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Numbers n such that 30*n^2 + 6 is a square.
4

%I #29 Sep 08 2022 08:45:31

%S 1,23,505,11087,243409,5343911,117322633,2575754015,56549265697,

%T 1241508091319,27256628743321,598404324261743,13137638505015025,

%U 288429642786068807,6332314502788498729,139022489418560903231

%N Numbers n such that 30*n^2 + 6 is a square.

%H Colin Barker, <a href="/A133283/b133283.txt">Table of n, a(n) for n = 1..700</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (22,-1).

%F a(n+2) = 22*a(n+1) - a(n); a(n+1) = 11*a(n) + 2*sqrt(30*a(n)^2 + 6).

%F a(n) = (sqrt(30)/10 - 1/2)*(11 + 2*sqrt(30))^n - (sqrt(30)/10 + 1/2) * (11 - 2*sqrt(30))^n. - _Emeric Deutsch_, Oct 24 2007

%F G.f.: x*(1+x)/(1-22*x+x^2). - _R. J. Mathar_, Nov 14 2007

%F a(n) = A077421(n) + A077421(n-1). - _R. J. Mathar_, Feb 19 2016

%F a(n) = Chebyshev(n-1, 11) + Chebyshev(n-2, 11). - _G. C. Greubel_, Jan 13 2020

%p a[1]:=1: a[2]:=23: for n to 14 do a[n+2]:=22*a[n+1]-a[n] end do: seq(a[n],n= 1..16); # _Emeric Deutsch_, Oct 24 2007

%t Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && 30*n^2+6 == k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 1, 20}] // Flatten // Union (* _Jean-François Alcover_, Dec 19 2013 *)

%t Rest@ CoefficientList[Series[x(1+x)/(1-22x+x^2), {x,0,20}], x] (* _Michael De Vlieger_, Jul 14 2016 *)

%t LinearRecurrence[{22,-1},{1,23},20] (* _Harvey P. Dale_, Sep 22 2017 *)

%t Table[ChebyshevU[n-1, 11] + ChebyshevU[n-2, 11], {n, 20}] (* _G. C. Greubel_, Jan 13 2020 *)

%o (PARI) Vec(x*(1+x)/(1-22*x+x^2) + O(x^20)) \\ _Colin Barker_, Jul 14 2016

%o (PARI) vector(20, n, polchebyshev(n-1,2,11) + polchebyshev(n-2,2,11) ) \\ _G. C. Greubel_, Jan 13 2020

%o (Magma) I:=[1,23]; [n le 2 select I[n] else 22*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Jan 13 2020

%o (Sage) [chebyshev_U(n-1,11) + chebyshev_U(n-2,11) for n in (1..20)] # _G. C. Greubel_, Jan 13 2020

%o (GAP) a:=[1,23];; for n in [3..20] do a[n]:=22*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 13 2020

%K nonn,easy

%O 1,2

%A _Richard Choulet_, Oct 16 2007

%E More terms from _Emeric Deutsch_, Oct 24 2007