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A267424
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Fibonacci numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.
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2
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55, 17711, 5702887, 1836311903, 591286729879, 190392490709135, 61305790721611591, 160500643816367088, 19740274219868223167, 6356306993006846248183, 2046711111473984623691759, 659034621587630041982498215, 212207101440105399533740733471, 68330027629092351019822533679447
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OFFSET
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1,1
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COMMENTS
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Fibonacci numbers that are the sum of 4 but no fewer nonzero squares. See first comment in 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, ...
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).
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LINKS
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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.
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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Select[Fibonacci[Range[160]], EvenQ[(e = IntegerExponent[#, 2])] && Mod[#/2^e, 8] == 7 &] (* Amiram Eldar, Jan 29 2022 *)
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PROG
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(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), ", ")));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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