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Preliminaries: Terminology and notation
A palindrome is any series of elements that is the same forwards and backwards. Each palindrome thus consists of two parts, which we shall call the head and the tail. The tail consists of the same series of elements as the head, but in reverse order. For example, in the palindromic word “noon,” the string “no” is the head, and “on” is the tail. In “noon,” the head and tail are entirely separate, but in a palindrome with an odd number of elements, the end of the head will overlap with the beginning of the tail. For example, the head of the palindromic word “level” is “lev,” and the tail is “vel”; a single “v” does double duty as the last element of the head and the first element of the tail.
No special notation is required to write a simple palindrome such as “noon” or “level,” but we are concerned in this post with repeating palindromic series (RPSs). If a palindrome consists of a head followed by a tail (possibly overlapping), an RPS is a head followed by a tail, then the head again, then the tail again, and so on to infinity. In the notation we will be using here, an RPS is represented by "RPS" followed by its head enclosed in parens (round brackets), thus:
- RPS (no) = noonnoonnoonnoonnoon...
If the end of the head overlaps with the beginning of the tail, an additional open-paren is placed before the first element of the tail, thus:
- RPS (le(v) = levellevellevellevellevel...
In an RPS, unlike a simple palindrome, overlap at the other end is also possible. For example, the word “grammar” is not itself a palindrome, but “grammar” endlessly repeated is an RPS. The letter “g” is both the beginning of the head and the end of the tail. This sort of overlap is indicated by placing a close-paren after the last element of the tail, thus:
- RPS (g)ram) = grammargrammargrammar...
It is possible to have overlap at both ends, as in this example.
- RPS (v)oo(d) = voodoovoodoovoodoo...
Overlap need not be limited to a single element; it can be a series of elements, provided that series is itself a palindrome. For example, consider the RPS created by endlessly repeating the word “sestet.”
- RPS (ses)t(e) = sestetsestetsestetsestet...
I should also mention that, much like a repeating decimal, an RPS has a beginning but not an end. As with a decimal like 0.16666666..., there may be a non-repeating segment at the beginning, before the RPS proper starts. Our notation can deal with this by putting this non-repeating segment before the first paren, as in this example.
- RPS n(eve)(r) = neverevereverevereverever...
One final note about the notation: If each element in the RPS can be represented by a single character (a letter, a one-digit number, etc.), it can be written as in the examples above, without commas or spaces. If the elements of an RPS are words, multidigit numbers, etc., the elements should be separated by commas and spaces.
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Several posts on this blog have dealt with the fact that, for any modulus k, the sequence of triangular numbers reduced modulo k will be an RPS. Two different proofs of this have been given (here and here). The pattern is easiest to see when k = 10, since (in the decimal system) any number reduced modulo 10 is equal to the final digit of that number.
The first several triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...
If we reduce that sequence modulo 10 (by replacing each number in the sequence with its final digit), we get: 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6...
This reduced sequence is RPS (0136051865). Any other modulus will also yield an RPS.
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Square numbers
Only some three months after proving the RPS theorem for triangular numbers did I notice that the sequence of square numbers shows a similar pattern. Below are the square numbers from 02 to 502. Start at the upper left, follow the zigzag down to the bottom, and then come back up the zigzag on the right.
Numbers in the same column have the same last digit (i.e., are congruent modulo 10). Numbers in the same row have the same last two digits (i.e., are congruent modulo 100).
Square numbers reduced modulo 10 = RPS (0)1496(5).
Square numbers reduced modulo 100 = RPS (0), 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 21, 44, 69, 96, 25, 56, 89, 24, 61, 0, 41, 84, 29, 76, (25).
I haven't checked if other moduli also yield RPSs, but, based on my experience with triangular numbers, and on the general principle that there is nothing mathematically special about powers of 10, I feel quite certain that they do.
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Other non-centered figurate numbers
Pentagonal numbers: 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187...
Pentagonal numbers reduced modulo 10 = RPS (01522510)275607)
Hexagonal numbers: 0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, 3003, 3160, 3321, 3486, 3655, 3828, 4005, 4186, 4371, 4560
Hexagonal numbers reduced modulo 10 = RPS (016585610(3)
Heptagonal numbers: 0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688
Heptagonal numbers reduced modulo 10 = 01784512895623906734 endlessly repeated -- not a palindrome!
Octagonal numbers reduced modulo 10 = RPS (01810)56(3)
Enneagonal numbers reduced modulo 10 = RPS (0194651441564910)69)
Decagonal numbers: 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326
Decagonal numbers reduced modulo 10 = RPS (010)725(6)
This is so bizarre that I almost think I must have made some mistake, but I'm pretty sure I haven't. Inductively, it looks like virtually all non-centered figurate numbers reduce to RPSs, but the heptagonal numbers are an exception! Why? Are there other exceptions?
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Centered figurate numbers
Centered triangular numbers: 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529
Centered triangular numbers reduced modulo 10 = RPS (1409164596)
Centered square numbers: 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, 4513
Centered square numbers reduced modulo 10 = RPS (15(3)
Centered pentagonal numbers: 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976, 3151, 3331, 3516, 3706, 3901, 4101, 4306, 4516, 4731, 4951, 5176, 5406
Centered pentagonal numbers reduced modulo 10 = RPS (16)
Hex numbers: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487
Hex numbers reduced modulo 10 = RPS (17(9)
Centered heptagonal numbers: 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246
Centered heptagonal numbers reduced modulo 10 = RPS (1823168736)
Centered octagonal numbers (i.e., odd squares): 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569
Centered octagonal numbers reduced modulo 10 = RPS (19(5)
Centered enneagonal numbers (i.e., every third triangular number): 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316
Centered enneagonal numbers reduced modulo 10 = RPS (1085160356)
Centered decagonal numbers: 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901
Centered decagonal numbers reduced modulo 10 = RPS (1)
Centered decagonal numbers reduced modulo 100 = RPS (1, 11, 31, 61, 1, 51, 11, 81, 61, 51)
Star numbers (i.e., centered dodecagonal numbers): 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837
Star numbers reduced modulo 10 = RPS (13(7)
It certainly looks as if all such sequences reduce to RPSs, but the unexpected exception of the (non-centered) heptagonal numbers makes me hesitant to jump to that conclusion.
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So the new mission (paging Kevin McCall!) is to come up with a general proof that almost all figurate number sequences reduce to RPSs -- a proof that makes it clear what the exceptions are and why. There's obviously a pattern here that goes beyond the triangular numbers, and it should be possible to express that pattern mathematically.
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