A195021 - OEIS

%I #62 Jan 15 2025 10:23:32

%S 0,3,34,93,180,295,438,609,808,1035,1290,1573,1884,2223,2590,2985,

%T 3408,3859,4338,4845,5380,5943,6534,7153,7800,8475,9178,9909,10668,

%U 11455,12270,13113,13984,14883,15810,16765,17748,18759,19798,20865,21960,23083,24234,25413

%N a(n) = n*(14*n - 11).

%C Sequence found by reading the first two vertices [0, 3] together with the line from 34, in the direction 34, 93, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020, which is related to the primitive Pythagorean triple [3, 4, 5]. For another version see A195030.

%H Vincenzo Librandi, <a href="/A195021/b195021.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 - 11*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*(3+25*x)/(1-x)^3. (End)

%F E.g.f.: exp(x)*x*(3 + 14*x). - _Elmo R. Oliveira_, Dec 30 2024

%t Table[n*(14*n - 11), {n, 0, 50}] (* _Paolo Xausa_, Jan 15 2025 *)

%o (Magma) [14*n^2 - 11*n: n in [0..50]]; // _Vincenzo Librandi_, Oct 14 2011

%o (PARI) a(n)=n*(14*n-11) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A144555, A185019, A193053, A195019, A195020, A195023, A195024, A195025, A195030, A195320, A198017.

%Y Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=28: see Comments lines of A226492.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Sep 07 2011

%E Edited by _Bruno Berselli_, Oct 18 2011