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a(n) = 3*n*(3*n + 7)/2 + 4.
5

%I #33 Sep 08 2022 08:46:18

%S 4,19,43,76,118,169,229,298,376,463,559,664,778,901,1033,1174,1324,

%T 1483,1651,1828,2014,2209,2413,2626,2848,3079,3319,3568,3826,4093,

%U 4369,4654,4948,5251,5563,5884,6214,6553,6901,7258,7624,7999,8383,8776,9178,9589,10009

%N a(n) = 3*n*(3*n + 7)/2 + 4.

%C Sum_{k = 0..n} (3*k + r)^3 is divisible by 3*n*(3*n + 2*r + 3)/2 + r^2: the sequence corresponds to the case r = 2 of this formula (other cases are listed in Crossrefs section).

%C Also, Sum_{k = 0..n} (3*k + 2)^3 / a(n) gives 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, ... (A005449).

%C a(n) is even if n belongs to A014601. No term is divisible by 3, 5, 7 and 11.

%H Bruno Berselli, <a href="/A283394/b283394.txt">Table of n, a(n) for n = 0..1000</a>

%H Nickolas Arustamyan, Christopher Cox, Erik Lundberg, Sean Perry, and Zvi Rosen, <a href="https://arxiv.org/abs/2106.11416">On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force</a>, arXiv:2106.11416 [math-ph], 2021.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F O.g.f.: (4 + 7*x - 2*x^2)/(1 - x)^3.

%F E.g.f.: (8 + 30*x + 9*x^2)*exp(x)/2.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F a(n) = A081271(-n-2).

%F a(n) = 3*A095794(n+1) + 1.

%F a(n) = A034856(3*n+2) = A101881(6*n+2) = A165157(6*n+3) = A186349(6*n+3).

%F The inverse binomial transform yields 4, 15, 9, 0 (0 continued), therefore:

%F a(n) = 4*binomial(n,0) + 15*binomial(n,1) + 9*binomial(n,2).

%t Table[3 n (3 n + 7)/2 + 4, {n, 0, 50}]

%t LinearRecurrence[{3,-3,1},{4,19,43},50] (* _Harvey P. Dale_, Mar 02 2019 *)

%o (Python) [3*n*(3*n+7)/2+4 for n in range(50)]

%o (Sage) [3*n*(3*n+7)/2+4 for n in range(50)]

%o (Maxima) makelist(3*n*(3*n+7)/2+4, n, 0, 50);

%o (Magma) [3*n*(3*n+7)/2+4: n in [0..50]];

%o (PARI) a(n) = 3*n*(3*n + 7)/2 + 4; \\ _Indranil Ghosh_, Mar 24 2017

%Y Cf. A005449, A034856, A095794, A101881, A165157, A186349.

%Y Sequences with formula 3*n*(3*n + 2*r + 3)/2 + r^2: A038764 (r=-1), A027468 (r=0), A081271 (r=1), this sequence (r=2), A027468 (r=3; offset: -1), A080855 (r=4; offset: -2).

%K nonn,easy

%O 0,1

%A _Bruno Berselli_, Mar 23 2017