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%I #45 Sep 08 2022 08:46:17
%S 0,8,9,17,18,26,27,35,36,44,45,53,54,62,63,71,72,80,81,89,90,98,99,
%T 107,108,116,117,125,126,134,135,143,144,152,153,161,162,170,171,179,
%U 180,188,189,197,198,206,207,215,216,224,225,233,234,242,243,251,252,260,261,269
%N Numbers m such that 9 divides m*(m + 1).
%C Equivalently, numbers congruent to 0 or 8 mod 9.
%C Terms of A007494 with indices in A047264. Also, terms of A060464 with indices in A047335.
%H Bruno Berselli, <a href="/A274406/b274406.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F G.f.: x^2*(8 + x)/((1 + x)*(1 - x)^2).
%F a(n) = (18*n + 7*(-1)^n - 11)/4. Therefore: a(2*m) = 9*m-1, a(2*m+1) = 9*m. It follows that a(j)+a(k) and a(j)*a(k) belong to the sequence if j and k are not both even.
%F a(n) = -A090570(-n+2).
%F a(n) = a(n-1) + a(n-2) - a(n-3).
%F a(2*r+1) + a(2*r+s+1) = a(4*r+s+1) and a(2*r) + a(2*r+2*s+1) = a(4*r+2*s). A particular case provided by these identities: a(n) = a(n - 2*floor(n/6)) + a(2*floor(n/6) + 1).
%F E.g.f.: 1 + ((9*x - 2)*cosh(x) + 9*(x - 1)*sinh(x))/2. - _Stefano Spezia_, Apr 24 2021
%t Select[Range[0, 300], Divisible[# (# + 1), 9] &]
%o (PARI) for(n=0, 300, if(n*(n+1)%9==0, print1(n", ")))
%o (Sage) [n for n in range(300) if 9.divides(n*(n+1))]
%o (Magma) [n: n in [0..300] | IsDivisibleBy(n*(n+1),9)];
%Y Cf. A008591 (first bisection), A010689 (first differences), A017257 (second bisection).
%Y Cf. similar sequences in which m*(m+1) is divisible by k: A014601 (k=4), A047208 (k=5), A007494 (k=3 and 6), A047335 (k=7), A047521 (k=8), this sequence (k=9).
%Y Cf. A301451: numbers congruent to {1, 7} mod 9; A193910: numbers congruent to {2, 6} mod 9.
%K nonn,easy
%O 1,2
%A _Bruno Berselli_, Jun 20 2016
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