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A063490 a(n) = (2*n - 1)*(7*n^2 - 7*n + 6)/6. 20

%I

%S 1,10,40,105,219,396,650,995,1445,2014,2716,3565,4575,5760,7134,8711,

%T 10505,12530,14800,17329,20131,23220,26610,30315,34349,38726,43460,

%U 48565,54055,59944,66246,72975,80145,87770,95864,104441,113515

%N a(n) = (2*n - 1)*(7*n^2 - 7*n + 6)/6.

%C From _Omar E. Pol_, Oct 23 2019: (Start)

%C a(n) is also the sum of terms that are in the n-th finite row and in the n-th finite column of the square [1,n]x[1,n] of the natural number array A000027; e.g., the [1,3]x[1,3] square is

%C 1..3..6

%C 2..5..9

%C 4..8..13,

%C so that a(1) = 1, a(2) = 2 + 3 + 5 = 10, a(3) = 4 + 6 + 8 + 9 + 13 = 40.

%C Hence the partial sums give A185505. (End)

%H Harry J. Smith, <a href="/A063490/b063490.txt">Table of n, a(n) for n = 1..1000</a>

%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (10).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: x*(1+x)*(1+5*x+x^2)/(1-x)^4. - _Colin Barker_, Mar 02 2012

%F a(n) = Sum_{k = n^2-2*n+2..n^2} A064788(k). - _Lior Manor_, Jan 13 2013

%F From _G. C. Greubel_, Dec 01 2017: (Start)

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

%F E.g.f.: (-6 + 12*x + 21*x^2 + 14*x^3)*exp(x)/6 + 1. (End)

%t Table[(2*n-1)*(7*n^2-7*n+6)/6, {n,1,50}] (* or *) LinearRecurrenc[{4,-6,4,-1}, {1,10,40,105}, 50] (* _G. C. Greubel_, Dec 01 2017 *)

%o (PARI) { for (n=1, 1000, write("b063490.txt", n, " ", (2*n - 1)*(7*n^2 - 7*n + 6)/6) ) } \\ _Harry J. Smith_, Aug 23 2009

%o (PARI) x='x+O('x^30); Vec(serlaplace((-6 + 12*x + 21*x^2 + 14*x^3 )*exp(x)/6 + 1)) \\ _G. C. Greubel_, Dec 01 2017

%o (MAGMA) [(2*n-1)*(7*n^2-7*n+6)/6: n in [1..30]]; // _G. C. Greubel_, Dec 01 2017

%Y 1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

%Y Cf. A064788, A185505.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Aug 01 2001

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Last modified March 3 14:58 EST 2021. Contains 341762 sequences. (Running on oeis4.)