A267534 Indices of Lucas numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

4, 8, 11, 16, 20, 21, 23, 28, 32, 35, 40, 44, 45, 47, 52, 56, 59, 64, 68, 69, 71, 76, 80, 83, 88, 92, 93, 95, 100, 104, 107, 112, 116, 117, 119, 124, 128, 131, 136, 140, 141, 143, 148, 152, 155, 160, 164, 165, 167, 172, 176, 179, 184, 188, 189, 191, 196, 200, 203, 208, 212

Offset

1,1

Comments

First differences of this sequence are 4, 3, 5, 4, 1, 2, 5, 4, 3, 5, 4, 1, 2, 5, 4, 3, 5, 4, 1, 2, 5, 4, 3, 5, 4, 1, 2, 5, 4, 3, 5, 4, 1, 2, 5, ...

So "4, 3, 5, 4, 1, 2, 5" appears periodically in first differences.

Corresponding Lucas numbers are 7, 47, 199, 2207, 15127, 24476, 64079, 710647, 4870847, 20633239, 228826127, 1568397607, 2537720636, 6643838879, 73681302247, 505019158607, 2139295485799, 23725150497407, 162614600673847, ...

Links

Colin Barker, Table of n, a(n) for n = 1..500

Formula

Conjectures from Colin Barker, Jan 30 2016: (Start)

a(n) = a(n-1)+a(n-7)-a(n-8) for n>8.

G.f.: x*(2 +x)*(2 +x +x^2 +2*x^3 +x^4 +x^6) / ((1 -x)^2*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)).

(End)

Example

4 is a term because A000032(4) = 7 and 7 = x^2 + y^2 + z^2 has no solution for integer values of x, y and z.

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); }
l(n) = fibonacci(n+1) + fibonacci(n-1);
for(n=0, 1e3, if(isA004215(l(n)), print1(n, ", ")));

Crossrefs

Cf. A000032, A004215, A267424.

Sequence in context: A311058 A311059 A311060 * A311061 A317331 A311062

Adjacent sequences: A267531 A267532 A267533 * A267535 A267536 A267537

Keyword

nonn,easy

Author

Altug Alkan, Jan 16 2016

Status

approved