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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

Iain Fox, Antidiagonals n = 0..20 of array, flattened

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<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

CROSSREFS

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).

Sequence in context: A153869 A128541 A122908 * A091008 A111006 A046742

Adjacent sequences:  A296438 A296439 A296440 * A296442 A296443 A296444

KEYWORD

base,nonn,tabl

AUTHOR

Iain Fox, Dec 12 2017

STATUS

approved

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