A098602 a(n) = A001652(n) * A046090(n).

0, 12, 420, 14280, 485112, 16479540, 559819260, 19017375312, 646030941360, 21946034630940, 745519146510612, 25325704946729880, 860328449042305320, 29225841562491651012, 992818284675673829100, 33726595837410418538400, 1145711440187278556476512

Offset: 0

Author

Charlie Marion, Oct 26 2004

Keywords

• nonn
• easy

Comments

From Ron Knott, Nov 25 2013: (Start)

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

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

2*a(n)+1 and 4*a(n)+1 are both square. - Paul Cleary, Jun 23 2014

Examples

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..255
• Nikola Adžaga, Andrej Dujella, Dijana Kreso, Petra Tadić, On Diophantine m-tuples and D(n)-sets, 2018.
• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Crossrefs

Cf. A002378, A053141.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(12*x/((1-x)*(x^2-34*x+1)))); // G. C. Greubel, Jul 15 2018

Mathematica

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 *)
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 *)
LinearRecurrence[{35, -35, 1}, {0, 12, 420}, 25] (* T. D. Noe, Nov 25 2013 *)
Table[(LucasL[4*n+2, 2] - 6)/16, {n,0,30}] (* G. C. Greubel, Jul 15 2018 *)

PARI

concat(0, Vec(12*x/((1-x)*(1-34*x+x^2)) + O(x^20))) \\ Colin Barker, Mar 02 2016

PARI

{a=1+sqrt(2); b=1-sqrt(2); Q(n) = a^n + b^n};
for(n=0, 30, print1(round((Q(4*n+2) - 6)/16), ", ")) \\ G. C. Greubel, Jul 15 2018

Formula

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

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

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

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

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

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

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

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

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

From Ron Knott, Nov 25 2013: (Start)

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

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

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

Extensions

More terms from Ray Chandler, Nov 10 2004

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