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A188644 Array of ((k^n)+(k^(-n)))/2 where k=((x^2-1)^(1/2)+x)^2 for integers x>=1 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Conjecture by C Hohn: Given function f(x, y)=((x^2+y)^(1/2)+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

LINKS

Table of n, a(n) for n=0..38.

FORMULA

a(n)=(A188646(n-1)+A188646(n))/2

EXAMPLE

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1, 7, 97, 1351, 18817, 262087, 3650401, ...

1, 17, 577, 19601, 665857, 22619537, ...

1, 31, 1921, 119071, 7380481, 457470751, ...

1, 49, 4801, 470449, 46099201, ...

1, 71, 10081, 1431431, 203253121, ...

1, 97, 18817, 3650401, 708158977, ...

1, 127, 32257, 8193151, 2081028097, ...

1, 161, 51841, 16692641, 5374978561, ...

1, 199, 79201, 31521799, 12545596801, ...

1, 241, 116161, 55989361, 26986755841, ...

1, 287, 164737, 94558751, 54276558337, ...

1, 337, 227137, 153090001, 103182433537, ...

1, 391, 305761, 239104711, 186979578241, ...

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-Fran├žois 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 2 is A056220 (difference in starting term), (column 2)*2 is A060626.

A188645 (f(x, y) as above with y=1).

Sequence in context: A015118 A174691 A156692 * A111830 A212943 A174588

Adjacent sequences:  A188641 A188642 A188643 * A188645 A188646 A188647

KEYWORD

nonn,tabl

AUTHOR

Charles L. Hohn, Apr 06 2011

STATUS

approved

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Last modified November 21 03:01 EST 2018. Contains 317427 sequences. (Running on oeis4.)