A265227 - OEIS

%I #25 Sep 22 2025 16:01:23

%S 0,1,3,6,8,9,10,12,15,17,18,19,21,24,26,27,28,30,33,35,36,37,39,42,44,

%T 45,46,48,51,53,54,55,57,60,62,63,64,66,69,71,72,73,75,78,80,81,82,84,

%U 87,89,90,91,93,96,98,99,100,102,105,107,108,109,111,114

%N Nonnegative m for which k*floor(m^2/9) = floor(k*m^2/9), with 2 < k < 9.

%C Also, nonnegative m congruent to 0, 1, 3, 6 or 8 (mod 9). The product of any two terms belongs to the sequence and so also a(n)^2, a(n)^3, a(n)^4, etc.

%C Integers x >= 0 satisfying k*floor(x^2/9) = floor(k*x^2/9) with k >= 0:

%C k = 0, 1:  x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...   (A001477);

%C k = 2:     x = 0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 15, ... (A060464);

%C k = 3..8:  x = 0, 1, 3, 6, 8, 9, 10, 12, 15, 17, 18, ...   (this sequence);

%C k > 8:     x = 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... (A008585).

%C Primes in sequence: 3, 17, 19, 37, 53, 71, 73, 89, 107, 109, 127, ...

%H Bruno Berselli, <a href="/A265227/b265227.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

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

%F a(n) = a(n-1) + a(n-5) - a(n-6) for n>6.

%t Select[Range[0, 120], 3 Floor[#^2/9] == Floor[3 #^2/9] &]

%t Select[Range[0, 120], MemberQ[{0, 1, 3, 6, 8}, Mod[#, 9]] &]

%t LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 3, 6, 8, 9}, 70]

%o (SageMath) [n for n in (0..120) if 3*floor(n^2/9) == floor(3*n^2/9)]

%o (Magma) [n: n in [0..120] | 3*Floor(n^2/9) eq Floor(3*n^2/9)]; /* or, by the definition: */ K:=[3..8]; [<k,[n: n in [0..30] | k*Floor(n^2/9) eq Floor(k*n^2/9)]>: k in K];

%Y Cf. A001477, A008585, A060464.

%Y Cf. similar sequences listed in A265188.

%K nonn,easy

%O 1,3

%A _Bruno Berselli_, Dec 06 2015