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A269966
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Integers n such that the n-th golden rectangle number is the sum of 2 nonzero squares.
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0
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2, 5, 6, 11, 12, 13, 14, 25, 26, 37, 38, 61, 62, 73, 74, 85, 86, 97, 98, 121, 122, 133, 134, 145, 146, 157, 158, 181, 182, 221, 222, 253, 254, 325, 326, 337, 338, 365, 366, 397, 398, 445, 446, 613, 614, 625, 626, 697, 698, 721, 722, 793, 794, 865, 866
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OFFSET
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1,1
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COMMENTS
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Corresponding golden rectangle numbers are 2, 40, 104, 12816, 33552, 87841, 229970, 9107509825, 23843770274, 944284833567073, 2472169789339634, ...
Initial terms of first differences are 3, 1, 5, 1, 1, 1, 11, 1, 11, 1, 23, 1, 11, 1, 11, 1, 11, 1, 23, ...
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LINKS
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EXAMPLE
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5 is a term because 1^2 + 1^2 + 2^2 + 3^2 + 5^2 = 5*8 = 40 = 2^2 + 6^2.
6 is a term because 1^2 + 1^2 + 2^2 + 3^2 + 5^2 + 8^2 = 8*13 = 104 = 2^2 + 10^2.
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MATHEMATICA
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Rest@ Select[Range@ 200, SquaresR[2, #] > 0 &[Fibonacci[#] Fibonacci[# + 1]] &] (* Michael De Vlieger, Mar 09 2016 *)
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PROG
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(PARI) isA000404(n)= for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))
a001654(n) = fibonacci(n)*fibonacci(n+1);
for(n=1, 1e2, if(isA000404(a001654(n)), print1(n, ", ")));
(PARI) has(f)=for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(0))); 1
isA009003(f)=for(i=1, #f~, if(f[i, 1]%4==1, return(1))); 0
is(n)=my(f, g); has(f=factor(fibonacci(n))) && has(g=factor(fibonacci(n+1))) && (n%3!=1 || isA009003(f) || isA009003(g)) \\ Charles R Greathouse IV, Mar 08 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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