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COMMENTS
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A nontrivial palindrome is a palindrome of length 2 or greater. (E.g., "1" is a trivial palindrome, but "11" and "121" are nontrivial palindromes.)
For example, 0062 is a string of length 4 over a seven letter alphabet that begins with a nontrivial palindrome (00).
7 divides a(n) for all n.
Number of walks of n steps that begin with a palindromic sequence on the complete graph K_7 with loops. (E.g., 0, 1, 1, 0, 4, 1, 2 is a valid walk with 7 steps and begins with the palindromic sequence '0110'.)
lim n -> infinity a(n)/7^n ~ 0.30271398450898696 is the probability that a random, infinite base-7 string begins with a nontrivial palindrome.
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EXAMPLE
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For n=3 the a(3) = 91 solutions are: 000, 001, 002, 003, 004, 005, 006, 010, 020, 030, 040, 050, 060, 101, 110, 111, 112, 113, 114, 115, 116, 121, 131, 141, 151, 161, 202, 212, 220, 221, 222, 223, 224, 225, 226, 232, 242, 252, 262, 303, 313, 323, 330, 331, 332, 333, 334, 335, 336, 343, 353, 363, 404, 414, 424, 434, 440, 441, 442, 443, 444, 445, 446, 454, 464, 505, 515, 525, 535, 545, 550, 551, 552, 553, 554, 555, 556, 565, 606, 616, 626, 636, 646, 656, 660, 661, 662, 663, 664, 665, 666
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