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%I #34 Dec 11 2019 23:41:34
%S 0,2,16,98,576,3362,19600,114242,665856,3880898,22619536,131836322,
%T 768398400,4478554082,26102926096,152139002498,886731088896,
%U 5168247530882,30122754096400,175568277047522,1023286908188736
%N Numbers n such that 2*n*(n+2) is a square.
%C Even-indexed terms are squares. Their square roots form sequence A005319. Odd-indexed terms divided by 2 are squares. Their square roots form the sequence A002315. (Index starts at 0.)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F a(n) = A001541(n) - 1.
%F a(n) = (1/2)*(s^n + t^n) - 1, where s = 3 + 2*sqrt(2), t = 3 - 2*sqrt(2). Note: s=1/t. a(n) = 6*a(n-1) - a(n-2) + 4, a(0)=0, a(1)=2.
%F a(n) = 1/kappa(sqrt(2)/A001542(n)); a(n) = 1/kappa(sqrt(8)/A005319(n)) where kappa(x) is the sum of successive remainders by computing the Euclidean algorithm for (1, x). - _Thomas Baruchel_, Nov 29 2003
%F G.f.: -2*x^2*(x+1)/((x-1)*(x^2-6*x+1)). - _Colin Barker_, Nov 22 2012
%t a[0] = 0; a[1] = 2; a[n_] := a[n] = 6a[n - 1] - a[n - 2] + 4; Table[ a[n], {n, 0, 20}]
%t LinearRecurrence[{7,-7,1},{0,2,16},30] (* _Harvey P. Dale_, Nov 21 2015 *)
%Y Cf. A002315, A005319.
%K easy,nonn
%O 1,2
%A _James R. Buddenhagen_, May 15 2003
%E More terms from _Robert G. Wilson v_, May 15 2003
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