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.

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 *)

A272575 Perfect powers that are the sum of two Fibonacci numbers.

Original entry on oeis.org

1, 4, 8, 9, 16, 36, 144, 1000, 1600, 14930496
Offset: 1

Views

Author

Altug Alkan, May 03 2016

Keywords

Comments

Intersection of A001597 and A084176.
Listed terms are 1, 2^2, 2^3, 3^2, 2^4, 6^2, 12^2, 10^3, 40^2, 3864^2.
First five terms are also members of A000961.
Conjecture: there are no more terms in this sequence. Any remaining terms must have over 10000 digits. - Charles R Greathouse IV, May 04 2016

Examples

			8 is a term because 2^3 = 3 + 5.

Crossrefs

Cf. A000045, A001597, A084176, A111378.

Programs

  • Mathematica
    Select[Range[10^4], Function[k, Or[k == 1, GCD @@ Map[Last, FactorInteger@ k] > 1] && Total@ Map[Times @@ Boole@ Map[MemberQ[s, #] &, #] &, Transpose@ {#, k - #} &@ Range[0, Floor[k/2]]] > 0]] (* Michael De Vlieger, May 03 2016 *)
  • PARI
    list(lim)=my(upper=log(lim*sqrt(5))\log((1+sqrt(5))/2)+1, t, tt, v=List([1])); if(fibonacci(t)>lim, t--); for(i=3, upper, t=fibonacci(i); for(j=2, i-1, tt=t+fibonacci(j); if(tt>lim, break); if(ispower(tt), listput(v, tt)))); Set(v) \\ Charles R Greathouse IV, May 03 2016

A362295 Sums of two Fibonacci numbers that are also sums of two squares.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 18, 26, 29, 34, 36, 37, 58, 68, 89, 90, 97, 144, 145, 146, 149, 157, 178, 233, 234, 241, 288, 377, 466, 521, 610, 612, 613, 754, 1000, 1021, 1042, 1076, 1220, 1597, 1600, 1602, 1618, 1741, 2592, 2597, 2605, 2817, 3194, 4181, 4194, 4325, 6770, 6773, 6778, 6786
Offset: 1

Views

Author

Robert Israel, Apr 14 2023

Keywords

Comments

Intersection of A001481 and A084176.

Examples

			a(5) = 5 is a term because 5 = 2 + 3 = A000045(3) + A000045(4) = 2^2 + 1^2.

Links

Crossrefs

Cf. A000045, A001481, A059389, A084176, A111378.

Programs

  • Maple
    ss:= proc(n) local F,t;
  F:= ifactors(n)[2];
  andmap(t ->  t[1] mod 4 <> 3 or t[2]::even, F)
end proc:
fibs:= map(combinat:-fibonacci, {$0..25}):
N:= max(fibs):
fib2:= {seq(seq(fibs[i]+fibs[j],i=1..j),j=1..nops(fibs))}:
sort(convert(select(t -> t <= N and ss(t), fib2),list));
  • Mathematica
    max = 150; (* max = 150 gives 1670 terms *)
Join[{0, 1}, Select[Union[Total /@ Tuples[Fibonacci[Range[2, max]], {2}]], # <= Fibonacci[max] && SquaresR[2, #] != 0&]] (* Jean-François Alcover, Sep 29 2024, after Harvey P. Dale in A059389 *)
  • Python
    from itertools import islice
from sympy import factorint
def A362295_gen(): # generator of terms
    yield from (0,1,2)
    a = [1,2]
    while True:
        b = a[-1]+a[-2]
        c = a[-1]<<1
        flag = True
        for d in a:
            n = b+d
            if flag and n>=c:
                if n>c:
                   f = factorint(c)
                   if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
                       yield c
                flag = False
            f = factorint(n)
            if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
                yield n
        a.append(b)
A362295_list = list(islice(A362295_gen(),60)) # Chai Wah Wu, Apr 16 2023

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 1-4 of 4 results.

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