A267797 Lucas numbers of the form (x^3 + y^3) / 2 where x and y are distinct positive integers.

76, 1364, 24476, 439204, 7881196, 141422324, 2537720636, 45537549124, 817138163596, 14662949395604, 263115950957276, 4721424167835364, 84722519070079276, 1520283919093591604, 27280388024614569596, 489526700523968661124, 8784200221406821330636

Offset: 1

Author

Altug Alkan, Jan 24 2016

Keywords

• nonn
• easy

Comments

Lucas numbers that are the averages of 2 distinct positive cubes.

Inspired by relation between sequence A024851 and A188378.

Corresponding indices are 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, ...

6*n + 3 is the corresponding form of indices.

Corresponding y values are listed by A188378, for n > 0. Note that corresponding x values are A188378(n) - 2, for n > 0.

Examples

			Lucas number 76 is a term because 76 = (3^3 + 5^3) / 2.
Lucas number 1364 is a term because 1364 = (10^3 + 12^3) / 2.
Lucas number 24476 is a term because 24476 = (28^3 + 30^3) / 2.
Lucas number 439204 is a term because 439204 = (75^3 + 77^3) / 2.
Lucas number 7881196 is a term because 7881196 = (198^3 + 200^3) / 2.
Lucas number 141422324 is a term because 141422324 = (520^3 + 522^3) / 2.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..750
• Index entries for linear recurrences with constant coefficients, signature (18,-1).

Crossrefs

Cf. A000032, A016945, A024670, A024851, A049629, A188378.

Programs

Magma

[Fibonacci(6*n+4)+Fibonacci(6*n+2): n in [1..20]]; // Vincenzo Librandi, Jan 24 2016

Mathematica

Table[Fibonacci[6 n + 4] + Fibonacci[6 n + 2], {n, 1, 20}] (* Vincenzo Librandi, Jan 24 2016 *)
LinearRecurrence[{18,-1},{76,1364},20] (* Harvey P. Dale, Jul 23 2024 *)

PARI

l(n) = fibonacci(n+1) + fibonacci(n-1);
is(n) = for(i=ceil(sqrtn(n\2+1, 3)), sqrtn(n-.5, 3), ispower(n-i^3, 3) && return(1));
for(n=1, 120, if(is(2*l(n)), print1(l(n), ", ")));

PARI

a(n) = ((5*fibonacci(n)*fibonacci(n+1) + 1 + (-1)^n)^3 + (5*fibonacci(n)*fibonacci(n+1) - 1 + (-1)^n)^3) / 2;

PARI

a(n) = (fibonacci(6*n+4) + fibonacci(6*n+2));

PARI

Vec(4*x*(19-x)/(1-18*x+x^2) + O(x^20)) \\ Colin Barker, Jan 24 2016

Formula

a(n) = A000032(A016945(n)), for n > 0.

a(n) = A188378(n)^3 - 3*A188378(n)^2 + 6*A188378(n) - 4, for n > 0.

From Colin Barker, Jan 24 2016: (Start)

a(n) = (9+4*sqrt(5))^(-n)*(2-sqrt(5)+(2+sqrt(5))*(9+4*sqrt(5))^(2*n)).

a(n) = 18*a(n-1)-a(n-2) for n>2.

G.f.: 4*x*(19-x) / (1-18*x+x^2).

(End)