A221471 Integers n such that n^2 is the difference of two Lucas numbers (A000032).

0, 1, 2, 3, 4, 5, 6, 11, 14, 29, 57, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371

Offset

1,3

Comments

This sequence, growing exponentially, is interesting because the corresponding sequence for Fibonacci numbers (A219114) appears to be finite. However, except the 7 numbers in A221472, it appears that the squares of all these numbers have the form 0, L(5) - L(0), or L(4n+2) - L(0), where L(n) denotes the n-th Lucas number.

Links

T. D. Noe, Table of n, a(n) for n = 1..100

Formula

Conjecture: a(n) = 3*a(n-1)-a(n-2) = A002878(n-8) for n>13. G.f.: x^2*(28*x^11-66*x^10-16*x^9-2*x^8-13*x^7-2*x^6-5*x^5-4*x^4-3*x^3-2*x^2-x+1) / (x^2-3*x+1). [Colin Barker, Feb 17 2013]

Mathematica

t = Union[Flatten[Abs[Table[LucasL[n] - LucasL[i], {n, 0, 120}, {i, n}]]]]; t2 = Select[t, IntegerQ[Sqrt[#]] &]; Sqrt[t2]

Crossrefs

Cf. A000032 (Lucas numbers), A113191 (difference of two Lucas numbers).

Cf. A219114 (corresponding sequence for Fibonacci numbers).

Sequence in context: A088411 A075073 A157420 * A058243 A264665 A308776

Adjacent sequences: A221468 A221469 A221470 * A221472 A221473 A221474

Keyword

nonn

Author

T. D. Noe, Feb 13 2013

Status

approved