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A316466 a(n) = 2*n*(7*n - 3). 9

%I

%S 0,8,44,108,200,320,468,644,848,1080,1340,1628,1944,2288,2660,3060,

%T 3488,3944,4428,4940,5480,6048,6644,7268,7920,8600,9308,10044,10808,

%U 11600,12420,13268,14144,15048,15980,16940,17928,18944,19988,21060,22160,23288,24444,25628,26840

%N a(n) = 2*n*(7*n - 3).

%C This is the case k = 9 of Sum_{i = 2..k} P(i,n) = (k - 1)*n*((k - 2)*n - (k - 6))/4, where P(k,n) = n*((k - 2)*n - (k - 4))/2 (see Crossrefs for similar sequences and "Square array in A139600" in Links section).

%C 14*x + 9 is a square for x = a(n) or x = a(-n).

%H Colin Barker, <a href="/A316466/b316466.txt">Table of n, a(n) for n = 0..1000</a>

%H Bruno Berselli, <a href="/A316466/a316466.jpg">Square array in A139600</a>.

%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*x*(2 + 5*x)/(1 - x)^3.

%F E.g.f.: 2*x*(4 + 7*x)*exp(x).

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

%F a(n) = 4*A218471(n).

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

%o (PARI) vector(50, n, n--; 2*n*(7*n-3))

%o (PARI) concat(0, Vec(4*x*(2 + 5*x)/(1 - x)^3 + O(x^40))) \\ _Colin Barker_, Jul 05 2018

%o (Sage) [2*n*(7*n-3) for n in (0..50)]

%o (Maxima) makelist(2*n*(7*n-3), n, 0, 50);

%o (GAP) List([0..50], n -> 2*n*(7*n-3));

%o (MAGMA) [2*n*(7*n-3): n in [0..50]];

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

%o (Julia) [2*n*(7*n-3) for n in 0:50] |> println

%Y Cf. A139600, A218471.

%Y Similar sequences (see the first comment): A000096 (k = 3), A045943 (k = 4), A049451 (k = 5), A033429 (k = 6), A167469 (k = 7), A152744 (k = 8), this sequence (k = 9), A152994 (k = 10).

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Jul 04 2018

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Last modified March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)