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%I #23 Mar 22 2023 15:18:03
%S 0,1,7,9,39,42,126,130,310,315,645,651,1197,1204,2044,2052,3276,3285,
%T 4995,5005,7315,7326,10362,10374,14274,14287,19201,19215,25305,25320,
%U 32760,32776,41752,41769,52479,52497,65151,65170,79990,80010,97230,97251,117117
%N Numbers of the form k*(k^3 +- 1)/2.
%C This sequence is also the integers k such that x^6 - 8*k*x^4 - 1 is reducible over the integers, and, for k > 0 such that the factors are two irreducible cubics. - _James R. Buddenhagen_, May 29 2010
%C Integer solutions of x + y = (x - y)^4. If x = a(n) then y = a(n - (-1)^n). - _Thomas Scheuerle_, Mar 06 2023
%H Colin Barker, <a href="/A057590/b057590.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).
%F G.f.: -x^2*(x^4+6*x^3-2*x^2+6*x+1) / ((x-1)^5*(x+1)^4). - _Colin Barker_, Apr 28 2015
%F From _Robert Israel_, Apr 28 2015: (Start)
%F a(n) = (2*n^4 + 4*n^3 + 6*n^2 + 4*n - 7)/64 + (-1)^n * (7 - 2*n - 2*n^2)*(1 + 2*n)/64.
%F a(n+8) - 4*a(n+6) + 6*a(n+4) - 4*a(n+2) + a(n) = 12. (End)
%p map(k -> (k*(k^3-1)/2, k*(k^3+1)/2), [$1..100]); # _Robert Israel_, Apr 28 2015
%o (PARI) concat(0, Vec(-x^2*(x^4+6*x^3-2*x^2+6*x+1)/((x-1)^5*(x+1)^4) + O(x^100))) \\ _Colin Barker_, Apr 28 2015
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Oct 05 2000
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