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 A296441 Array A(n, k) = G_k(n) where G_k(n) is the k-th term of the Goodstein sequence of n, read by antidiagonals. 1
 0, 0, 1, 0, 0, 2, 0, 0, 2, 3, 0, 0, 1, 3, 4, 0, 0, 0, 3, 26, 5, 0, 0, 0, 2, 41, 27, 6, 0, 0, 0, 1, 60, 255, 29, 7, 0, 0, 0, 0, 83, 467, 257, 30, 8, 0, 0, 0, 0, 109, 775, 3125, 259, 80, 9, 0, 0, 0, 0, 139, 1197, 46655, 3127, 553, 81, 10, 0, 0, 0, 0, 173, 1751, 98039, 46657, 6310, 1023, 83, 11 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS G_0(n) = n. To get to the second term in the row, convert n to hereditary base 2 representation (see links), replace each 2 with a 3, and subtract 1. For the third term, convert the second term (G_1(n)) into hereditary base 3 notation, replace each 3 with a 4, and subtract one. This pattern continues until the sequence converges to 0, which, by Goodstein's Theorem, occurs for all n. LINKS Eric Weisstein's World of Mathematics, Hereditary Representation Eric Weisstein's World of Mathematics, Goodstein Sequence Eric Weisstein's World of Mathematics, Goodstein's Theorem Wikipedia, Hereditary base-n notation Wikipedia, Goodstein sequence Wikipedia, Goodstein's Theorem EXAMPLE | n\k |  0   1    2     3      4      5       6       7       8       9  ... |-----|------------------------------------------------------------------------ |  0  |  0,  0,   0,    0,     0,     0,      0,      0,      0,      0, ... |  1  |  1,  0,   0,    0,     0,     0,      0,      0,      0,      0, ... |  2  |  2,  2,   1,    0,     0,     0,      0,      0,      0,      0, ... |  3  |  3,  3,   3,    2,     1,     0,      0,      0,      0,      0, ... |  4  |  4, 26,  41,   60,    83,   109,    139,    173,    211,    253, ... |  5  |  5, 27, 255,  467,   775,  1197,   1751,   2454,   3325,   4382, ... |  6  |  6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775, ... | ... | PROG (PARI) B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n

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Last modified July 17 16:58 EDT 2019. Contains 325107 sequences. (Running on oeis4.)