A267424 Fibonacci numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

55, 17711, 5702887, 1836311903, 591286729879, 190392490709135, 61305790721611591, 160500643816367088, 19740274219868223167, 6356306993006846248183, 2046711111473984623691759, 659034621587630041982498215, 212207101440105399533740733471, 68330027629092351019822533679447

Offset

1,1

Comments

Fibonacci numbers that are the sum of 4 but no fewer nonzero squares. See first comment in A004215.

Intersection of A000045 and A004215.

Corresponding index of Fibonacci numbers are 10, 22, 34, 46, 58, 70, 82, 84, 94, 106, 118, 130, 142, 154, 166, 178, 180, 190, 202, 214, 226, 238, 250, 262, 274, 276, 286, 298, 310, 322, ...

First differences of corresponding indices are 12, 12, 12, 12, 12, 12, 2, 10, 12, 12, 12, 12, 12, 12, 12, 2, 10, 12, 12, 12, 12, ...

From Robert Israel, Jan 17 2016: (Start)

A000045(10+12*k) == 7 (mod 8), so these are in the sequence.

A000045(84+96*k) == 4^2*7 (mod 4^2*8), so these are in the sequence.

It appears that A000045((84+96*k)*4^j) == 4^(j+2)*7 (mod 4^(j+2)*8) for j,k>=0, so these are in the sequence. (End).

Links

Robert Israel, Table of n, a(n) for n = 1..970

R. M. Grassl, Problem B-226, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 10, No. 2 (1972), p. 218; Fibonacci Sum of Four Squares, Solution to Problem B-226 by Paul S. Bruckman, ibid., Vol. 11, No. 2 (1973), p. 106.

Example

Fibonacci number 21 is not a term of this sequence because 21 = 1^2 + 2^2 + 4^2.
55 is a term because it is a Fibonacci number and there is no integer values of x, y and z for the equation 55 = x^2 + y^2 + z^2.

Maple

is004215:= proc(n) local t;
  t:= padic:-ordp(n, 2);
  if t::odd then false else evalb(n/2^t mod 8 = 7) fi
end proc:
select(is004215, [seq(combinat:-fibonacci(n), n=2..1000)]); # Robert Israel, Jan 17 2016

Mathematica

Select[Fibonacci[Range[160]], EvenQ[(e = IntegerExponent[#, 2])] && Mod[#/2^e, 8] == 7 &] (* Amiram Eldar, Jan 29 2022 *)

Prog

(PARI) isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
f(n) = fibonacci(n);
for(n=0, 1e3, if(isA004215(f(n)), print1(f(n), ", ")));

Crossrefs

Cf. A000045, A004215.

Sequence in context: A268167 A363869 A266031 * A221772 A221123 A231826

Adjacent sequences: A267421 A267422 A267423 * A267425 A267426 A267427

Keyword

nonn

Author

Altug Alkan, Jan 14 2016

Status

approved