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 1-5 of 5 results.

A103324 Square array T(n,k) read by antidiagonals: powers of Lucas numbers.

Original entry on oeis.org

2, 4, 1, 8, 1, 3, 16, 1, 9, 4, 32, 1, 27, 16, 7, 64, 1, 81, 64, 49, 11, 128, 1, 243, 256, 343, 121, 18, 256, 1, 729, 1024, 2401, 1331, 324, 29, 512, 1, 2187, 4096, 16807, 14641, 5832, 841, 47, 1024, 1, 6561, 16384, 117649, 161051, 104976, 24389, 2209, 76
Offset: 1

Views

Author

Ralf Stephan, Feb 03 2005

Keywords

Examples

			2,1,3,4,7,11,18,
4,1,9,16,49,121,324,
8,1,27,64,343,1331,5832,
16,1,81,256,2401,14641,104976,
32,1,243,1024,16807,161051,1889568,
64,1,729,4096,117649,1771561,34012224,

References

  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 140.

Crossrefs

Rows include A000032, A001254, A075155, A099923, A103325.

Formula

T(n, k) = A000032(k)^n, n>=1, k>=0.
T(n, k) = Sum[i_1>=0, Sum[i_2>=0, ... Sum[i_{k-1}>=0, 2^i_1*C(n, i_1)*C(n-i_1, i_2)*C(n-i_2, i_3)*...*C(n-i_{k-2}, i_{k-1}) ] ... ]].

A200381 Products of Lucas numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 21, 22, 24, 27, 28, 29, 32, 33, 36, 42, 44, 47, 48, 49, 54, 56, 58, 63, 64, 66, 72, 76, 77, 81, 84, 87, 88, 94, 96, 98, 99, 108, 112, 116, 121, 123, 126, 128, 132, 141, 144, 147, 152, 154, 162, 168, 174, 176, 188
Offset: 1

Views

Author

R. J. Mathar, Nov 17 2011

Keywords

Comments

Complement of A200995.
Contains A000032, A001254, A075155, A099923, A103325 .. and any products of their members.

Examples

			Contains 9=3^2, 14=2*7, 21=3*7, 27=3^3, 42=2*3*7, 141 =3*47 etc.

Links

Crossrefs

Cf. A000032, A181156. Complement of A200995.

Programs

  • Mathematica
    lim = 11; luc = LucasL[Range[0, lim]]; luc = Delete[luc, 2]; last = luc[[-1]]; t = {1}; Do[t2 = luc[[n]]^Range[Floor[Log[last]/Log[luc[[n]]]]]; s = Select[Union[Flatten[Outer[Times, t, t2]]], # <= last &];  t = Union[t, s], {n, lim}]; t (* T. D. Noe, Nov 17 2011 *)

A075151 a(n)=L(n)^2*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (comment to A061084).

Original entry on oeis.org

8, -1, 27, -64, 343, -1331, 5832, -24389, 103823, -438976, 1860867, -7880599, 33386248, -141420761, 599077107, -2537716544, 10749963743, -45537538411, 192900170952, -817138135549, 3461452853383, -14662949322176, 62113250509227, -263115950765039, 1114577054530568
Offset: 0

Views

Author

Mario Catalani (mario.catalani(AT)unito.it), Sep 05 2002

Keywords

Links

Crossrefs

Cf. A000032, A061084, A001254, A075155.

Programs

  • Magma
    [((-1)^n*Lucas(n))^3: n in [0..30]]; // Vincenzo Librandi, Apr 22 2018
  • Mathematica
    CoefficientList[Series[(8+23*x-24*x^2-x^3)/(1+3*x-6*x^2-3*x^3+x^4), {x, 0, 25}], x]
LinearRecurrence[{-3,6,3,-1},{8,-1,27,-64},30] (* Harvey P. Dale, Apr 06 2013 *)
Table[LucasL[-n]^3, {n, 0, 25}] (* Vincenzo Librandi, Apr 22 2018 *)

Formula

a(n) = 3*L(n)+(-1)^n*L(3n).
a(n) = -3a(n-1)+6a(n-2)+3a(n-3)-a(n-4), n>3.
G.f.: ( 8+23*x-24*x^2-x^3 ) / ( (x^2+x-1)*(x^2-4*x-1) ).
a(n) is asymptotic to (-phi)^(3n) where phi is the golden ratio (1+sqrt(5))/2. - Benoit Cloitre, Sep 07 2002
a(n) = ((-1)^n*L(n))^3 = L(-n)^3. - Ehren Metcalfe, Apr 21 2018

A105941 Powers of Lucas numbers.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 11, 16, 18, 27, 29, 32, 47, 49, 64, 76, 81, 121, 123, 128, 199, 243, 256, 322, 324, 343, 512, 521, 729, 841, 843, 1024, 1331, 1364, 2048, 2187, 2207, 2209, 2401, 3571, 4096, 5776, 5778, 5832, 6561, 8192, 9349, 14641, 15127, 15129, 16384
Offset: 1

Views

Author

Jonathan Vos Post, Apr 27 2005

Keywords

References

  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 56.
  • Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001.
  • V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

Links

Crossrefs

A000032 Lucas numbers. A001254 Squares of Lucas numbers. A075155 Cubes of Lucas numbers. A099923 Fourth powers of Lucas numbers. A103325 Fifth powers of Lucas numbers. A103324 Square array T(n, k) read by antidiagonals: powers of Lucas numbers. A105317 Powers of Fibonacci numbers.
Cf. A000032, A001254, A075155, A099923, A103325, A103324, A105317.

Programs

  • Mathematica
    lim = 10^5; t = Table[f = LucasL[n]; If[f == 1, {1}, f^Range[Floor[Log[lim]/Log[f]]]], {n, 0, Floor[Log[GoldenRatio, lim]]}]; Union[Flatten[t]] (* T. D. Noe, Sep 27 2011 *)

Formula

{a(n)} = {A000204} U {A001254} U {A075155} U {A099923} U {A103325}... L(n)^2 = L(2n) + 2(-1)^n = L(n-1)*L(n+1) + 5(-1)^n. L(n)^3 = L(3n) + 3(-1)^n*L(n). L(n)^4 = L(4n) + 4(-1)^n*L(2n) + 6. L(n)^5 = L(5n) + 5(-1)^n*L(3n) + 10L(n).

Extensions

Corrected by T. D. Noe, Sep 26 2011

A105949 Powers of 3-Step Lucas numbers (A001644).

Original entry on oeis.org

1, 3, 9, 11, 21, 27, 39, 49, 71, 81, 121, 131, 241, 243, 343, 441, 443, 729, 815, 1331, 1499, 1521, 2187, 2401, 2757, 5041, 5071, 6561, 9261, 9327, 14641, 16807, 17155, 17161, 19683, 31553, 58035, 58081, 59049, 59319, 106743, 117649, 161051, 177147
Offset: 1

Views

Author

Jonathan Vos Post, Apr 27 2005

Keywords

Comments

A001644 3-Step Lucas numbers. A000032 Lucas numbers. A001254 Squares of Lucas numbers. A075155 Cubes of Lucas numbers. A099923 Fourth powers of Lucas numbers. A103325 Fifth powers of Lucas numbers. A103324 Square array T(n,k) read by antidiagonals: powers of Lucas numbers. A105317 Powers of Fibonacci numbers.

Links

Crossrefs

Cf. A001644, A105317.

Formula

{A001644(n)} U {A001644(n)^2} U {A001644(n)^3}...
Showing 1-5 of 5 results.

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