A209257 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.

4, 7, 10, 16, 28, 52, 97, 193, 301, 493, 1150, 1162, 3076, 2386, 3283, 10423, 5827, 20659, 9646, 37852, 15112, 18592, 83692, 27331, 133660, 38857, 45832, 251050, 62566, 367318, 83527, 523315, 109375, 124351, 852826, 158872, 1152508, 200140, 223561, 1754809

Offset: 0

Author

Raphie Frank, Jan 14 2013

Keywords

• nonn
• easy

Comments

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

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

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

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):

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

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

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

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

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

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

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

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

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

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:

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

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

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.

Examples

			Z_1 = 3*((1 - 1 +  1 -  1 +  1) + (-1 * 1)) + 4 =   4,
Z_2 = 3*((1 + 0 +  0 +  0 +  0) +  (0 * 0)) + 4 =   7,
Z_3 = 3*((1 + 1 +  0 +  0 +  0) +  (0 * 0)) + 4 =  10,
Z_4 = 3*((1 + 2 +  1 +  0 +  0) +  (0 * 1)) + 4 =  16,
Z_5 = 3*((1 + 3 +  3 +  1 +  0) +  (0 * 0)) + 4 =  28,
Z_6 = 3*((1 + 4 +  6 +  4 +  1) +  (0 * 1)) + 4 =  52,
Z_7 = 3*((1 + 5 + 10 + 10 +  5) +  (1 * 0)) + 4 =  97,
Z_8 = 3*((1 + 6 + 15 + 20 + 15) +  (6 * 1)) + 4 = 193,
Z_9 = 3*((1 + 7 + 21 + 35 + 35) + (21 * 0)) + 4 = 301.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• J. Baez, This Week's Finds in Mathematical Physics (Week 234)
• H. Chang, Titius-Bode’s Relation and Distribution of Exoplanets
• D. M. F. Chapman, The Titius-Bode Rule, Part 2: Science or Numerology?
• Chemteam, The Balmer Formula
• A. L. Kholodenko, Role of general relativity and quantum mechanics in dynamics of Solar System
• Wikipedia, Dwarf planet
• Wikipedia, Planet
• Wikipedia, Titius-Bode law

Crossrefs

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

Programs

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

Mathematica

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 *)

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};
for(n=0,30, print1(z(n), ", ")) \\ G. C. Greubel, Jan 07 2018

Formula

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.

Extensions

a(18) corrected by G. C. Greubel, Jan 07 2018