%I #20 Jul 04 2025 10:06:08

%S 1,2,8,15,16,18,22,32,47,48,53,58,64,70,77,78,80,94,95,96,103,106,128,

%T 138,163,199,206,256,257,266,269,311,326,330,352,358,385,397,398,401,

%U 419,421,499,512,514,538,587,599,617,622,640,672,683,757,768,770,773

%N Numbers k such that arithmetic mean of squares of the first k tribonacci numbers is an integer.

%C Can the arithmetic mean of the tribonacci numbers (T(0)+...+T(k-1)) / k be an integer?

%C The arithmetic means are integers for the first 1, 2, 47, 53, 94, 103, 106, ... (A141579) tribonacci numbers. - _R. J. Mathar_, Aug 04 2008

%H Amiram Eldar, <a href="/A140973/b140973.txt">Table of n, a(n) for n = 1..10000</a>

%F Numbers k such that (T(0)^2+ T(1)^2+ ... + T(k-1)^2) / k is an integer, where T(i) = i-th tribonacci number.

%t With[{m = 1000}, Position[Accumulate[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, m]^2] / Range[m], _?IntegerQ] // Flatten] (* _Amiram Eldar_, Jul 04 2025 *)

%Y Cf. A000073, A141579.

%Y Cf. A107239. - _R. J. Mathar_, Aug 04 2008

%K easy,nonn

%O 1,2

%A _Ctibor O. Zizka_, Jul 27 2008

%E a(1)-a(2) inserted and a(32) onwards added by _R. J. Mathar_, Aug 04 2008