%I #56 Sep 08 2022 08:45:15

%S 0,12,420,14280,485112,16479540,559819260,19017375312,646030941360,

%T 21946034630940,745519146510612,25325704946729880,860328449042305320,

%U 29225841562491651012,992818284675673829100,33726595837410418538400,1145711440187278556476512

%N a(n) = A001652(n) * A046090(n).

%C From _Ron Knott_, Nov 25 2013: (Start)

%C a(n) = 2*r*(r+1) which is also of form s(s+1) where the s is in A053141.

%C a(n) is an oblong number (A002378) which is twice another oblong number. (End)

%C 2*a(n)+1 and 4*a(n)+1 are both square. - _Paul Cleary_, Jun 23 2014

%H Reinhard Zumkeller, <a href="/A098602/b098602.txt">Table of n, a(n) for n = 0..255</a>

%H Nikola Adžaga, Andrej Dujella, Dijana Kreso, Petra Tadić, <a href="https://www.researchgate.net/publication/323934704_On_Diophantine_m-tuples_and_Dn-sets">On Diophantine m-tuples and D(n)-sets</a>, 2018.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).

%F a(n) = 2*A029549(n) = 2*A001109(n)*A001109(n+1).

%F a(n) = (A001653(n)^2 - 1)/2.

%F a(n) = A053141(n)^2 + A011900(n)^2 - 1.

%F For n>0, a(n) = A053141(2n) - 2*A001109(n-1)^2.

%F For n>0, a(n) = 3*(A001542(n)^2 - A001542(n-1)^2).

%F For n>0, a(n) = A053141(2n-1) + 2*(A001653(2n-1) - A001109(n-1)^2).

%F a(n+1) + a(n) = 3*A001542(n+1)^2.

%F a(n+1) - a(n) = A001542(2*n).

%F a(n+1)*a(n) = 4*(A001109(n)^4 - A001109(n)^2) = 4*A001110(n)*(A001110(n) - 1).

%F From _Ron Knott_, Nov 25 2013: (Start)

%F a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3).

%F G.f.: 12*x / ((1-x)*(x^2-34*x+1)). (End)

%F a(n) = (-6 + (3-2*sqrt(2))*(17+12*sqrt(2))^(-n)+(3+2*sqrt(2))*(17+12*sqrt(2))^n)/16. - _Colin Barker_, Mar 02 2016

%e a(1) = 12 = 2(2*3) = 3*4, a(2) = 420 = 2(14*15) = 20*21.

%t 2*Table[ Floor[(Sqrt[2] + 1)^(4n + 2)/32], {n, 0, 20} ] (* _Ray Chandler_, Nov 10 2004, copied incorrect program from A029549, revised Jul 09 2015 *)

%t RecurrenceTable[{a[n+3] == 35 a[n+2] - 35 a[n+1] + a[n], a[1] == 0, a[2] == 12, a[3] == 420}, a, {n, 1, 10}] (* _Ron Knott_, Nov 25 2013 *)

%t LinearRecurrence[{35, -35, 1}, {0, 12, 420}, 25] (* _T. D. Noe_, Nov 25 2013 *)

%t Table[(LucasL[4*n+2, 2] - 6)/16, {n,0,30}] (* _G. C. Greubel_, Jul 15 2018 *)

%o (PARI) concat(0, Vec(12*x/((1-x)*(1-34*x+x^2)) + O(x^20))) \\ _Colin Barker_, Mar 02 2016

%o (PARI) {a=1+sqrt(2); b=1-sqrt(2); Q(n) = a^n + b^n};

%o for(n=0, 30, print1(round((Q(4*n+2) - 6)/16), ", ")) \\ _G. C. Greubel_, Jul 15 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(12*x/((1-x)*(x^2-34*x+1)))); // _G. C. Greubel_, Jul 15 2018

%Y Cf. A002378, A053141.

%K nonn,easy

%O 0,2

%A _Charlie Marion_, Oct 26 2004

%E More terms from _Ray Chandler_, Nov 10 2004

%E Corrected by Bill Lam (bill_lam(AT)myrealbox.com), Feb 27 2006