A371902 Positive integers whose binary form follows the periodic pattern 1101110: the concatenation of halftones 2 2 1 2 2 2 1, diminished by one, between successive pitches in the Ionian Major Scale.

1, 3, 6, 13, 27, 55, 110, 221, 443, 886, 1773, 3547, 7095, 14190, 28381, 56763, 113526, 227053, 454107, 908215, 1816430, 3632861, 7265723, 14531446, 29062893, 58125787, 116251575, 232503150, 465006301, 930012603, 1860025206, 3720050413

Offset: 1

Author

Federico Provvedi, Apr 13 2024

Keywords

• nonn
• easy

Comments

The periodic binary digits of 55/107 is the pattern sequence A291454(n)-1 which is the new bit introduced into a(n): a(n+1) = 2*a(n) + A291454(n) - 1.

Examples

			For n=10, playing 10 + 1 = 11 notes of the major scale (in Ionian mode), the 10 intervals between the pitches C D E F G A B C' D' E' F' expressed in halftones are 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, whose values diminished by one give the binary form '1101110110', which in decimal is 886, hence a(10) = 886.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,0,1,-2).

Crossrefs

Cf. A083026, A291454, A000225, A234046

Programs

Mathematica

Floor[110/127*2^Range[50]] (* Paolo Xausa, Jun 21 2024 *)

Formula

a(n) = floor((110/127)*2^n).

D.g.f.: z^2*(z^5 + z^4 + z^2 + z + 1)/((2 - z) (1 - z^7)) = z * Dgf(A000225) * Dgf(A234046).

G.f.: x*(1 + x + x^3 + x^4 + x^5)/((1 - 2*x)*(1 - x^7)). - Stefano Spezia, May 04 2024