A371902 - OEIS

%I #51 Jun 21 2024 16:04:03

%S 1,3,6,13,27,55,110,221,443,886,1773,3547,7095,14190,28381,56763,

%T 113526,227053,454107,908215,1816430,3632861,7265723,14531446,

%U 29062893,58125787,116251575,232503150,465006301,930012603,1860025206,3720050413

%N 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.

%C 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.

%H Paolo Xausa, <a href="/A371902/b371902.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,0,0,0,1,-2).

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

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

%F G.f.: x*(1 + x + x^3 + x^4 + x^5)/((1 - 2*x)*(1 - x^7)). - _Stefano Spezia_, May 04 2024

%e 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.

%t Floor[110/127*2^Range[50]] (* _Paolo Xausa_, Jun 21 2024 *)

%Y Cf. A083026, A291454, A000225, A234046

%K nonn,easy

%O 1,2

%A _Federico Provvedi_, Apr 13 2024