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A062783 a(n) = 3*n*(4*n-1).

%I A062783 #25 Oct 31 2024 20:40:16
%S A062783 0,9,42,99,180,285,414,567,744,945,1170,1419,1692,1989,2310,2655,3024,
%T A062783 3417,3834,4275,4740,5229,5742,6279,6840,7425,8034,8667,9324,10005,
%U A062783 10710,11439,12192,12969,13770,14595,15444,16317,17214,18135,19080
%N A062783 a(n) = 3*n*(4*n-1).
%C A062783 Write 1, 2, 3, 4, ... counterclockwise in a hexagonal spiral around 0 starting left down, then a(n) is the sequence found by reading from 0 in the vertical downward direction.
%C A062783 Polygonal number connection: 2He_n + 7S_n where He_n is the n-th Heptagonal number and S_n is the n-th Square number. - _William A. Tedeschi_, Sep 12 2010
%D A062783 L. Berzolari, Allgemeine Theorie der Höeren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906. p. 341.
%H A062783 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A062783 a(n) = 24*n + a(n-1) - 15 with n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 07 2010
%F A062783 G.f.: 3*x*(3 + 5*x)/(1-x)^3. - _Colin Barker_, Feb 28 2012
%F A062783 From _Elmo R. Oliveira_, Oct 31 2024: (Start)
%F A062783 E.g.f.: 3*x*(3 + 4*x)*exp(x).
%F A062783 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%e A062783 The spiral begins:
%e A062783 ......16..15..14
%e A062783 ....17..5...4...13
%e A062783 ..18..6...0...3...12
%e A062783 19..7...1...2...11..26
%e A062783 ..20..8...9...10..25
%e A062783 ....21..22..23..24
%t A062783 s=0;lst={s};Do[s+=n++ +9;AppendTo[lst, s], {n, 0, 8!, 24}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 17 2008 *)
%o A062783 (PARI) a(n)=3*n*(4*n-1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A062783 Equals 3*A033991.
%Y A062783 Cf. A000567, A063436.
%K A062783 easy,nonn
%O A062783 0,2
%A A062783 _Floor van Lamoen_, Jul 21 2001