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User:Jon Awbrey/TABLES
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A061396
Plain 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 |