OFFSET
1,1
COMMENTS
Lucas numbers that are the averages of 2 distinct positive cubes.
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.
LINKS
Colin Barker, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (18,-1).
FORMULA
EXAMPLE
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.
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 *)
PROG
(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
(Magma) [Fibonacci(6*n+4)+Fibonacci(6*n+2): n in [1..20]]; // Vincenzo Librandi, Jan 24 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Jan 24 2016
STATUS
approved