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%I #58 Sep 08 2019 02:04:05
%S 0,2,7,13,21,32,44,58,75,93,113,136,160,186,215,245,277,312,348,386,
%T 427,469,513,560,608,658,711,765,821,880,940,1002,1067,1133,1201,1272,
%U 1344,1418,1495,1573,1653,1736,1820,1906,1995,2085,2177,2272,2368,2466
%N a(n+3) = a(n) + 6*n + 13, a(0)=0, a(1)=2, a(2)=7.
%C Companion of A240438 extended from right to left:
%C ..., 21, 13, 7, 2, 0, 0, 1, 5, 11, 18, ...
%C ..., -8, -6, -5, -2, 0, 1, 4, 6, 7, 10, ... see A047267 and A047234
%C ..., 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, ... .
%C The last digit of a(n) is of period 30. Like A240438.
%C Is there a definition equivalent to the NAME of A240438?
%H Colin Barker, <a href="/A262523/b262523.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = A000290(n+1) - A004523(n+2).
%F a(n) = A240438(n+1) + A004523(n+1).
%F a(n) = A240438(n) + A047395(n+1).
%F a(n+2) - 2*a(n+1) + a(n) = period 3: repeat (3, 1, 2).
%F a(n+3) = a(n-3) + 4*(2 + 3*n). [Thus, a(n+3m) = a(n-3m) + 4m*(2 + 3n), and a(6m+k) = 4m*(9m + 3k + 2) + a(k): explicit formula for a(n) in terms of a(k), 0 <= k <= 5. - _M. F. Hasler_, Jun 06 2017]
%F O.g.f.: -x*(x+1)*(x+2) / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Sep 25 2015
%F E.g.f.: (x/3)*(3*x+7)*exp(x) - (2/(3*sqrt(3)))*exp(-x/2)*sin((sqrt(3)*x)/2). - _G. C. Greubel_, Sep 28 2015
%F (a(n+3) - a(n)) mod 2 = 1; (a(n+6) - a(n)) mod 2 = 0. - _Altug Alkan_, Sep 28 2015
%F (a(n) mod 2) = (0, 0, 1, 1, 1, 0) repeated. (a(n) mod 3) = (0, 2, 1, 1, 0, 2, 2, 1, 0) repeated. (a(n) mod 4) = (0, 2, 3, 1, 1, 0) repeated. (a(n) mod m) has a period of length 3*m, but for m = 4, 8, 12, ... also of length 3*m/2. - _M. F. Hasler_, Jun 06 2017
%F a(n) = n*(n+1) + floor((n+1)/3). - _Bruno Berselli_, Jun 06 2017
%t LinearRecurrence[{2, -1, 1, -2, 1}, {0, 2, 7, 13, 21}, 101] (* _Ray Chandler_, Sep 24 2015 *)
%t RecurrenceTable[{a[n+3] == a[n] + 6 n + 13, a[0]==0, a[1]==2, a[2]==7}, a, {n, 0, 500}] (* _G. C. Greubel_, Sep 28 2015 *)
%t Table[n (n + 1) + Floor[(n + 1)/3], {n, 0, 50}] (* _Bruno Berselli_, Jun 06 2017 *)
%o (PARI) concat(0, Vec(-x*(x+1)*(x+2)/ ((x-1)^3*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Sep 25 2015
%o (PARI) A262523(n)=(2+[9,3]*n=divrem(n,6))*4*n[1]+[0,2,7,13,21,32][n[2]+1] \\ _M. F. Hasler_, Jun 06 2017
%Y Cf. A000290, A004523, A010882, A016789, A042965, A047234, A047267, A047395, A240438, A262397.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Sep 24 2015
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