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%I #34 Sep 08 2022 08:44:36
%S 0,48,1680,57120,1940448,65918160,2239277040,76069501248,
%T 2584123765440,87784138523760,2982076586042448,101302819786919520,
%U 3441313796169221280,116903366249966604048,3971273138702695316400,134906383349641674153600,4582845760749114225906048
%N Numbers k such that k+1 and k/2+1 are squares.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.
%H Colin Barker, <a href="/A008845/b008845.txt">Table of n, a(n) for n = 0..650</a>
%H Henry Ernest Dudeney, <a href="https://archive.org/stream/AmusementsInMathematicspdf/AmusementsInMathematics#page/n29/mode/2up">Amusements in Mathematics</a>, 1917. See problem 114, "Curious numbers".
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F a(n) = 2*(A008844(n)-1) = 16*A075528(n) = 48*A029546(n). - corrected by _Sean A. Irvine_, Apr 07 2018
%F a(0)=0, a(1)=48, a(2)=1680, a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). - _Harvey P. Dale_, May 24 2014
%F From _Colin Barker_, Mar 02 2016: (Start)
%F a(n) = (-6+(3-2*sqrt(2))*(17+12*sqrt(2))^(-n)+(3+2*sqrt(2))*(17+12*sqrt(2))^n)/4.
%F G.f.: 48*x / ((1-x)*(1-34*x+x^2)).
%F (End)
%F a(n) = 34*a(n-1) - a(n-2) + 48. - _Vincenzo Librandi_, Mar 03 2016
%e 48+1 = 49 = 7^2 and 48/2+1 = 24+1 = 25 = 5^2.
%p seq(coeff(series(48*x/((1-x)*(1-34*x+x^2)), x, n+1), x, n), n = 0..20); # _G. C. Greubel_, Sep 13 2019
%t LinearRecurrence[{35,-35,1},{0,48,1680},20] (* _Harvey P. Dale_, May 24 2014 *)
%o (PARI) concat(0, Vec(48*x/((1-x)*(1-34*x+x^2)) + O(x^20))) \\ _Colin Barker_, Mar 02 2016
%o (Magma) I:=[0,48]; [n le 2 select I[n] else 34*Self(n-1) - Self(n-2)+48: n in [1..20]]; // _Vincenzo Librandi_, Mar 03 2016
%o (Sage)
%o def A008845_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(48*x/((1-x)*(1-34*x+x^2))).list()
%o A008845_list(20) # _G. C. Greubel_, Sep 13 2019
%o (GAP) a:=[0,48,1680];; for n in [4..20] do a[n]:=35*a[n-1]-35*a[n-2] +a[n-3]; od; a; # _G. C. Greubel_, Sep 13 2019
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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