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A062725 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, ...

%I A062725 #49 Dec 12 2024 16:33:18
%S A062725 0,7,23,48,82,125,177,238,308,387,475,572,678,793,917,1050,1192,1343,
%T A062725 1503,1672,1850,2037,2233,2438,2652,2875,3107,3348,3598,3857,4125,
%U A062725 4402,4688,4983,5287,5600,5922,6253,6593,6942,7300,7667,8043,8428,8822,9225,9637,10058
%N A062725 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 A062725 Central terms of triangle A245300. - _Reinhard Zumkeller_, Jul 17 2014
%C A062725 Digital root of a(n) = A180597(n). - _Gionata Neri_, Apr 29 2015
%H A062725 Reinhard Zumkeller, <a href="/A062725/b062725.txt">Table of n, a(n) for n = 0..10000</a>
%H A062725 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 A062725 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A062725 a(n) = n*(9*n+5)/2.
%F A062725 a(n) = 9*n + a(n-1) - 2 with a(0)=0. - _Vincenzo Librandi_, Aug 07 2010
%F A062725 From _Colin Barker_, Jul 07 2012: (Start)
%F A062725 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A062725 G.f.: x*(7+2*x)/(1-x)^3. (End)
%F A062725 a(n) = A218470(9*n+6). - _Philippe Deléham_, Mar 27 2013
%F A062725 a(n) = a(n-1) + A017245(n-1), a(0)=0. - _Gionata Neri_, Apr 30 2015
%F A062725 E.g.f.: exp(x)*x*(14 + 9*x)/2. - _Elmo R. Oliveira_, Dec 12 2024
%e A062725 The spiral begins:
%e A062725 .
%e A062725             15
%e A062725             / \
%e A062725           16  14
%e A062725           /     \
%e A062725         17   3  13
%e A062725         /   / \   \
%e A062725       18   4   2  12
%e A062725       /   /     \   \
%e A062725     19   5   0---1  11
%e A062725     /   /             \
%e A062725   20   6---7---8---9--10
%e A062725 .
%t A062725 s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 9}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008 *)
%t A062725 CoefficientList[Series[x (7 + 2 x)/(1 - x)^3, {x, 0, 45}], x] (* _Michael De Vlieger_, Jan 11 2020 *)
%o A062725 (Haskell)
%o A062725 a062725 n = n * (9 * n + 5) `div` 2 -- _Reinhard Zumkeller_, Jul 17 2014
%o A062725 (PARI) a(n) = n*(9*n+5)/2 \\ _Charles R Greathouse IV_, Apr 30 2015
%Y A062725 Cf. A017245, A051682, A180597, A218470, A245300.
%K A062725 nonn,easy
%O A062725 0,2
%A A062725 _Floor van Lamoen_, Jul 21 2001
%E A062725 Formula that confused indices corrected by _R. J. Mathar_, Jun 04 2010