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A061396

Plain Wiki Table

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Nested Wiki Table

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Old ASCII Version

Illustration of initial terms of A061396
Jon Awbrey (jawbrey(AT)oakland.edu)

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_(p_p)   @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p_(p^p)   @     o      @            ((((())))())
|                                        ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =                    /   \       |
|           p_1^p_(p_1)       p^p_p     @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =                  /           |
|           p_(p_1)^(p_1)     p_p^p     @            @            ((())((())))
|                                        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =                       ^          o---o
|           p_1^(p_1^2) =                /           |
|           p_1^(p_1^p_1)     p^(p^p)   @            @            (((((())))))
|
o--------------------------------------------------------------------------------

Further Comments:

Here are a couple more pages from my notes,
where it looks like I first arrived at the
generating function, and also carried out
some brute force enumerations of riffs.

I am going to experiment with a different way of
transcribing indices and powers into a plaintext.

|                jj
|              p<
|      j      /  ji
|    p<     p<         etc.
|      i      \  ij
|              p<
|                ii

-------------------------------------------------------

1978-11-06

Generating Function

| R(x) = 1 + x + 2x^2 + ...
|
|      =   1 + x.x^0 (1 + x + 2x^2 + ...)
|        . 1 + x.x^1 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . ...
|
|      = 1 + x + 2x^2 + ...
|
| Product over (i = 0 to infinity) of (1 + x.x^i.R(x))^R_i  =  R(x)

-------------------------------------------------------

1978-11-10

Brute force enumeration of R_n

| 4 p's
|
|       p
|     p<        p_p                 p                    p
|   p<        p<        p p_p     p<_p     p_p_p     p_p<
| p<        p<        p<        p<       p<        p<
|
|
|       p
|     p<        p_p                 p                    p
| p_p<      p_p<      p<        p_p<_p   p_p_p_p   p_p_p<
|                       p p_p
|
|
|     p
|   p<        p_p       p         p        p           p
| p<        p<        p<        p<       p<  p<    p p<
|   p         p         p_p       p^p          p       p
|
|
| p p_p_p   p p<
|               p^p
|

Altogether, 20 riffs of weight 4.

| o---------------------o---------------------o---------------------o
| | 3                   | 4                   | 5                   |
| o---------------------o---------------------o---------------------|
| | // // 2             | 10, 3, 1, 6         | 36, 10, 2, 3, 2, 20 |
| o---------------------o---------------------o---------------------|
| |                     | 0^1 4^1,            |                     |
| |                     | 1^1 3^1,            |                     |
| |                     | 2^2,                |                     |
| |                     | 4^1 0^1             |                     |
| o---------------------o---------------------o---------------------o
| | 6                   | 20                  | 73                  |
| o---------------------o---------------------o---------------------o
|

-------------------------------------------------------

Here are the number values of the riffs on 4 nodes:

o----------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
|   p<        p<        p p_p     p<_p     p_p_p     p_p<
| p<        p<        p<        p<       p<        p<
|
| 2^16      2^8       2^6       2^9      2^5       2^7
| 65536     256       64        512      32        128
o----------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
| p_p<      p_p<      p<        p_p<_p   p_p_p_p   p_p_p<
|                       p p_p
|
| p_16      p_8       p_6       p_9      p_5       p_7
| 53        19        13        23       11        17
o----------------------------------------------------------------------
|
|     p
|   p<        p_p       p         p                    p
| p<        p<        p<        p<       p^p p_p   p p<
|   p         p         p_p       p^p                  p
|
| 3^4       3^3       5^2       7^2
| 81        27        25        49       12        18
o----------------------------------------------------------------------
|
| p p_p_p   p p<
|               p^p
|
| 10        14 
o----------------------------------------------------------------------

For ease of reference, I include the previous table
of smaller riffs and rotes, redone in the new style.

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_p_p     @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p<        @     o      @            ((((())))())
|                               p^p      ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =           p_p      /   \       |
|           p_1^p_(p_1)       p<        @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =         p        /           |
|           p_(p_1)^(p_1)     p<        @            @            ((())((())))
|                               p        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =               p       ^          o---o
|           p_1^(p_1^2) =       p<       /           |
|           p_1^(p_1^p_1)     p<        @            @            (((((())))))
|
o--------------------------------------------------------------------------------

(later)

Expanded version of first table:

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_p_p     @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p<        @     o      @            ((((())))())
|                               p^p      ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =           p_p      /   \       |
|           p_1^p_(p_1)       p<        @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =         p        /           |
|           p_(p_1)^(p_1)     p<        @            @            ((())((())))
|                               p        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =               p       ^          o---o
|           p_1^(p_1^2) =       p<       /           |
|           p_1^(p_1^p_1)     p<        @            @            (((((())))))
|
o--------------------------------------------------------------------------------

o================================================================================
|
|       p
|     p<        p          p_p         p
|   p<        p<_p       p<        p_p<      p p_p     p_p_p
| p<        p<         p<        p<        p<        p<
|
| 2^16      2^9        2^8       2^7       2^6       2^5
| 65536     512        256       128       64        32
|
o--------------------------------------------------------------------------------
|
|       p
|     p<        p          p_p         p
| p_p<      p_p<_p     p_p<      p_p_p<    p<        p_p_p_p
|                                            p p_p
|
| p_16      p_9        p_8       p_7       p_6       p_5
| 53        23         19        17        13        11
|
o--------------------------------------------------------------------------------
|
|   p^p       p_p        p         p
| p<        p<         p<        p<
|   p         p          p^p       p_p
|
| 3^4       3^3        7^2       5^2
| 81        27         49        25
|
o--------------------------------------------------------------------------------
|
|     p
| p p<      p p<       p^p p_p   p p_p_p
|     p         p^p
|
| 18        14         12        10
|
o================================================================================

Triangle in which k-th row lists natural number
values for the collection of riffs with k nodes.

k | natural numbers n such that |riff(n)| = k
--o------------------------------------------------
0 | 1;
1 | 2;
2 | 3, 4;
3 | 5, 6, 7, 8, 9, 16;
4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27,
  | 32, 49, 53, 64, 81, 128, 256, 512, 65536;

The natural number values for the riffs with
at most 3 pts are as follows (@'s are roots):

|                  o       o  o       o
|                  |       ^  |       ^
|                  v       |  v       |
|            o  o  o    o  o  o  o o  o
|            |  ^  |    |  |  ^  | ^  ^
|            v  |  v    v  v  |  v/   |
| Riff:   @; @, @; @, @ @, @, @, @,   @;
|
| Value:  2; 3, 4; 5,  6 , 7, 8, 9,  16;

---------------------------------------------------

1, 2, 3, 4, 5, 6, 7, 8, 9, 16,
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536,

---------------------------------------------------

1; 2; 3, 4; 5, 6, 7, 8, 9, 16;
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536;

---------------------------------------------------

Example of a Raw HTML Table

 
 
   
   
     
     
 
 

Figurate Numbers

Regular Polygonal Numbers

Data copied from Figurate numbers#Regular polygonal numbers

Wiki Table

–(A067998) 0-gonal
numbers
1 0 –3 –8 –15 –24 –35 –48 –63 –80 –99 –120 –143 –168 –195 –224 –255 –288 –323 –360
A080956 1-gonal
numbers
1 1 0 –2 –5 –9 –14 –20 –27 –35 –44 –54 –65 –77 –90 –104 –119 –135 –152 –170
A000027 2-gonal
numbers
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A000217 Triangular
numbers
1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210
A000290 Square
numbers
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400
A000326 Pentagonal
numbers
1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 376 425 477 532 590
A000384 Hexagonal
numbers
1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 496 561 630 703 780
A000566 Heptagonal
numbers
1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 616 697 783 874 970
A000567 Octagonal
numbers
1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 736 833 936 1045 1160
A001106 Nonagonal
numbers
1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 856 969 1089 1216 1350
A001107 Decagonal
numbers
1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 976 1105 1242 1387 1540
A051682 11-gonal
numbers
1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 1096 1241 1395 1558 1730
A051624 12-gonal
numbers
1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 1216 1377 1548 1729 1920
A051865 13-gonal
numbers
1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 1336 1513 1701 1900 2110
A051866 14-gonal
numbers
1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 1456 1649 1854 2071 2300
A051867 15-gonal
numbers
1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 1576 1785 2007 2242 2490
A051868 16-gonal
numbers
1 16 45 88 145 216 301 400 513 640 781 936 1105 1288 1485 1696 1921 2160 2413 2680
A051869 17-gonal
numbers
1 17 48 94 155 231 322 428 549 685 836 1002 1183 1379 1590 1816 2057 2313 2584 2870
A051870 18-gonal
numbers
1 18 51 100 165 246 343 456 585 730 891 1068 1261 1470 1695 1936 2193 2466 2755 3060
A051871 19-gonal
numbers
1 19 54 106 175 261 364 484 621 775 946 1134 1339 1561 1800 2056 2329 2619 2926 3250
A051872 20-gonal
numbers
1 20 57 112 185 276 385 512 657 820 1001 1200 1417 1652 1905 2176 2465 2772 3097 3440

Centered Polygonal Numbers

Data copied from Figurate numbers#Centered polygonal numbers

Wiki Table

A000012 Centered
0-gonal
numbers
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000124 Centered
1-gonal
numbers
1 2 4 7 11 16 22 29 37 46 56 67 79 92 106 121 137 154 172 191
A002061 Centered
2-gonal
numbers
1 3 7 13 21 31 43 57 73 91 111 133 157 183 211 241 273 307 343 381
A005448 Centered
triangular
numbers
1 4 10 19 31 46 64 85 109 136 166 199 235 274 316 361 409 460 514 571
A001844 Centered
square
numbers
1 5 13 25 41 61 85 113 145 181 221 265 313 365 421 481 545 613 685 761
A005891 Centered
pentagonal
numbers
1 6 16 31 51 76 106 141 181 226 276 331 391 456 526 601 681 766 856 951
A003215 Centered
hexagonal
numbers
1 7 19 37 61 91 127 169 217 271 331 397 469 547 631 721 817 919 1027 1141
A069099 Centered
heptagonal
numbers
1 8 22 43 71 106 148 197 253 316 386 463 547 638 736 841 953 1072 1198 1331
A016754 Centered
octagonal
numbers
1 9 25 49 81 121 169 225 289 361 441 529 625 729 841 961 1089 1225 1369 1521
A060544 Centered
nonagonal
numbers
1 10 28 55 91 136 190 253 325 406 496 595 703 820 946 1081 1225 1378 1540 1711
A062786 Centered
decagonal
numbers
1 11 31 61 101 151 211 281 361 451 551 661 781 911 1051 1201 1361 1531 1711 1901
A069125 Centered
11-gonal
numbers
1 12 34 67 111 166 232 309 397 496 606 727 859 1002 1156 1321 1497 1684 1882 2091
A003154 Centered
12-gonal
numbers
1 13 37 73 121 181 253 337 433 541 661 793 937 1093 1261 1441 1633 1837 2053 2281
A069126 Centered
13-gonal
numbers
1 14 40 79 131 196 274 365 469 586 716 859 1015 1184 1366 1561 1769 1990 2224 2471
A069127 Centered
14-gonal
numbers
1 15 43 85 141 211 295 393 505 631 771 925 1093 1275 1471 1681 1905 2143 2395 2661
A069128 Centered
15-gonal
numbers
1 16 46 91 151 226 316 421 541 676 826 991 1171 1366 1576 1801 2041 2296 2566 2851
A069129 Centered
16-gonal
numbers
1 17 49 97 161 241 337 449 577 721 881 1057 1249 1457 1681 1921 2177 2449 2737 3041
A069130 Centered
17-gonal
numbers
1 18 52 103 171 256 358 477 613 766 936 1123 1327 1548 1786 2041 2313 2602 2908 3231
A069131 Centered
18-gonal
numbers
1 19 55 109 181 271 379 505 649 811 991 1189 1405 1639 1891 2161 2449 2755 3079 3421
A069132 Centered
19-gonal
numbers
1 20 58 115 191 286 400 533 685 856 1046 1255 1483 1730 1996 2281 2585 2908 3250 3611
A069133 Centered
20-gonal
numbers
1 21 61 121 201 301 421 561 721 901 1101 1321 1561 1821 2101 2401 2721 3061 3421 3801