%I #30 Sep 08 2022 08:46:01

%S 4,7,10,16,28,52,97,193,301,493,1150,1162,3076,2386,3283,10423,5827,

%T 20659,9646,37852,15112,18592,83692,27331,133660,38857,45832,251050,

%U 62566,367318,83527,523315,109375,124351,852826,158872,1152508,200140

%N A musically inspired Titius-Bode-like sequence based on the geometric division of 4- and 5-dimensional space: Z_(n+1) = 3 * (C(n-1, 0) + C(n-1, 1) + C(n-1, 2) + C(n-1, 3) + C(n-1, 4) + C(n-1, 5)*A059620(n+6)) + 4.

%C The classical Titius-Bode version of this sequence is given in A003461.

%C C(n, 0) + C(n, 1) + C(n, 2) + C(n, 3) + C(n, 4) = A000127(n) = A059173(n+1)/2.

%C C(n, 0) + C(n, 1) + C(n, 2) + C(n, 3) + C(n, 4) + C(n, 5) = A006261(n) = A059174(n+1)/2.

%C Where planetary and dwarf-planetary distances from the Sun at semi-major axis are expressed in astronomical units/10, then compare the following (noting that the running correlation coefficient, r, trends upwards as the population size increases):

%C n = 0, Mercury @ semi-major = 3.8710 vs. 4.0 --> 96.78%.

%C n = 1, Venus @ semi-major = 7.2333 vs. 7.0 --> 103.33%.

%C n = 2, Earth @ semi-major = 10.0000 vs. 10.0 --> 100.00%, r = 0.998430.

%C n = 3, Mars @ semi-major = 15.2368 vs. 16.0 --> 95.23%, r = 0.998356.

%C n = 4, Ceres @ semi-major = 27.654 vs. 28.0 --> 98.76%, r = 0.999412.

%C n = 5, Jupiter @ semi-major = 52.0427 vs. 52.0 --> 100.08%, r = 0.999809.

%C n = 6, Saturn @ semi-major = 95.8202 vs. 97.0 --> 98.78%, r = 0.999937.

%C n = 7, Uranus @ semi-major = 192.2941 vs. 193.0 --> 99.63%, r = 0.999981.

%C n = 8, Neptune @ semi-major = 301.0366 vs. 301.0 --> 100.01%, r = 0.999990.

%C The correspondence between this sequence and planetary distances breaks down subsequent to Neptune unless one adopts the conceit of considering the outer four dwarf planets -- Pluto, Haumea, MakeMake and Eris -- as one unit occupying one "planetary band" (note that Eris @ perihelion is inside the Kuiper Belt). Then:

%C n = 9, Pluto/Haumea/MakeMake/Eris @ semi-major ~ 490.492 average vs. 493.0 --> 99.49%, r = 0.999994.

%C Empirical source: Wikipedia planet pages as of Jan 14 2013.

%C This sequence originated as part of an attempt to compare and contrast the "good" numerology of Johann Balmer to the "bad" numerology of Titius-Bode. Coincidentally, (Totient(C(31, 0) + C(31, 1) + C(31, 2) + C(31, 3) + C(31, 4)))/10^11 equals 3.6456*10^-7, in meters, the Balmer constant as given by Johann Balmer in 1885.

%H G. C. Greubel, <a href="/A209257/b209257.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Baez, <a href="http://math.ucr.edu/home/baez/week234.html">This Week's Finds in Mathematical Physics (Week 234)</a>

%H H. Chang, <a href="http://www.janss.kr/Upload/files/JASS/27_1_01.pdf">Titius-Bode’s Relation and Distribution of Exoplanets </a>

%H D. M. F. Chapman, <a href="http://astronomia.udea.edu.co/sitios/astronomia-2.0/pages/metcomp.rs/files/Tareas/Tarea5/71755174/2001JRASC..95..189C.pdf">The Titius-Bode Rule, Part 2: Science or Numerology?</a>

%H Chemteam, <a href="http://www.chemteam.info/Electrons/Balmer-Formula.html">The Balmer Formula</a>

%H A. L. Kholodenko, <a href="http://arxiv.org/abs/1003.5696">Role of general relativity and quantum mechanics in dynamics of Solar System</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dwarf_planet">Dwarf planet</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Planet">Planet</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Titius-Bode_law">Titius-Bode law</a>

%F Z_(n+1) = 3 * (C(n-1, 0) + C(n-1, 1) + C(n-1, 2) + C(n-1, 3) + C(n-1, 4) + C(n-1, 5)*(floor((5*(n+6)+7)/12) - floor((5*(n+6)+2)/12)))) + 4.

%e Z_1 = 3*((1 - 1 + 1 - 1 + 1) + (-1 * 1)) + 4 = 4,

%e Z_2 = 3*((1 + 0 + 0 + 0 + 0) + (0 * 0)) + 4 = 7,

%e Z_3 = 3*((1 + 1 + 0 + 0 + 0) + (0 * 0)) + 4 = 10,

%e Z_4 = 3*((1 + 2 + 1 + 0 + 0) + (0 * 1)) + 4 = 16,

%e Z_5 = 3*((1 + 3 + 3 + 1 + 0) + (0 * 0)) + 4 = 28,

%e Z_6 = 3*((1 + 4 + 6 + 4 + 1) + (0 * 1)) + 4 = 52,

%e Z_7 = 3*((1 + 5 + 10 + 10 + 5) + (1 * 0)) + 4 = 97,

%e Z_8 = 3*((1 + 6 + 15 + 20 + 15) + (6 * 1)) + 4 = 193,

%e Z_9 = 3*((1 + 7 + 21 + 35 + 35) + (21 * 0)) + 4 = 301.

%t Z[n_]:= 3*(Binomial[n - 1, 0] + Binomial[n - 1, 1] + Binomial[n - 1, 2] + Binomial[n - 1, 3] + Binomial[n - 1, 4] + Binomial[n - 1, 5]*(Floor[(5 (n + 6) + 7)/12] - Floor[(5 (n + 6) + 2)/12])) + 4; Table[Z[n], {n, 0, 50}] (* _G. C. Greubel_, Jan 07 2018 *)

%o (PARI) {z(n) = 3*(binomial(n-1,0) + binomial(n-1,1) + binomial(n-1,2) + binomial(n-1,3) + binomial(n-1,4) + binomial(n-1,5)*(floor((5*(n+6) + 7)/12) - floor((5*(n+6)+2)/12))) + 4};

%o for(n=0,30, print1(z(n), ", ")) \\ _G. C. Greubel_, Jan 07 2018

%o (Magma) [3*(Binomial(n-1,0) + Binomial(n-1,1) + Binomial(n-1,2) + Binomial(n-1,3) + Binomial(n-1,4) + Binomial(n-1,5)*(Floor((5*(n+6) + 7)/12) - Floor((5*(n+6)+2)/12))) + 4: n in [0..30]]; // _G. C. Greubel_, Jan 07 2018

%Y Cf. A003461, A059620, A000127, A059173, A006261, A059174.

%K nonn

%O 0,1

%A _Raphie Frank_, Jan 14 2013

%E a(18) corrected by _G. C. Greubel_, Jan 07 2018