This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188644 #65 Jan 01 2019 12:01:14
%S A188644 1,1,1,1,7,1,1,97,17,1,1,1351,577,31,1,1,18817,19601,1921,49,1,1,
%T A188644 262087,665857,119071,4801,71,1,1,3650401,22619537,7380481,470449,
%U A188644 10081,97,1,1,50843527,768398401,457470751,46099201,1431431,18817,127,1
%N A188644 Array of (k^n + k^(-n))/2 where k = (sqrt(x^2-1) + x)^2 for integers x >= 1.
%C A188644 Conjecture: Given the function f(x,y) = (sqrt(x^2+y) + x)^2 and constant k=f(x,y), then for all integers x >= 1 and y=[+-]1, k may be irrational, but (k^n + k^(-n))/2 always produces integer sequences; y=-1 results shown here; y=1 results are A188645.
%C A188644 Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{2*k}(x), evaluated at x=n. - _Seiichi Manyama_, Dec 30 2018
%H A188644 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H A188644 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A188644 A(n,k) = (A188646(n,k-1) + A188646(n,k))/2.
%F A188644 A(n,k) = Sum_{j=0..k} binomial(2*k,2*j)*(n^2-1)^(k-j)*n^(2*j). - _Seiichi Manyama_, Jan 01 2019
%e A188644 Row 2 gives {( (2+sqrt(3))^(2*n) + (2-sqrt(3))^(2*n) )/2}.
%e A188644 Square array begins:
%e A188644 | 0 1 2 3 4
%e A188644 -----+---------------------------------------------
%e A188644 1 | 1, 1, 1, 1, 1, ...
%e A188644 2 | 1, 7, 97, 1351, 18817, ...
%e A188644 3 | 1, 17, 577, 19601, 665857, ...
%e A188644 4 | 1, 31, 1921, 119071, 7380481, ...
%e A188644 5 | 1, 49, 4801, 470449, 46099201, ...
%e A188644 6 | 1, 71, 10081, 1431431, 203253121, ...
%e A188644 7 | 1, 97, 18817, 3650401, 708158977, ...
%e A188644 8 | 1, 127, 32257, 8193151, 2081028097, ...
%e A188644 9 | 1, 161, 51841, 16692641, 5374978561, ...
%e A188644 10 | 1, 199, 79201, 31521799, 12545596801, ...
%e A188644 11 | 1, 241, 116161, 55989361, 26986755841, ...
%e A188644 12 | 1, 287, 164737, 94558751, 54276558337, ...
%e A188644 13 | 1, 337, 227137, 153090001, 103182433537, ...
%e A188644 14 | 1, 391, 305761, 239104711, 186979578241, ...
%e A188644 15 | 1, 449, 403201, 362074049, 325142092801, ...
%e A188644 ...
%t A188644 max = 9; y = -1; t = Table[k = ((x^2 + y)^(1/2) + x)^2; ((k^n) + (k^(-n)))/2 // FullSimplify, {n, 0, max - 1}, {x, 1, max}]; Table[ t[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 17 2013 *)
%Y A188644 Row 2 is A011943, row 3 is A056771, row 8 is A175633, (row 2)*2 is A067902, (row 9)*2 is A089775.
%Y A188644 Column 0-5 give A000012, A056220, A144130, A243132, A243134, A243136.
%Y A188644 (column 1)*2 is A060626.
%Y A188644 Cf. A188645 (f(x, y) as above with y=1).
%Y A188644 Diagonals give A173129, A322899.
%Y A188644 Cf. A188646, A322836.
%K A188644 nonn,tabl
%O A188644 0,5
%A A188644 _Charles L. Hohn_, Apr 06 2011
%E A188644 Edited by _Seiichi Manyama_, Dec 30 2018
%E A188644 More terms from _Seiichi Manyama_, Jan 01 2019