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Numbers k such that k+1 and k/2+1 are squares.
1

%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_