%I #41 Sep 08 2022 08:45:59

%S 0,13,54,123,220,345,498,679,888,1125,1390,1683,2004,2353,2730,3135,

%T 3568,4029,4518,5035,5580,6153,6754,7383,8040,8725,9438,10179,10948,

%U 11745,12570,13423,14304,15213,16150,17115,18108,19129,20178,21255,22360

%N a(n) = n*(14*n - 1).

%C Related to the primitive Pythagorean triple [3, 4, 5].

%C Sequence found by reading the line from 0, in the direction 0, 13, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral.

%C Also sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the generalized 9-gonal numbers A118277. - _Omar E. Pol_, Jul 28 2012

%H Vincenzo Librandi, <a href="/A195024/b195024.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = 14*n^2 - n.

%F From _Colin Barker_, Apr 09 2012: (Start)

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

%F G.f.: x*(13+15*x)/(1-x)^3. (End)

%t Table[n(14n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,13,54},50] (* _Harvey P. Dale_, Jul 28 2012 *)

%o (Magma) [14*n^2 - n: n in [0..50]]; // _Vincenzo Librandi_, Oct 14 2011

%o (PARI) a(n)=n*(14*n-1) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A144555, A152760, A195019, A195020, A195320, A185019, A193053, A198017.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Oct 13 2011