A385643 -id:A385643 - cp's OEIS frontend

A385587 Galileo sequence with ratio k = 4: 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, 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

Stefano Spezia, Jul 03 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)) = ...

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

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

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, A385610, A005408 [Galileo, 1615], this sequence, A385643.

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]

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.

Original entry on oeis.org

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]

cp's OEIS Frontend

A385587 Galileo sequence with ratio k = 4: 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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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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cp's OEIS Frontend

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.

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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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A385587 Galileo sequence with ratio k = 4: 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.

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.