|
%I #45 Dec 31 2017 13:26:11
%S 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,
%T 60,255,29,7,0,0,0,0,83,467,257,30,8,0,0,0,0,109,775,3125,259,80,9,0,
%U 0,0,0,139,1197,46655,3127,553,81,10,0,0,0,0,173,1751,98039,46657,6310,1023,83,11
%N 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.
%C 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.
%H Iain Fox, <a href="/A296441/b296441.txt">Antidiagonals n = 0..20 of array, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HereditaryRepresentation.html">Hereditary Representation</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein Sequence</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoodsteinsTheorem.html">Goodstein's Theorem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein's_theorem#Hereditary_base-n_notation">Hereditary base-n notation</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_sequences">Goodstein sequence</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein's_theorem">Goodstein's Theorem</a>
%e | n\k | 0 1 2 3 4 5 6 7 8 9 ...
%e |-----|------------------------------------------------------------------------
%e | 0 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e | 1 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e | 2 | 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, ...
%e | 3 | 3, 3, 3, 2, 1, 0, 0, 0, 0, 0, ...
%e | 4 | 4, 26, 41, 60, 83, 109, 139, 173, 211, 253, ...
%e | 5 | 5, 27, 255, 467, 775, 1197, 1751, 2454, 3325, 4382, ...
%e | 6 | 6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775, ...
%e | ... |
%o (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)))
%o A(n, k) = for(i=1, k, if(n==0, break()); n=B(n, i+1)-1); n
%Y 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).
%Y 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).
%Y G_n(n) = A266201(n) (main diagonal of array).
%K base,nonn,tabl
%O 0,6
%A _Iain Fox_, Dec 12 2017
|
