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%I #22 Aug 22 2021 22:38:05
%S 0,1,2,2,3,3,3,3,4,4,3,4,4,5,4,4,4,5,5,5,5,4,4,5,6,5,6,5,4,4,5,6,6,6,
%T 6,5,4,5,5,6,6,7,6,6,5,5,5,6,6,6,7,7,6,6,6,5,4,6,7,6,7,7,7,6,7,6,4,5,
%U 5,7,7,7,7,7,7,7,7,5,5,5,6,6,7,8,7,7,7,8,7,6,6,5,6,6,7,6,8,8,7,7,8,8,6,7,6,6
%N Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.
%C The fixed point of RASTxx transformation. The repeated applications of RASTxx starting from A072643 seem to converge toward this sequence. Compare to A072768 from which this differs first time at the position n=37, where A072768(37) = 4, while A071673(37) = 5.
%C Each term k occurs A000108(k) times, and maximal position where k occurs is A072638(k).
%C The size of each Catalan structure encoded by the corresponding terms in triangles A071671 & A071672 (i.e., the number of digits / 2), as obtained with the global ranking/unranking scheme presented in A071651-A071654.
%H Antti Karttunen, <a href="/A071673/b071673.txt">Table of n, a(n) for n = 0..10440 (rows 0..144 of the triangle, flattened)</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a> (Maple code for RASTxx transform)
%F a(0) = 0, a(n) = 1 + a(A025581(n-1)) + a(A002262(n-1)) = 1 + a(A004736(n)) + a(A002260(n)).
%e The first 15 rows of this irregular triangular table:
%e 0,
%e 1,
%e 2, 2,
%e 3, 3, 3,
%e 3, 4, 4, 3,
%e 4, 4, 5, 4, 4,
%e 4, 5, 5, 5, 5, 4,
%e 4, 5, 6, 5, 6, 5, 4,
%e 4, 5, 6, 6, 6, 6, 5, 4,
%e 5, 5, 6, 6, 7, 6, 6, 5, 5,
%e 5, 6, 6, 6, 7, 7, 6, 6, 6, 5,
%e 4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4,
%e 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5,
%e 5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5,
%e 6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6
%e etc.
%e E.g., we have
%e a(1) = T(0,0) = a(0) + a(0) + 1 = 1,
%e a(2) = T(1,0) = a(1) + a(0) + 1 = 2,
%e a(3) = T(0,1) = a(0) + a(1) + 1 = 2,
%e a(4) = T(2,0) = a(2) + a(0) + 1 = 3, etc.
%o (PARI)
%o up_to = 105;
%o A002260(n) = (n-binomial((sqrtint(8*n)+1)\2, 2)); \\ From A002260
%o A004736(n) = (1-n+(n=sqrtint(8*n)\/2)*(n+1)\2); \\ From A004736
%o A071673list(up_to) = { my(v=vector(1+up_to)); v[1] = 0; for(n=1,up_to,v[1+n] = 1 + v[A004736(n)] + v[A002260(n)]); (v); };
%o v071673 = A071673list(up_to);
%o A071673(n) = v071673[1+n]; \\ _Antti Karttunen_, Aug 17 2021
%o (Scheme) (define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))
%Y Same triangle computed modulo 2: A071674.
%Y Permutations of this sequence include: A072643, A072644, A072645, A072660, A072768, A072789, A075167.
%Y Cf. also A000108, A002260, A004736, A002262, A025581, A072638.
%K nonn,tabf,eigen
%O 0,3
%A _Antti Karttunen_, May 30 2002. Self-referential definition added Jun 03 2002.
%E Term a(0) = 0 prepended and the Example-section amended by _Antti Karttunen_, Aug 17 2021
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