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%I #45 Jan 11 2020 15:28:31
%S 0,7,23,48,82,125,177,238,308,387,475,572,678,793,917,1050,1192,1343,
%T 1503,1672,1850,2037,2233,2438,2652,2875,3107,3348,3598,3857,4125,
%U 4402,4688,4983,5287,5600,5922,6253,6593,6942,7300,7667,8043,8428,8822,9225
%N Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the terms along the line from 0 in the direction 0,7,...
%C Central terms of triangle A245300. - _Reinhard Zumkeller_, Jul 17 2014
%C Digital root of a(n) = A180597(n). - _Gionata Neri_, Apr 29 2015
%H Reinhard Zumkeller, <a href="/A062725/b062725.txt">Table of n, a(n) for n = 0..10000</a>
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = n*(9*n+5)/2.
%F a(n) = 9*n + a(n-1) - 2 with a(0)=0. - _Vincenzo Librandi_, Aug 07 2010
%F From _Colin Barker_, Jul 07 2012: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F G.f.: x*(7+2*x)/(1-x)^3. (End)
%F a(n) = A218470(9*n+6). - _Philippe Deléham_, Mar 27 2013
%F a(n) = a(n-1) + A017245(n-1), a(0)=0. - _Gionata Neri_, Apr 30 2015
%e The spiral begins:
%e .
%e 15
%e / \
%e 16 14
%e / \
%e 17 3 13
%e / / \ \
%e 18 4 2 12
%e / / \ \
%e 19 5 0---1 11
%e / / \
%e 20 6---7---8---9--10
%e .
%t s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 9}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008 *)
%t CoefficientList[Series[x (7 + 2 x)/(1 - x)^3, {x, 0, 45}], x] (* _Michael De Vlieger_, Jan 11 2020 *)
%o (Haskell)
%o a062725 n = n * (9 * n + 5) `div` 2 -- _Reinhard Zumkeller_, Jul 17 2014
%o (PARI) a(n) = n*(9*n+5)/2 \\ _Charles R Greathouse IV_, Apr 30 2015
%Y Cf. A051682.
%K nonn,easy
%O 0,2
%A _Floor van Lamoen_, Jul 21 2001
%E Formula that confused indices corrected by _R. J. Mathar_, Jun 04 2010
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