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A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.

%I A071673 #24 Jul 09 2025 01:41:42
%S A071673 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 A071673 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 A071673 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 A071673 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 A071673 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 A071673 Each term k occurs A000108(k) times, and maximal position where k occurs is A072638(k).
%C A071673 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 A071673 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 A071673 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a> (Maple code for RASTxx transform)
%F A071673 a(0) = 0, a(n) = 1 + a(A025581(n-1)) + a(A002262(n-1)) = 1 + a(A004736(n)) + a(A002260(n)).
%e A071673 The first 15 rows of this irregular triangular table:
%e A071673                0,
%e A071673                1,
%e A071673               2, 2,
%e A071673              3, 3, 3,
%e A071673             3, 4, 4, 3,
%e A071673            4, 4, 5, 4, 4,
%e A071673           4, 5, 5, 5, 5, 4,
%e A071673          4, 5, 6, 5, 6, 5, 4,
%e A071673         4, 5, 6, 6, 6, 6, 5, 4,
%e A071673        5, 5, 6, 6, 7, 6, 6, 5, 5,
%e A071673       5, 6, 6, 6, 7, 7, 6, 6, 6, 5,
%e A071673      4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4,
%e A071673     5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5,
%e A071673    5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5,
%e A071673   6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6
%e A071673 etc.
%e A071673 E.g., we have
%e A071673   a(1) = T(0,0) = a(0) + a(0) + 1 = 1,
%e A071673   a(2) = T(1,0) = a(1) + a(0) + 1 = 2,
%e A071673   a(3) = T(0,1) = a(0) + a(1) + 1 = 2,
%e A071673   a(4) = T(2,0) = a(2) + a(0) + 1 = 3, etc.
%o A071673 (PARI)
%o A071673 up_to = 105;
%o A071673 A002260(n) = (n-binomial((sqrtint(8*n)+1)\2, 2)); \\ From A002260
%o A071673 A004736(n) = (1-n+(n=sqrtint(8*n)\/2)*(n+1)\2); \\ From A004736
%o A071673 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 A071673 v071673 = A071673list(up_to);
%o A071673 A071673(n) = v071673[1+n]; \\ _Antti Karttunen_, Aug 17 2021
%o A071673 (Scheme) (define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))
%Y A071673 Same triangle computed modulo 2: A071674.
%Y A071673 Permutations of this sequence include: A072643, A072644, A072645, A072660, A072768, A072789, A075167.
%Y A071673 Cf. also A000108, A002260, A004736, A002262, A025581, A072638.
%K A071673 nonn,tabf,eigen
%O A071673 0,3
%A A071673 _Antti Karttunen_, May 30 2002
%E A071673 Self-referential definition added Jun 03 2002
%E A071673 Term a(0) = 0 prepended and the Example-section amended by _Antti Karttunen_, Aug 17 2021