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A321782 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321768(n, k) + A321770(n, k))/2).

%I A321782 #23 Mar 10 2022 04:09:50
%S A321782 2,3,5,4,4,8,7,8,12,9,7,9,6,5,11,10,13,19,14,12,16,11,11,21,18,19,29,
%T A321782 22,16,20,13,10,18,15,14,22,17,11,13,8,6,14,13,18,26,19,17,23,16,18,
%U A321782 34,29,30,46,35,25,31,20,17,31,26,25,39,30,20,24,15,14,30
%N A321782 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321768(n, k) + A321770(n, k))/2).
%C A321782 This sequence and A321783 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.
%C A321782 This sequence is "i" from the construction in A321768.  It takes ternary digits of k-1 from most to least significant.  Here the result is the same going instead least to most, due to how the relevant matrix product is related to its reversal.  As a flat sequence this means a(A351702(n)) = a(n) unchanged. - _Kevin Ryde_, Mar 10 2022
%H A321782 Rémy Sigrist, <a href="/A321782/b321782.txt">Rows n = 1..9, flattened</a>
%H A321782 Kevin Ryde, <a href="http://user42.tuxfamily.org/triples/index.html">Trees of Primitive Pythagorean Triples</a>, section UAD Tree, "row-wise p".
%H A321782 Robert Saunders and Trevor Randall, <a href="http://www.jstor.org/stable/3618576">The Family Tree of the Pythagorean Triplets Revisited</a>, Mathematical Gazette, item 78.12, volume 78, July 1994, pages 190-193, see page 192 tree terms "m" by columns.
%H A321782 <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples</a>
%F A321782 Empirically:
%F A321782 - T(n, 1) = n + 1,
%F A321782 - T(n, (3^(n-1) + 1)/2) = A000129(n + 1),
%F A321782 - T(n, 3^(n-1)) = 2 * n.
%e A321782 The first rows are:
%e A321782    2
%e A321782    3, 5, 4
%e A321782    4, 8, 7, 8, 12, 9, 7, 9, 6
%o A321782 (PARI) M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
%o A321782 T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (sqrtint((t[1, 1] + t[3, 1])/2))
%Y A321782 Cf. A000129, A321768, A321770, A321783.
%Y A321782 Cf. A001542 (row sums).
%Y A321782 Cf. A351702 (product reversal permutation).
%K A321782 nonn,tabf
%O A321782 1,1
%A A321782 _Rémy Sigrist_, Nov 18 2018