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A296441
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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.
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1
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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
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OFFSET
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0,6
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COMMENTS
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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.
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LINKS
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EXAMPLE
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| 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, ...
| ... |
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PROG
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(PARI) B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n<b+i, #n-i, B(#n-i, b)))
A(n, k) = for(i=1, k, if(n==0, break()); n=B(n, i+1)-1); n
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CROSSREFS
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n-th row: A000004 (n=0), A000007 (n=1), A215409 (n=3), A056193 (n=4), A266204 (n=5), A266205 (n=6), A271554 (n=7), A271555 (n=8), A271556 (n=9), A271557 (n=10), A271558 (n=11), A271559 (n=12), A271560 (n=13), A271561 (n=14), A222117 (n=15), A059933 (n=16), A271562 (n=17), A271975 (n=18) A211378 (n=19), A271976 (n=20).
k-th column: A001477 (k=0), A056004 (k=1), A057650 (k=2), A059934 (k=3), A059935 (k=4), A059936 (k=5), A271977 (k=6), A271978 (k=7), A271979 (k=8), A271985 (k=9), A271986 (k=10).
G_n(n) = A266201(n) (main diagonal of array).
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KEYWORD
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AUTHOR
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STATUS
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approved
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