cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A113192 Primes that are the difference of two Lucas numbers; primes in A113191.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 29, 43, 47, 73, 181, 197, 199, 293, 311, 503, 521, 839, 1361, 2131, 2203, 2207, 3571, 5749, 9349, 13763, 23633, 24469, 24473, 38239, 103483, 103681, 161983, 167759, 271367, 399601, 439081, 439157, 709283, 1692737, 3010349
Offset: 1

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Author

T. D. Noe, Oct 17 2005

Keywords

Comments

The difference L(i)-L(j) equals the sum L(j+1)+...+L(i+2).

Examples

			The prime 181 is here because it is L(11)-L(6).

Links

Crossrefs

Cf. A000032 (Lucas numbers), A001606 (Lucas(n) is prime), A113193 (number of times that Lucas(n)-Lucas(i) is prime for i=0..n-3).

Programs

  • Mathematica
    Lucas[n_] := Fibonacci[n+1]+Fibonacci[n-1]; lst={}; Do[p=Lucas[n]-Lucas[i]; If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 40}, {i, 0, n-2}]; Union[lst]

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

Original entry on oeis.org

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

Views

Author

T. D. Noe, Feb 13 2013

Keywords

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.
It is easy to show that L(4n+2) - L(0) = L(2n+1)^2. - Zhi-Wei Sun, Jan 02 2015

Links

Crossrefs

Cf. A000032 (Lucas numbers), A113191 (difference of two Lucas numbers).
Cf. A219114 (corresponding sequence for Fibonacci numbers).

Programs

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

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]

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

Views

Author

T. D. Noe, Feb 13 2013

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A000032 (Lucas numbers), A113191 (difference of two Lucas numbers).
Cf. A219114 (corresponding sequence for Fibonacci numbers).

Programs

  • Mathematica
    t = Union[Flatten[Abs[Table[LucasL[n] - LucasL[i], {n, 120}, {i, n}]]]]; t2 = Select[t, IntegerQ[Sqrt[#]] &]; Sqrt[t2]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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