OFFSET
1,2
COMMENTS
From a problem in A269254. For detailed theory, see [Hone].
From Charles L. Hohn, Sep 28 2024: (Start)
For rows n >= 3, values x >= 3 where (x^2-4)/(n^2-4) is a square.
For rows n >= 3, Lim_{k->oo}(T(n, k+1)/T(n, k)) = (sqrt(n^2-4)+n)/2. (End)
LINKS
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.
FORMULA
A(n,k) = T_k(n), n >= 1, k >= 1, where T_j(x) = x*T_{j-1}(x) - T_{j-2}(x), j >= 2, T_0(x) = 2, T_1(x) = x, (dilated Chebyshev polynomials of the first kind).
EXAMPLE
Array begins:
1 -1 -2 -1 1 2 1 -1 -2 -1
2 2 2 2 2 2 2 2 2 2
3 7 18 47 123 322 843 2207 5778 15127
4 14 52 194 724 2702 10084 37634 140452 524174
5 23 110 527 2525 12098 57965 277727 1330670 6375623
6 34 198 1154 6726 39202 228486 1331714 7761798 45239074
7 47 322 2207 15127 103682 710647 4870847 33385282 228826127
8 62 488 3842 30248 238142 1874888 14760962 116212808 914941502
9 79 702 6239 55449 492802 4379769 38925119 345946302 3074591599
10 98 970 9602 95050 940898 9313930 92198402 912670090 9034502498
MATHEMATICA
t[n_, 0] := 2; t[n_, 1] := n; t[n_, k_] := n*t[n, k - 1] - t[n, k - 2]; Table[t[n, k], {n, 10}, {k, 10}] // Grid
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
STATUS
approved