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A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n+7)/2.

%I A062728 #55 Jun 20 2025 16:37:23
%S A062728 0,8,25,51,86,130,183,245,316,396,485,583,690,806,931,1065,1208,1360,
%T A062728 1521,1691,1870,2058,2255,2461,2676,2900,3133,3375,3626,3886,4155,
%U A062728 4433,4720,5016,5321,5635,5958,6290,6631,6981,7340,7708,8085,8471,8866,9270
%N A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n+7)/2.
%C A062728 Old name: Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the line from 0 in the direction 0,8,...
%C A062728 Sequence found by reading the line from 0, in the direction 0, 25, ... and the line from 8, in the direction 8, 51, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - _Omar E. Pol_, Jul 24 2012
%H A062728 Ivan Panchenko, <a href="/A062728/b062728.txt">Table of n, a(n) for n = 0..1000</a>
%H A062728 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A062728 a(n) = n*(9*n+7)/2.
%F A062728 a(n) = 9*n + a(n-1) - 1 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F A062728 From _Bruno Berselli_, Jan 13 2011: (Start)
%F A062728 G.f.: x*(8 + x)/(1 - x)^3.
%F A062728 a(n) = Sum_{i=0..n-1} A017257(i) for n > 0. (End)
%F A062728 a(n) = A218470(9n+7). - _Philippe Deléham_, Mar 27 2013
%F A062728 E.g.f.: x*(16 + 9*x)*exp(x)/2. - _G. C. Greubel_, May 24 2019
%e A062728 The spiral begins:
%e A062728           15
%e A062728           / \
%e A062728         16  14
%e A062728         /     \
%e A062728       17   3  13
%e A062728       /   / \   \
%e A062728     18   4   2  12
%e A062728     /   /     \   \
%e A062728   19   5   0---1  11
%e A062728   /   /             \
%e A062728 20   6---7---8---9--10
%t A062728 Table[n*(9*n+7)/2, {n,0,50}] (* _G. C. Greubel_, May 24 2019 *)
%t A062728 LinearRecurrence[{3,-3,1},{0,8,25},50] (* _Harvey P. Dale_, Sep 06 2019 *)
%o A062728 (PARI) a(n)=n*(9*n+7)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%o A062728 (Magma) [n*(9*n+7)/2: n in [0..50]]; // _G. C. Greubel_, May 24 2019
%o A062728 (Sage) [n*(9*n+7)/2 for n in (0..50)] # _G. C. Greubel_, May 24 2019
%o A062728 (GAP) List([0..50], n-> n*(9*n+7)/2); # _G. C. Greubel_, May 24 2019
%Y A062728 Cf. A051682.
%Y A062728 Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, this sequence, A135705.
%K A062728 nonn,easy
%O A062728 0,2
%A A062728 _Floor van Lamoen_, Jul 21 2001
%E A062728 New name from _Bruno Berselli_ (with the original formula), Jan 13 2011