A385587 -id:A385587 - cp's OEIS frontend

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

Stefano Spezia, Jul 04 2025

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)) = ...

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

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

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

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

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]

A385643 Galileo sequence with ratio k = 5: a(1) = 1, a(2) = k, a(2n-1) = floor(((k + 1)a(n) -1)/2), and a(2n) = floor((k + 1)a(n)/2) + 1 for n > 2.

Original entry on oeis.org

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

Stefano Spezia, Jul 06 2025

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)) = ...

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

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 23, 35.

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

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

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]

cp's OEIS Frontend

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

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A385643 Galileo sequence with ratio k = 5: a(1) = 1, a(2) = k, a(2n-1) = floor(((k + 1)a(n) -1)/2), and a(2n) = floor((k + 1)a(n)/2) + 1 for n > 2.

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cp's OEIS Frontend

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.

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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.

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

A385643 Galileo sequence with ratio k = 5: a(1) = 1, a(2) = k, a(2n-1) = floor(((k + 1)a(n) -1)/2), and a(2n) = floor((k + 1)a(n)/2) + 1 for n > 2.