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.

A007298 Sums of consecutive Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 16, 18, 19, 20, 21, 26, 29, 31, 32, 33, 34, 42, 47, 50, 52, 53, 54, 55, 68, 76, 81, 84, 86, 87, 88, 89, 110, 123, 131, 136, 139, 141, 142, 143, 144, 178, 199, 212, 220, 225, 228, 230, 231, 232, 233, 288, 322
Offset: 1

Views

Author

N. J. A. Sloane, Jan 02 2000

Keywords

Comments

Also the differences between two Fibonacci numbers, because the difference F(i+2) - F(j+1) equals the sum F(j) + ... + F(i). - T. D. Noe, Oct 17 2005; corrected by Patrick Capelle, Mar 01 2008

Links

Crossrefs

Cf. A000045, A050939.
Cf. A113188 (primes that are the difference of two Fibonacci numbers).
Cf. A219114 (numbers whose squares are here).

Programs

  • Maple
    isA007298 := proc(n)
    local i,Fi,j,Fj ;
    for i from 0 do
        Fi := combinat[fibonacci](i) ;
        for j from i do
            Fj :=combinat[fibonacci](j) ;
            if Fj-Fi = n then
                return true;
            elif Fj-Fi > n then
                break;
            end if;
        end do:
        Fj :=combinat[fibonacci](i+1) ;
        if Fj-Fi > n then
            return false;
        end if;
    end do:
end proc:
for n from 0 to 100 do
    if isA007298(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 25 2016
  • Mathematica
    Union[Flatten[Table[Fibonacci[n]-Fibonacci[i], {n, 14}, {i, n}]]] (* T. D. Noe, Oct 17 2005 *)
isA007298[n_] := Module[{i, Fi, j, Fj}, For[i = 0, True, i++, Fi = Fibonacci[i]; For[j = i, True, j++, Fj = Fibonacci[j]; Which[Fj - Fi == n, Return@True, Fj - Fi > n, Break[]]]; Fj := Fibonacci[i + 1]; If[Fj - Fi > n, Return@False]]];
Select[Range[0, 1000], isA007298] (* Jean-François Alcover, Nov 16 2023, after R. J. Mathar *)
  • PARI
    A130233(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2)
list(lim)=my(v=List([0]),F=vector(A130233(lim),i,fibonacci(i)),s,t); for(i=1,#F, s=0; forstep(j=i,1,-1, s+=F[j]; if(s>lim, break); listput(v,s))); Set(v) \\ Charles R Greathouse IV, Oct 06 2016

Formula

log a(n) >> sqrt(n). - Charles R Greathouse IV, Oct 06 2016

A221471 Integers n such that n^2 is the difference of two Lucas numbers (A000032).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 14, 29, 57, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371
Offset: 1

Views

Author

T. D. Noe, Feb 13 2013

Keywords

Comments

This sequence, growing exponentially, is interesting because the corresponding sequence for Fibonacci numbers (A219114) appears to be finite. However, except the 7 numbers in A221472, it appears that the squares of all these numbers have the form 0, L(5) - L(0), or L(4n+2) - L(0), where L(n) denotes the n-th Lucas number.
It is easy to show that L(4n+2) - L(0) = L(2n+1)^2. - Zhi-Wei Sun, Jan 02 2015

Links

Crossrefs

Cf. A000032 (Lucas numbers), A113191 (difference of two Lucas numbers).
Cf. A219114 (corresponding sequence for Fibonacci numbers).

Programs

  • Mathematica
    t = Union[Flatten[Abs[Table[LucasL[n] - LucasL[i], {n, 0, 120}, {i, n}]]]]; t2 = Select[t, IntegerQ[Sqrt[#]] &]; Sqrt[t2]

Formula

Conjecture: a(n) = 3*a(n-1)-a(n-2) = A002878(n-8) for n>13. G.f.: x^2*(28*x^11-66*x^10-16*x^9-2*x^8-13*x^7-2*x^6-5*x^5-4*x^4-3*x^3-2*x^2-x+1) / (x^2-3*x+1). [Colin Barker, Feb 17 2013]

A221472 Integers n such that n^2 is the difference of two Lucas numbers (A000204).

Original entry on oeis.org

0, 1, 2, 5, 6, 14, 57
Offset: 1

Views

Author

T. D. Noe, Feb 13 2013

Keywords

Comments

This sequence is similar to the one for Fibonacci numbers (A219114) and appears to be finite also. See A221471 for an infinite version of this sequence.

Examples

			The only known square differences of Lucas numbers:
1^2 = L(3)-L(2) = 4-3,
2^2 = L(4)-L(2) 7-3 = L(5)-L(4) = 11-7,
5^2 = L(7)-l(3) = 29-4,
6^2 = L(8)-L(5) = 47-11,
14^2 = L(11)-L(2) = 199-3,
57^2 = L(17)-L(12) = 3571-322.

Crossrefs

Cf. A000032 (Lucas numbers), A113191 (difference of two Lucas numbers).
Cf. A219114 (corresponding sequence for Fibonacci numbers).

Programs

  • Mathematica
    t = Union[Flatten[Abs[Table[LucasL[n] - LucasL[i], {n, 120}, {i, n}]]]]; t2 = Select[t, IntegerQ[Sqrt[#]] &]; Sqrt[t2]

A272712 Perfect powers that are the difference of two nonnegative Fibonacci numbers.

Original entry on oeis.org

1, 4, 8, 16, 32, 81, 144, 225, 343, 576
Offset: 1

Views

Author

Altug Alkan, May 05 2016

Keywords

Comments

Listed 10 terms are 1, 2^2, 2^3, 2^4, 2^5, 3^4, 12^2, 15^2, 3^5, 24^2.
1, 4, 8, 16, 32, 81, 343 are also members of A000961.
1, 4, 8, 16, 144 are in the intersection of this sequence and A272575.
Is this sequence finite?
If a(11) exists, it must be larger than 10^2000. - Giovanni Resta, May 25 2016

Examples

			32 is a term because 32 = 2^5 = 34 - 2 = Fibonacci(9) - Fibonacci(3).

Crossrefs

Cf. A000961, A007298, A001597, A219114, A272575.

Programs

  • Maple
    isA272712 := proc(n)
    isA001597(n) and isA007298(n) ; #uses code in A001597 and A007298
end proc:
for n from 1 do
    if isA272712(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, May 25 2016
  • Mathematica
    isA001597[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1;
isA007298[n_] := Module[{i, Fi, j, Fj}, For[i = 0, True, i++, Fi = Fibonacci[i]; For[j = i, True, j++, Fj = Fibonacci[j]; Which[Fj - Fi == n, Return@True, Fj - Fi > n, Break[]]]; Fj := Fibonacci[i + 1]; If[Fj - Fi > n, Return@False]]];
Select[Range[1000], isA001597[#] && isA007298[#]&] (* Jean-François Alcover, Nov 16 2023, after R. J. Mathar in A007298 *)

A344661 Integers k such that k^2 is the sum of two Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 12, 40, 3864
Offset: 1

Views

Author

Lamine Ngom, May 26 2021

Keywords

Comments

Is this sequence finite?
No other terms below 10^20899.

Examples

			These square sums of Fibonacci numbers correspond to:
     0^2 = F(0)  + F(0);
     1^2 = F(1)  + F(0)  = F(2) + F(0);
     2^2 = F(4)  + F(1)  = F(4) + F(2) = F(3) + F(3);
     3^2 = F(6)  + F(1)  = F(6) + F(2);
     4^2 = F(7)  + F(4)  = F(6) + F(6);
     6^2 = F(9)  + F(3);
    12^2 = F(11) + F(10) = F(12) + F(0);
    40^2 = F(17) + F(4);
  3864^2 = F(36) + F(12).

Links

Crossrefs

Cf. A000045, A007298, A084176, A111378, A219114.
Showing 1-5 of 5 results.

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

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