A269966 Integers n such that the n-th golden rectangle number is the sum of 2 nonzero squares.

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

Offset

1,1

Comments

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, ...

Links

Table of n, a(n) for n=1..55.

Example

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.

Mathematica

Rest@ Select[Range@ 200, SquaresR[2, #] > 0 &[Fibonacci[#] Fibonacci[# + 1]] &] (* Michael De Vlieger, Mar 09 2016 *)

Prog

(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

Crossrefs

Cf. A000404, A001654, A009003.

Sequence in context: A265716 A376762 A206332 * A026344 A284488 A322047

Adjacent sequences: A269963 A269964 A269965 * A269967 A269968 A269969

Keyword

nonn

Author

Altug Alkan, Mar 08 2016

Extensions

a(36)-a(55) from Charles R Greathouse IV, Mar 08 2016

Status

approved