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 1-2 of 2 results.

A249097 Ordered union of the sets {h^6, h >=1} and {3*k^6, k >=1}.

Original entry on oeis.org

1, 3, 64, 192, 729, 2187, 4096, 12288, 15625, 46656, 46875, 117649, 139968, 262144, 352947, 531441, 786432, 1000000, 1594323, 1771561, 2985984, 3000000, 4826809, 5314683, 7529536, 8957952, 11390625, 14480427, 16777216, 22588608, 24137569, 34012224, 34171875
Offset: 1

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Author

Clark Kimberling, Oct 21 2014

Keywords

Comments

Let S = {h^6, h >=1} and T = {3*k^6, k >=1}. Then S and T are disjoint. The position of n^6 in the ordered union of S and T is A249098(n), and the position of 3*n^6 is A249079(n).

Examples

			{h^6, h >=1} = {1, 64, 729, 4096, 15625, 46656, 117649, ...};
{3*k^6, k >=1} = {3, 192, 2187, 12288, 46875, 139968, ...};
so the union is {1, 3, 64, 192, 729, 2187, 4096, 12288, ...}
		

Crossrefs

Programs

  • PARI
    upto(n)=setunion(apply(k->k^6, [1..sqrtnint(n,6)]), apply(k->3*k^6, [1..sqrtnint(n\3,6)])) \\ Andrew Howroyd, Feb 18 2025

A256278 a(0)=1, a(1)=2, a(n) = 31*a(n-1) - 29*a(n-2).

Original entry on oeis.org

1, 2, 33, 965, 28958, 869713, 26121321, 784539274, 23563199185, 707707535789, 21255600833094, 638400107288033, 19173990901769297, 575880114843495250, 17296237823997043137, 519482849213446974997, 15602377428720941973934, 468608697663159238917041
Offset: 0

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Author

Karl V. Keller, Jr., Jun 02 2015

Keywords

Comments

The sequence A084330 is a(0)=0, a(1)=1, a(n)=31a(n-1)-29a(n-2), and the ratio A084330(n+1)/a(n) converges to phi^7 (~29.034441853748633...), where phi is the golden ratio (A001622).
The continued fraction for phi^7 is {29,{29}}, and 29 occurs in the following approximations for n*phi^7: A248786 (29*n+floor(n/29)+0^n-0^(n mod 29)) for A004922 (floor(n*phi^7)), A249079 (29*n+floor(n/29)+0^(1-floor((14+(n mod 29))/29))) for A004942 (round(n*phi^7)), and A248739 (29*n+ceiling(n/29)) for A004962 (ceiling(n*phi^7)).

Examples

			For n=3, 31*a(2)-29*a(1) = 31*(33)-29*(2) = 1023-58 = 965.
		

Crossrefs

Programs

  • Magma
    I:=[1,2]; [n le 2 select I[n] else 31*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 03 2015
  • Maple
    a:= n-> (<<0|1>, <-29|31>>^n. <<1, 2>>)[1,1]:
    seq(a(n), n=0..23);  # Alois P. Heinz, Dec 22 2023
  • Mathematica
    LinearRecurrence[{31, -29}, {1, 2}, 50] (* or *) CoefficientList[Series[(1 - 29 x)/(29 x^2 - 31 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 03 2015 *)
  • Python
    print(1, end=', ')
    print(2, end=', ')
    an = [1,2]
    for n in range(2,26):
      print(31*an[n-1]-29*an[n-2], end=', ')
      an.append(31*an[n-1]-29*an[n-2])
    

Formula

G.f.: (1-29*x)/(29*x^2-31*x+1). - Vincenzo Librandi, Jun 03 2015
E.g.f.: exp(31*x/2)*(65*cosh(13*sqrt(5)*x/2) - 27*sqrt(5)*sinh(13*sqrt(5)*x/2))/65. - Stefano Spezia, Aug 31 2025
Showing 1-2 of 2 results.