861, A385610, A005408 [Galileo, 1615], this sequence, A385643.

Sequence in context: A182244 A312847 A141365 * A179055 A277428 A002641

Adjacent sequences: A385584 A385585 A385586 * A385588 A385589 A385590

Keyword

nonn,easy,look

Author

Stefano Spezia, Jul 03 2025

Status

approved