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A195320 7 times hexagonal numbers: a(n) = 7*n*(2*n-1).

%I A195320 #42 Jan 29 2025 19:36:41
%S A195320 0,7,42,105,196,315,462,637,840,1071,1330,1617,1932,2275,2646,3045,
%T A195320 3472,3927,4410,4921,5460,6027,6622,7245,7896,8575,9282,10017,10780,
%U A195320 11571,12390,13237,14112,15015,15946,16905,17892,18907,19950,21021,22120,23247,24402,25585
%N A195320 7 times hexagonal numbers: a(n) = 7*n*(2*n-1).
%C A195320 Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277.
%C A195320 Also sequence found by reading the same line (mentioned above) 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, which is related to the primitive Pythagorean triple [3, 4, 5]. - _Omar E. Pol_, Oct 13 2011
%H A195320 Vincenzo Librandi, <a href="/A195320/b195320.txt">Table of n, a(n) for n = 0..10000</a>
%H A195320 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A195320 a(n) = 14*n^2 - 7*n = 7*A000384(n).
%F A195320 G.f.: -7*x*(1+3*x)/(x-1)^3. - _R. J. Mathar_, Sep 27 2011
%F A195320 From _Elmo R. Oliveira_, Dec 27 2024: (Start)
%F A195320 E.g.f.: 7*exp(x)*x*(2*x + 1).
%F A195320 a(n) = A316466(n) - n = A024966(2*n+1).
%F A195320 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A195320 Table[7*n*(2*n - 1), {n, 0, 50}] (* _Paolo Xausa_, Jan 29 2025 *)
%t A195320 7*PolygonalNumber[6, Range[0, 50]] (* _Paolo Xausa_, Jan 29 2025 *)
%o A195320 (Magma) [7*n*(2*n-1): n in [0..50]]; // _Vincenzo Librandi_, Sep 28 2011
%o A195320 (PARI) a(n)=7*n*(2*n-1) \\ _Charles R Greathouse IV_, Sep 28 2015
%Y A195320 Bisection of A024966.
%Y A195320 Cf. A000384, A118277, A152746, A152750, A185019, A193053, A195019, A195020, A198017, A316466.
%K A195320 nonn,easy
%O A195320 0,2
%A A195320 _Omar E. Pol_, Sep 18 2011