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.

A179244 Numbers that have 4 terms in their Zeckendorf representation.

Original entry on oeis.org

33, 46, 51, 53, 54, 67, 72, 74, 75, 80, 82, 83, 85, 86, 87, 101, 106, 108, 109, 114, 116, 117, 119, 120, 121, 127, 129, 130, 132, 133, 134, 137, 138, 139, 141, 156, 161, 163, 164, 169, 171, 172, 174, 175, 176, 182, 184, 185, 187, 188, 189, 192, 193, 194, 196
Offset: 1

Views

Author

Emeric Deutsch, Jul 05 2010

Keywords

Comments

A007895(a(n)) = 4. - Reinhard Zumkeller, Mar 10 2013

Examples

			33=21+8+3+1;
46=34+8+3+1;
51=34+13+3+1;
53=34+13+5+1;
54=34+13+5+2;

Links

Crossrefs

Cf. A035517, A007895, A111458, A135709, A179242 - A179253.

Programs

  • Haskell
    a179244 n = a179244_list !! (n-1)
a179244_list = filter ((== 4) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013
  • Maple
    with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(9)-1 to 200 do if B(i) = 4 then Q := `union`(Q, {i}) else end if end do: Q;
  • Mathematica
    zeck = DigitCount[Select[Range[2000], BitAnd[#, 2*#] == 0&], 2, 1];
Position[zeck, 4] // Flatten (* Jean-François Alcover, Jan 25 2018 *)

A111458 Numbers that cannot be represented as the sum of at most three Fibonacci numbers (with repetitions allowed).

Original entry on oeis.org

33, 46, 51, 53, 54, 67, 72, 74, 75, 80, 82, 83, 85, 86, 87, 88, 101, 106, 108, 109, 114, 116, 117, 119, 120, 121, 122, 127, 129, 130, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 156, 161, 163, 164, 169, 171, 172
Offset: 1

Views

Author

Stefan Steinerberger, Nov 15 2005

Keywords

Examples

			33 is neither a Fibonacci number nor can be written as the sum of two or three Fibonacci numbers.

Links

Crossrefs

Cf. A000045, A000119, A179244, A135709.

Programs

  • Maple
    isA111458 := proc(n) # returns true if n is in the sequence
    local xi,yi,x,y,z ;
    for xi from 0 do
        x := A000045(xi) ;
        if 3*x > n then
            return true;
        end if;
        for yi from xi do
            y := A000045(yi) ;
            if x+2*y > n then
                break;
            else
                z := n-x-y ;
                if isA000045(z) then # see isFib in A000045
                    return false;
                end if;
            end if;
        end do:
    end do:
end proc:
A111458 := proc(n)
    option remember;
    local a;
    if n = 0 then
        -1;
    else
        for a from procname(n-1)+1 do
            if isA111458(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 09 2015
  • Mathematica
    FibQ[n_] := IntegerQ[Sqrt[5n^2+4]] || IntegerQ[Sqrt[5n^2-4]];
P[n_] := IntegerPartitions[n, 3, Select[Range[n], FibQ]];
Select[Range[1000], P[#] == {}&] (* Jean-François Alcover, Jul 20 2023 *)

A135558 Sums of three distinct nonzero Fibonacci numbers.

Original entry on oeis.org

6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 76, 77, 78, 79, 81, 84, 89, 90, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Colm Mulcahy, Feb 23 2008, Mar 02 2008

Keywords

Comments

These numbers may have more than one such representation.

Links

Crossrefs

Cf. A000045, A135709 (Complement).

Programs

  • Maple
    isA135558 := proc(n) # returns true if n is in the sequence
    local xi,yi,x,y,z ;
    for xi from 2 do
        x := A000045(xi) ;
        if 3*x > n then
            return false;
        end if;
        for yi from xi+1 do
            y := A000045(yi) ;
            if x+2*y > n then
                break;
            else
                z := n-x-y ;
                if z >y and isA000045(z) then # see isFib in A000045
                    return true;
                end if;
            end if;
        end do:
    end do:
end proc:
A135558 := proc(n)
    option remember;
    local a;
    if n = 1 then
        6;
    else
        for a from procname(n-1)+1 do
            if isA135558(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 09 2015
  • Mathematica
    fibs[n_ /; n >= 6] := Reap[Module[{k = 1}, While[Fibonacci[k] < n, Sow[Fibonacci[k++]]]]][[2, 1]];
okQ[n_] := AnyTrue[IntegerPartitions[n, {3}, fibs[n]], Length[Union[#]] == 3&];
Select[Range[6, 100], okQ] (* Jean-François Alcover, Dec 12 2023 *)
Select[Total/@Subsets[Fibonacci[Range[2,12]],{3}]//Union,#<=100&] (* Harvey P. Dale, Jul 06 2025 *)

Extensions

More terms from N. J. A. Sloane, Mar 01 2008, corrected Mar 05 2008

A160238 Numbers n such that n^2 can be expressed as the sum of three different nonzero Fibonacci numbers.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 17, 18, 20, 23, 24, 25, 32, 33, 35, 37, 40, 47, 57, 86, 112, 123, 139, 216, 322, 843, 1161, 1476, 2207, 3864, 4999, 5778, 15127, 39603, 103682, 271443, 710647, 1244196, 1860498, 4870847, 12752043
Offset: 1

Views

Author

Carmine Suriano, May 05 2009

Keywords

Comments

There exist a proper subsequence b(i)of a(n): n=[1, 2, 8, 17, 21, 24, 25, 28,29, 30, 31, 32, 33, 34, ...] such that approximatively b(i+1)=b(i)*(1+phi) where phi is 1.618... is the golden ratio and the approximation holds as a limit when i goes to infinity. For such a subsequence b(i) we have the following formula for the corresponding term when squared b(i)*b(i)=Fib(4*i+1)+Fib(4*i-1)+Fib(3). In the previous example 4999=b(9).

Examples

			4999*4999=24990001=Fib(37)+Fib(35)+Fib(3)

Crossrefs

Cf. A000045, A135709, A135558.

Extensions

Inserted 4 (with 4^2=13+1+2), 6 (with 36=21+2+13), 12 (with 12^2=89+21+34) etc. Added "nonzero" to definition - R. J. Mathar, Oct 23 2010
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.