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%I #9 Jan 28 2019 04:06:26
%S 2,3,3,6,4,6,13,8,8,13,30,18,14,18,30,71,46,28,28,46,71,170,122,70,48,
%T 70,122,170,409,330,192,108,108,192,330,409,986,898,558,288,202,288,
%U 558,898,986,2379,2450,1652,868,466,466,868,1652,2450,2379,5742,6690,4934,2728,1338,880,1338,2728,4934,6690,5742
%N Square array read by antidiagonals (n >= 1, k >= 2): T(n,k) = b(n,k) + b(k-1,n+1), where b(n,k) = ((1 + sqrt(k))^n - (1 - sqrt(k))^n)/(2*sqrt(k)).
%e Square array begins:
%e n\k | 2 3 4 5 6 7 8 9 ...
%e -------------------------------------------------
%e 1 | 2 3 6 13 30 71 170 409 ...
%e 2 | 3 4 8 18 46 122 330 898 ...
%e 3 | 6 8 14 28 70 192 558 1652 ...
%e 4 | 13 18 28 48 108 288 868 2728 ...
%e 5 | 30 46 70 108 202 466 1338 4264 ...
%e 6 | 71 122 192 288 466 880 2174 6560 ...
%e 7 | 170 330 558 868 1338 2174 4286 10888 ...
%e 8 | 409 898 1652 2728 4264 6560 10888 21760 ...
%e ...
%t a[n_, q_] = ((1 + Sqrt[q])^n - (1 - Sqrt[q])^n)/(2*Sqrt[q]);
%t b = Table[Table[FullSimplify[ExpandAll[a[n, q]]], {n, 1, 11}], {q, 2, 12}];
%t c = (b + Transpose[b]);
%t Table[Table[c[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}] // Flatten
%o (Maxima)
%o b(n, q) := ((1 + sqrt(q))^n - (1 - sqrt(q))^n)/(2*sqrt(q))$
%o T(n, k) := b(n, k) + b(k - 1, n + 1)$
%o create_list(fullratsimp(T(k - 1, n - k + 3)), n, 1, 20, k, 2, n + 1);
%o /* _Franck Maminirina Ramaharo_, Jan 27 2019 */
%Y Cf. A173738.
%K nonn,easy,tabl
%O 1,1
%A _Roger L. Bagula_, Feb 23 2010
%E Edited by _Franck Maminirina Ramaharo_, Jan 27 2019
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