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Consider the sequence (Sloane's A118550)
defined this way:
a0=1.
an=an-1+n, if n is in the sequence.
an=an-1+1 if n is missing from the sequence.
The sequence features long sections of sequential elements (such as 92, 93,
94, 95, 96, 97, 98, 99, 100) which result in step climbs later on (a91=476,
a92=568, a93=661, a94=755, ..., a100=1340).
If you turn your head sideways and look at the second graph, you can kind of see
that. These step climbs, in turn, result in more long, nearly flat
stretches. A476 through A1340 is mostly sequential,
with only 9 big jumps.
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 I call this a "fractal sequence" because it has a shape that looks
flights of stairs, increasing in size with each flight. In addition, each
flight of stairs is the mirror image (reflected about the line y=mx, for some
number, m) of a previous flight.
To illustrate this self-similarity, note the small red rectangle at the lower
left of this graph. I have taken this section of the graph, reflected it
about the line 250a(n)=3500n, and blown it up to show the detail of the steps
inside it.
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 As you can
see, the same general shape, down to the number (five) of steps between
n=100 and n=250 are repeated in mirror image between n=1300 and n=2200.
This graph depicts the first 245 terms (a0 through a244),
which are 1, 2, 4, 5, 9, 14, 15, 16, 17, 26, 27, 28, 29, 30, 44, 59, 75, 92, 93, 94, 95, 96, 97, 98, 99, 100, 126, 153, 181, 210, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 568, 661, 755, 850, 946, 1043, 1141, 1240, 1340, 1341, 1342, 1343, 1344, 1345, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1361, 1362, 1363, 1364, 1365, 1491, 1492, 1493, 1494, 1495, 1496, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506, 1507, 1508, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1670, 1671, 1672, 1673, 1674, 1675, 1676, 1677, 1678, 1679, 1680, 1681, 1682, 1683, 1684, 1685, 1686, 1687, 1688, 1689, 1690, 1691, 1692, 1693, 1694, 1695, 1696, 1697, 1878, 1879, 1880, 1881, 1882, 1883, 1884, 1885, 1886, 1887, 1888, 1889, 1890, 1891, 1892, 1893, 1894, 1895, 1896, 1897, 1898, 1899, 1900, 1901, 1902, 1903, 1904, 1905, 1906, 2116, 2117, 2118, 2119, 2120, 2121, 2122, 2123, 2124, 2125, 2126, 2127, 2128, 2129, 2130, 2131, 2132, 2133, 2134, 2135, 2136, 2137, 2138, 2139, 2140, 2141, 2142, 2143, 2144, 2145, 2385, 2626, 2868, 3111, 3355. |
Internet References
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Recurrence Relations
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