cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A091307 a(n)=6*2^n+4 (Bode Number A003461(n+2)) except for a(1)=6.

Original entry on oeis.org

6, 28, 52, 100, 196, 388, 772, 1540, 3076, 6148, 12292, 24580, 49156, 98308, 196612, 393220, 786436, 1572868, 3145732, 6291460, 12582916, 25165828, 50331652, 100663300, 201326596, 402653188, 805306372, 1610612740, 3221225476, 6442450948, 12884901892
Offset: 1

Views

Author

Alford Arnold, Feb 21 2004

Keywords

Comments

Sequence similar to Bode Numbers relevant to A079946 and numeric partitions.
A053445 describes certain partitions which start triangular arrays of all other numeric partitions; e.g. - 22, 33, 222, 44, 332, 2222, ... A079946 provides the indices for these partitions. (cf. A090324 and A090774).
By expanding the terms of a(n) in a similar manner, the vertex partitions can be readily indexed by noting that the indices increase by eight as follows: 6 28 (one case), 52 60 (two cases), 100 108 116 124 (four cases), 196 204 212 220 228 236 244 252 (eight cases), 388 ...

Examples

			a(3) = 52 because we can write 52 = 2*28 - 4.

Links

Crossrefs

Except for initial term, same as A003461(n+2). Cf. A053445, A079946, A090774.

Programs

  • Mathematica
    CoefficientList[Series[2x (3+5x-10x^2)/((1-x)(1-2x)),{x,0,30}],x] (* or *) LinearRecurrence[{3,-2},{0,6,28,52},40] (* Harvey P. Dale, Sep 01 2021 *)

Formula

a(1) = 6, a(2) = 28, a(n) = 2*a(n-1) - 4 for n > 2.
G.f.: 2*x*(3+5*x-10*x^2)/((1-x)*(1-2*x)). - Colin Barker, Mar 12 2012

Extensions

Edited by M. F. Hasler, Apr 07 2009

A317764 T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1 or 2 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 8, 10, 10, 8, 16, 20, 16, 20, 16, 32, 42, 28, 28, 42, 32, 64, 89, 52, 43, 52, 89, 64, 128, 190, 100, 72, 72, 100, 190, 128, 256, 407, 196, 127, 109, 127, 196, 407, 256, 512, 873, 388, 232, 177, 177, 232, 388, 873, 512, 1024, 1874, 772, 432, 302, 266, 302
Offset: 1

Views

Author

R. H. Hardin, Aug 06 2018

Keywords

Comments

Table starts
...1...2...4...8..16...32...64..128..256..512.1024..2048..4096..8192.16384
...2...6..10..20..42...89..190..407..873.1874.4024..8642.18561.39866.85627
...4..10..16..28..52..100..196..388..772.1540.3076..6148.12292.24580.49156
...8..20..28..43..72..127..232..432..813.1539.2922..5557.10577.20141.38362
..16..42..52..72.109..177..302..532..955.1733.3164..5796.10637.19541.35918
..32..89.100.127.177..266..425..709.1217.2126.3753..6666.11882.21223.37952
..64.190.196.232.302..425..639.1012.1663.2801.4792..8278.14385.25088.43852
.128.407.388.432.532..709.1012.1529.2413.3927.6524.10984.18651.31842.54552
.256.873.772.813.955.1217.1663.2413.3674.5798.9381.15434.25672.43007.72386

Examples

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..1. .0..1..1..0
..1..1..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1. .1..1..0..0
..1..1..1..1. .0..0..0..0. .0..0..0..0. .0..1..1..1. .1..0..0..0
..1..1..1..1. .0..0..0..0. .0..0..0..0. .1..1..1..1. .0..0..0..0
..0..0..0..0. .1..1..1..1. .0..0..0..1. .0..0..0..0. .0..0..0..0

Links

Crossrefs

Column 1 is A000079(n-1).
Column 3 is A003461 for n>1.

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 2*a(n-1) +a(n-2) -a(n-3) -a(n-4) for n>6
k=3: a(n) = 3*a(n-1) -2*a(n-2) for n>3
k=4: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4) +a(n-5) for n>6
k=5: a(n) = 2*a(n-1) -a(n-4) for n>6
k=6: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4) +a(n-7) for n>10
k=7: a(n) = 3*a(n-1) -2*a(n-2) -a(n-4) +a(n-5) -a(n-6) +a(n-7) for n>11

A061654 a(n) = (3*16^n + 2)/5.

Original entry on oeis.org

1, 10, 154, 2458, 39322, 629146, 10066330, 161061274, 2576980378, 41231686042, 659706976666, 10555311626650, 168884986026394, 2702159776422298, 43234556422756762, 691752902764108186, 11068046444225730970, 177088743107611695514, 2833419889721787128218, 45334718235548594051482
Offset: 0

Views

Author

G. L. Honaker, Jr., Jun 16 2001

Keywords

Comments

Integers arising in Bode's sequence (3*2^m + 4)/10.

Links

Crossrefs

Cf. A003461.

Programs

  • Mathematica
    (3*16^Range[0,20]+2)/5 (* or *) LinearRecurrence[{17,-16},{1,10},20] (* Harvey P. Dale, Feb 04 2015 *)
  • PARI
    a(n) = { (3*16^n + 2)/5 } \\ Harry J. Smith, Jul 25 2009

Formula

From Colin Barker, Mar 13 2012: (Start)
a(n) = 17*a(n-1) - 16*a(n-2).
G.f.: (1-7*x)/((1-x)*(1-16*x)). (End)

A131500 Radii of orbits of planets in solar system, in units of radius of orbit of Mercury, multiplied by 4.

Original entry on oeis.org

4, 7, 10, 16, 28, 52, 100, 196, 292, 388
Offset: 1

Views

Author

Alexander R. Povolotsky, Aug 12 2007, Aug 20 2007

Keywords

Comments

A003461 is the classical version of this sequence. - N. J. A. Sloane, Aug 15 2007
(Titius
Bode
n Planet Conversion Result TitiusBode Result Result)*4
= ======= ==================== =================== =========
1 Mercury 0.387/0.387 = 1.00 (3/4)*0 + 1 = 1 4
2 Venus 0.723/0.387 = 1.87 (3/4)*1 + 1 = 1.75 7
3 Earth 1.000/0.387 = 2.58 (3/4)*2 + 1 = 2.5 10
4 Mars 1.524/0.387 = 3.94 (3/4)*4 + 1 = 4 16
5 AstrBlt 2.77 /0.387 = 7.16 (3/4)*8 + 1 = 7 28
6 Jupiter 5.203/0.387 = 13.44 (3/4)*16 + 1 = 13 52
7 Saturn 9.539/0.387 = 24.65 (3/4)*32 + 1 = 25 100
8 Uranus 19.18/0.387 = 49.56 (3/4)*64 + 1 = 49 196
9 Neptune 30.06/0.387 = 77.67 (3/4)*96 + 1 = 73 292
10 Pluto 39.44/0.387 = 101.91 (3/4)*128 + 1 = 97 388
Includes both Neptune's and Pluto's orbits, which are missing from A003461.

Extensions

Edited by N. J. A. Sloane, Oct 23 2009

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.

Original entry on oeis.org

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

Views

Author

Raphie Frank, Jan 14 2013

Keywords

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

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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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