A221471 -id:A221471 - cp's OEIS frontend

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

T. D. Noe, Oct 17 2005

nonn

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

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

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

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

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

A221472 Integers n such that n^2 is the difference of two Lucas numbers (A000204).

Original entry on oeis.org

0, 1, 2, 5, 6, 14, 57

Offset: 1

T. D. Noe, Feb 13 2013

nonn

This sequence is similar to the one for Fibonacci numbers (A219114) and appears to be finite also. See A221471 for an infinite version of this sequence.

			The only known square differences of Lucas numbers:
1^2 = L(3)-L(2) = 4-3,
2^2 = L(4)-L(2) 7-3 = L(5)-L(4) = 11-7,
5^2 = L(7)-l(3) = 29-4,
6^2 = L(8)-L(5) = 47-11,
14^2 = L(11)-L(2) = 199-3,
57^2 = L(17)-L(12) = 3571-322.

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

Cf. A219114 (corresponding sequence for Fibonacci numbers).

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

cp's OEIS Frontend

A113191 Difference of two Lucas numbers.

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