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 1-4 of 4 results.

A091697 Values of m corresponding to members of A088966.

Original entry on oeis.org

2, 3, 5, 9, 17, 33, 35, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577
Offset: 1

Views

Author

David Wasserman, Jan 29 2004

Keywords

Comments

For every k >= 0, 2^k + 1 is in the sequence.

Examples

			a(2) = 3, corresponding to A088966(2) = 8 because A007947(8) = A007947(3+1) and A007947(3) = A007947(8+1).

Links

Programs

  • Maple
    rad:= n -> convert(numtheory:-factorset(n), `*`):
count:= 0: lastr:= rad(1):
for n from 2 to 10^7 do
  newr:= rad(n);
  P[lastr, newr]:= n-1;
  if assigned(P[newr, lastr]) then
    count:= count+1; A[count]:= n-1; M[count]:= P[newr, lastr];
  fi;
  lastr:= newr;
od:
seq(M[n], n=1..count); # Robert Israel, Aug 11 2014

Extensions

a(13) from Robert Israel, Aug 11 2014
a(14)-a(22) from Hercher's paper by Martin Fuller, Jul 18 2025

A087914 Numbers m such that A007947(m) = A007947(k) and A007947(m+1) = A007947(k+1), for some k < m.

Original entry on oeis.org

8, 48, 224, 960, 1215, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224
Offset: 1

Views

Author

Naohiro Nomoto, Oct 26 2003

Keywords

Comments

For every k > 1, the sequence includes 4^k - 2^(k+1), with m = 2^k - 2. - David Wasserman, Jan 29 2004
a(12) <= 16769024. a(13) <= 67092480. a(14) <= 268402688. a(15) <= 1073676288. [Donovan Johnson, Dec 19 2008]

Examples

			A007947(8) = A007947(2) and A007947(9) = A007947(3), so 8 is in the sequence.

Crossrefs

Cf. A007947, A088966.

Programs

  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(m) = {my(rm = rad(m), sm = rad(m+1)); for (k=1, m-1, if ((rad(k) == rm) && (rad(k+1) == sm), return (1)););} \\ Michel Marcus, Apr 05 2021

Extensions

a(7)-a(11) from Donovan Johnson, Dec 19 2008
Name edited by Michel Marcus, Apr 06 2021
Confirmed a(12)-a(15) and extended with a(16) by Martin Ehrenstein, Apr 18 2021

A242985 a(n) = 4^n + 2^(n+1).

Original entry on oeis.org

3, 8, 24, 80, 288, 1088, 4224, 16640, 66048, 263168, 1050624, 4198400, 16785408, 67125248, 268468224, 1073807360, 4295098368, 17180131328, 68720001024, 274878955520, 1099513724928, 4398050705408, 17592194433024, 70368760954880, 281475010265088, 1125899973951488
Offset: 0

Views

Author

Michel Marcus, Aug 17 2014

Keywords

Comments

Subsequence of A088966.
For n > 1, number of connected dominating sets in the n-book graph. - Eric W. Weisstein, Jun 29 2017

Examples

			a(0) = 4^0 + 2^1 = 1 + 2 = 3.

Links

Crossrefs

Cf. A000079, A000302, A088966.

Programs

  • GAP
    List([0..30], n-> 4^n + 2^(n+1)); # G. C. Greubel, Jan 11 2020
  • Magma
    [4^n + 2^(n+1): n in [0..30]]; // G. C. Greubel, Jan 11 2020
  • Maple
    seq( 4^n + 2^(n+1), n=0..30); # G. C. Greubel, Jan 11 2020
  • Mathematica
    (* From Eric W. Weisstein, Jun 29 2017: (Start) *)
Table[4^n + 2^(n+1), {n, 0, 30}]
LinearRecurrence[{6,-8}, {8,24}, {0, 30}]
CoefficientList[Series[(3-10x)/((1-2x)(1-4x)), {x, 0, 30}], x] (* End *)
  • PARI
    a(n) = 4^n + 2^(n+1);
  • Sage
    [4^n + 2^(n+1) for n in (0..30)] # G. C. Greubel, Jan 11 2020

Formula

a(n) = A000302(n) + A000079(n+1).
G.f.: 1/(1-4*x) + 2/(1-2*x). - Robert Israel, Aug 17 2014
E.g.f.: exp(4*x) + 2*exp(2*x). - G. C. Greubel, Jan 11 2020

A290075 Number of monomials in c(n) where c(1) = x, c(2) = y, c(n+2) = c(n+1) + c(n)^2.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 24, 44, 80, 152, 288, 560, 1088, 2144, 4224, 8384, 16640, 33152, 66048, 131840, 263168, 525824, 1050624, 2100224, 4198400, 8394752, 16785408, 33566720, 67125248, 134242304, 268468224, 536920064, 1073807360, 2147581952, 4295098368
Offset: 1

Views

Author

Michael Somos, Jul 19 2017

Keywords

Examples

			G.f. = x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 14*x^7 + 24*x^8 + 44*x^9 + ...
c(3) = x^2 + y so a(3) = 2, c(4) = x^2 + (y + y^2) so a(4) = 3, c(5) = x^4 + x^2(2*y) + (y + 2*y^2) so a(5) = 5.

Links

Crossrefs

Cf. A088966, A103484, A211525, A257273.

Programs

  • Mathematica
    nn:=36; nn:=10; Rest[CoefficientList[Series[(x - x^2 - 2*x^3 + x^4 - x^5) / ((1 - 2*x) * (1 - 2*x^2)),{x, 0, nn}], x]] (* Georg Fischer, May 10 2020 *)
  • PARI
    {a(n) = if( n<3, n>0, my(e=n%2, m=2^((n+e)/2-2)); m * (m+2+e) / (1+e))};
  • PARI
    Vec(x*(1 - x - 2*x^2 + x^3 - x^4) / ((1 - 2*x)*(1 - 2*x^2)) + O(x^50)) \\ Colin Barker, Jul 22 2017

Formula

G.f.: (x - x^2 - 2*x^3 + x^4 - x^5) / ((1 - 2*x) * (1 - 2*x^2)).
0 = 4*a(n) - 2*a(n+1) - 2*a(n+2) + 1*a(n+3) for n>=3.
A088966(n) = a(2*n+2). A257273(n) = a(2*n+3). A211525(n) = a(n+8).
From Colin Barker, Jul 22 2017: (Start)
a(n) = 2^(n/2-1) + 2^(n-4) for n>2 and even.
a(n) = 3*2^((n-5)/2) + 2^(n-4) for n>2 and odd.
(End)
Given the sequence c(n, x, y), then the coefficients of: (1) c(n+2, sqrt(t), 0), (2) c(n+1, 0, t), and (3) c(n, t, t), each form the triangular sequence A103484. - Michael Somos, Jul 24 2017
Showing 1-4 of 4 results.

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