A113191 Difference of two Lucas numbers.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 22, 25, 26, 27, 28, 29, 36, 40, 43, 44, 45, 46, 47, 58, 65, 69, 72, 73, 74, 75, 76, 94, 105, 112, 116, 119, 120, 121, 122, 123, 152, 170, 181, 188, 192, 195, 196, 197, 198, 199, 246, 275, 293, 304

Offset: 1

Author

T. D. Noe, Oct 17 2005

Keywords

• nonn

Comments

Also the sum of consecutive Lucas numbers because the difference L(i) - L(j) equals the sum L(j+1) + ... + L(i+2).

Conjecture: L(m) - L(n) with m > 1 and m > n >= 0 is a perfect power but not a square only for (m,n) = (7,0), (5,2). This has been verified for n < m <= 500. Note that L(7) - L(0) = 29 - 2 = 3^3 and L(5) - L(2) = 11 - 3 = 2^3. - Zhi-Wei Sun, Jan 02 2025

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000032 (Lucas numbers), A007298 (difference of two Fibonacci numbers).

Cf. A221471, A221472 (square root of squares in this sequence).

Programs

Mathematica

Lucas[n_] := Fibonacci[n+1]+Fibonacci[n-1]; Union[Flatten[Table[Lucas[n]-Lucas[i], {n, 13}, {i, 0, n-2}]]]