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A188644
Array of (k^n + k^(-n))/2 where k = (sqrt(x^2-1) + x)^2 for integers x >= 1.
10
1, 1, 1, 1, 7, 1, 1, 97, 17, 1, 1, 1351, 577, 31, 1, 1, 18817, 19601, 1921, 49, 1, 1, 262087, 665857, 119071, 4801, 71, 1, 1, 3650401, 22619537, 7380481, 470449, 10081, 97, 1, 1, 50843527, 768398401, 457470751, 46099201, 1431431, 18817, 127, 1
OFFSET
0,5
COMMENTS
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.
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
FORMULA
A(n,k) = (A188646(n,k-1) + A188646(n,k))/2.
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
EXAMPLE
Row 2 gives {( (2+sqrt(3))^(2*n) + (2-sqrt(3))^(2*n) )/2}.
Square array begins:
| 0 1 2 3 4
-----+---------------------------------------------
1 | 1, 1, 1, 1, 1, ...
2 | 1, 7, 97, 1351, 18817, ...
3 | 1, 17, 577, 19601, 665857, ...
4 | 1, 31, 1921, 119071, 7380481, ...
5 | 1, 49, 4801, 470449, 46099201, ...
6 | 1, 71, 10081, 1431431, 203253121, ...
7 | 1, 97, 18817, 3650401, 708158977, ...
8 | 1, 127, 32257, 8193151, 2081028097, ...
9 | 1, 161, 51841, 16692641, 5374978561, ...
10 | 1, 199, 79201, 31521799, 12545596801, ...
11 | 1, 241, 116161, 55989361, 26986755841, ...
12 | 1, 287, 164737, 94558751, 54276558337, ...
13 | 1, 337, 227137, 153090001, 103182433537, ...
14 | 1, 391, 305761, 239104711, 186979578241, ...
15 | 1, 449, 403201, 362074049, 325142092801, ...
...
MATHEMATICA
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 *)
CROSSREFS
Row 2 is A011943, row 3 is A056771, row 8 is A175633, (row 2)*2 is A067902, (row 9)*2 is A089775.
(column 1)*2 is A060626.
Cf. A188645 (f(x, y) as above with y=1).
Diagonals give A173129, A322899.
Sequence in context: A340428 A174691 A156692 * A331899 A111830 A212943
KEYWORD
nonn,tabl
AUTHOR
Charles L. Hohn, Apr 06 2011
EXTENSIONS
Edited by Seiichi Manyama, Dec 30 2018
More terms from Seiichi Manyama, Jan 01 2019
STATUS
approved