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%I #40 Dec 31 2023 21:13:19
%S 2,8,12,24,30,48,56,80,90,120,132,168,182,224,240,288,306,360,380,440,
%T 462,528,552,624,650,728,756,840,870,960,992,1088,1122,1224,1260,1368,
%U 1406,1520,1560,1680,1722,1848,1892,2024,2070,2208,2256,2400,2450,2600
%N Positive nonsquare integers k such that each term of the regular continued fraction of sqrt(k) divides k.
%C a(n) = n*(n+1) if n is odd and n*(n+2) if n is even. - _Wesley Ivan Hurt_, Nov 19 2013
%C From _Rolf Knobel_, Dec 27 2023: (Start)
%C The first term of the continued fraction also divides k since it is half the last term of the periodic part. Except for a(1), the period is 2 (see A013642). -
%C Interleaving of A033996, A002939. Alternating mixture of A005563, A002378. (End)
%D H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, Table 1.
%D Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).
%H Harvey P. Dale, <a href="/A013654/b013654.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(2n) = 2*(n+1)*(2n), a(2n+1) = 2*(n+1)*(2n+1). - _Frank Ellermann_, Feb 22 2002
%F G.f.: (-2-6*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F a(n) = n * (n + 1 + (1+(-1)^n)/2). - _Wesley Ivan Hurt_, Nov 19 2013
%p A013654:=n->n*(n+1+(1+(-1)^n)/2); seq(A013654(n), n=1..100); # _Wesley Ivan Hurt_, Nov 19 2013
%t Table[n(n+1+(1+(-1)^n)/2), {n,100}] (* _Wesley Ivan Hurt_, Nov 19 2013 *)
%t Table[If[OddQ[n],n(n+1),n(n+2)],{n,50}] (* _Harvey P. Dale_, May 15 2021 *)
%Y Cf. A002378, A002939, A005563, A013642, A033996.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_
%E More terms from _David W. Wilson_
%E Name clarified by _Rolf Knobel_, Dec 27 2023
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