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 Math Help > Geometry > Fractals > Fractal Sequence

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.

 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.
 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.

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