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.

A081071 a(n) = Lucas(4*n+2)-2 = Lucas(2*n+1)^2.

Original entry on oeis.org

1, 16, 121, 841, 5776, 39601, 271441, 1860496, 12752041, 87403801, 599074576, 4106118241, 28143753121, 192900153616, 1322157322201, 9062201101801, 62113250390416, 425730551631121, 2918000611027441, 20000273725560976
Offset: 0

Views

Author

R. K. Guy, Mar 04 2003

Keywords

Comments

Conjecture: a(n) = Fibonacci(4*n+3) + Sum_{k=2..2*n} Fibonacci(2*k). - Alex Ratushnyak, May 06 2012
The above conjecture is true for n >= 1. - Nguyen Tuan Anh, Aug 02 2025

References

  • Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Links

Crossrefs

Cf. A000032 (Lucas numbers), A000045, A001622, A002878 is Lucas(2n+1), A081069.

Programs

  • Magma
    I:=[1, 16, 121]; [n le 3 select I[n] else 8*Self(n-1)-8*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012
  • Maple
    luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d,`,luc(4*n+2)-2) od: # James Sellers, Mar 05 2003
  • Mathematica
    CoefficientList[Series[-(1+8*x+x^2)/((x-1)*(x^2-7*x+1)),{x,0,40}],x] (* or *) LinearRecurrence[{8,-8,1},{1,16,121},50] (* Vincenzo Librandi, Jun 26 2012 *)
LucasL[4*Range[0,20]+2]-2 (* Harvey P. Dale, Nov 25 2012 *)
  • PARI
    x='x+O('x^30); Vec((1+8*x+x^2)/((1-x)*(x^2-7*x+1))) \\ G. C. Greubel, Dec 21 2017

Formula

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: -(1+8*x+x^2)/((x-1)*(x^2-7*x+1)). - Colin Barker, Jun 26 2012
From Peter Bala, Nov 19 2019: (Start)
Sum_{n >= 1} 1/(a(n) + 5) = (3*sqrt(5) - 5)/30.
Sum_{n >= 1} 1/(a(n) - 5) = (15 - 4*sqrt(5) )/60.
Sum_{n >= 1} (-1)^(n+1)/(a(n) - 5) = 1/12.
Sum_{n >= 1} (-1)^(n+1)/(a(n) - 25/a(n)) = (5 + 2*sqrt(5))/120. (End)
Sum_{n>=0} 1/a(n) = (1/sqrt(5)) * Sum_{n>=1} n/F(2*n), where F(n) is the n-th Fibonacci number (A000045). - Amiram Eldar, Oct 05 2020
Product_{n>=1} (1 - 5/a(n)) = phi^2/4, where phi is the golden ratio (A001622) (Davlianidze, 2020). - Amiram Eldar, Dec 04 2024
From Enrique Navarrete, Mar 24 2025: (Start)
20 + 5*a(n) = A106729(n)^2.
Limit_{n->oo} a(n+1)/a(n) = (7 + 3*sqrt(5))/2. (End)

Extensions

More terms from James Sellers, Mar 05 2003

A179334 Squares that are the sum of three positive Fibonacci numbers.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 64, 81, 100, 144, 256, 289, 324, 400, 529, 576, 625, 1024, 1089, 1225, 1369, 1600, 2209, 3249, 7396, 12544, 15129, 19321, 46656, 103684, 710649, 1347921, 2178576, 4870849, 14930496, 24990001, 33385284, 228826129, 1568397609, 10749957124
Offset: 1

Views

Author

Carmine Suriano, Jan 12 2011

Keywords

Comments

There are infinitely many such numbers, because L_{2n}^2 = F_{4n+1} + F_{4n-1} + F_3 (observation of Ingrid Vukusic). - Jeffrey Shallit, Aug 19 2025
Squares k > 1 such that A007895(k) <= 3. - Robert Israel, Aug 20 2025

Examples

			a(5) = 36 = 1+1+34 = Fib(1)+Fib(2)+Fib(9).

Links

Crossrefs

Cf. A000032, A000045, A007895, A081069, A111378, A160238.

Programs

  • Maple
    phi:= 1/2 + sqrt(5)/2:
fib:= combinat:-fibonacci:
invfib := proc(x::posint)
  local q, n;
  q:= evalf((ln(x+1/2) + ln(5)/2)/ln(phi));
  n:= floor(q);
  if fib(n) <= x then
    while fib(n+1) <= x do
      n := n+1
    end do
  else
    while fib(n) > x do
      n := n-1
    end do
  end if;
  n
end:
g:= proc(n)  local ct,x,y,R;
  ct:= 0; x:= n^2; R:=NULL;
  while x > 0 do
    y:= invfib(x);
    ct:= ct+1;
    if ct = 4 then return [false, max(n+1,isqrt(fib(R[1])+fib(R[2]) + fib(R[3]+1)))]  fi;
    R:= R, y;
    x:= x - fib(y)
  od;
  if ct < 3 then [true,n+1] else [true, max(n+1,isqrt(fib(R[1])+fib(R[2])+fib(R[3]+1)))] fi
end proc:
R:= NULL: count:= 0:
n:= 2:
while count < 40 do
  V:= g(n);
  if V[1] then R:= R, n^2; count:= count+1; fi;
  n:= V[2];
od:
R; # Robert Israel, Aug 20 2025
  • Mathematica
    f=Fibonacci[Range[40]]; Select[Union[Flatten[Outer[Plus, f, f, f]]], #Harvey P. Dale, Apr 29 2015 *)

A286810 Number of non-attacking bishop positions on a cylindrical 2 X 2n chessboard.

Original entry on oeis.org

1, 9, 49, 324, 2209, 15129, 103684, 710649, 4870849, 33385284, 228826129, 1568397609, 10749957124, 73681302249, 505019158609, 3461452808004, 23725150497409, 162614600673849, 1114577054219524, 7639424778862809, 52361396397820129, 358890350005878084, 2459871053643326449, 16860207025497407049
Offset: 0

Views

Author

Richard M. Low, May 20 2017

Keywords

Comments

Essentially the same as A081069. - R. J. Mathar, May 25 2017

Links

Programs

  • PARI
    Vec((1 + x - 15*x^2 + 3*x^3) / ((1 - x)*(1 - 7*x + x^2)) + O(x^30)) \\ Colin Barker, May 21 2017

Formula

G.f.: (1+x^2-15*x^4+3*x^6) / (1-8*x^2+8*x^4-x^6).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>3. - Colin Barker, May 21 2017

A014730 Squares of odd Lucas numbers.

Original entry on oeis.org

1, 9, 49, 121, 841, 2209, 15129, 39601, 271441, 710649, 4870849, 12752041, 87403801, 228826129, 1568397609, 4106118241, 28143753121, 73681302249, 505019158609, 1322157322201, 9062201101801, 23725150497409, 162614600673849, 425730551631121, 2918000611027441
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A014447, A081069.

Programs

  • Mathematica
    Select[LucasL[Range[50]],OddQ]^2 (* Harvey P. Dale, Nov 13 2021 *)
  • PARI
    Vec(-(x-1)*(x^4+10*x^3+42*x^2+10*x+1)/((x^2-4*x-1)*(x^2+1)*(x^2+4*x-1)) + O(x^100)) \\ Colin Barker, May 14 2014

Formula

a(n) = 17*a(n-2)+17*a(n-4)-a(n-6). - R. J. Mathar, Feb 10 2012
G.f.: -(x-1)*(x^4+10*x^3+42*x^2+10*x+1) / ((x^2-4*x-1)*(x^2+1)*(x^2+4*x-1)). - Colin Barker, May 14 2014

Extensions

More terms from Colin Barker, May 14 2014
Showing 1-4 of 4 results.

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