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A195145 Concentric 14-gonal numbers.

%I A195145 #52 Nov 30 2024 09:49:07
%S A195145 0,1,14,29,56,85,126,169,224,281,350,421,504,589,686,785,896,1009,
%T A195145 1134,1261,1400,1541,1694,1849,2016,2185,2366,2549,2744,2941,3150,
%U A195145 3361,3584,3809,4046,4285,4536,4789,5054,5321,5600,5881,6174,6469,6776,7085,7406
%N A195145 Concentric 14-gonal numbers.
%C A195145 Also concentric tetradecagonal numbers or concentric tetrakaidecagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 14, ..., and the same line from 1, in the direction 1, 29, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Main axis, perpendicular to A024966 in the same spiral.
%C A195145 Partial sums of A113801. - _Reinhard Zumkeller_, Jan 07 2012
%H A195145 Vincenzo Librandi, <a href="/A195145/b195145.txt">Table of n, a(n) for n = 0..10000</a>
%H A195145 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A195145 G.f.: -x*(1+12*x+x^2) / ( (1+x)*(x-1)^3 ). - _R. J. Mathar_, Sep 18 2011
%F A195145 From _Vincenzo Librandi_, Sep 27 2011: (Start)
%F A195145 a(n) = (14*n^2 + 5*(-1)^n - 5)/4;
%F A195145 a(n) = a(-n) = -a(n-1) + 7*n^2 - 7*n + 1. (End)
%F A195145 Sum_{n>=1} 1/a(n) = Pi^2/84 + tan(sqrt(5/7)*Pi/2)*Pi/(2*sqrt(35)). - _Amiram Eldar_, Jan 16 2023
%F A195145 E.g.f.: (7*x*(x + 1)*cosh(x) + (7*x^2 + 7*x - 5)*sinh(x))/2. - _Stefano Spezia_, Nov 30 2024
%t A195145 LinearRecurrence[{2, 0, -2, 1}, {0, 1, 14, 29}, 50] (* _Amiram Eldar_, Jan 16 2023 *)
%o A195145 (Magma) [(14*n^2+5*(-1)^n-5)/4: n in [0..50]]; // _Vincenzo Librandi_, Sep 27 2011
%o A195145 (Haskell)
%o A195145 a195145 n = a195145_list !! n
%o A195145 a195145_list = scanl (+) 0 a113801_list
%o A195145 -- _Reinhard Zumkeller_, Jan 07 2012
%Y A195145 A144555 and A195314 interleaved.
%Y A195145 Cf. A024966, A032527, A032528, A077221, A113801, A195045, A195046, A195142, A195143, A195146, A195147, A195148, A195149.
%Y A195145 Column 14 of A195040. - _Omar E. Pol_, Sep 28 2011
%K A195145 nonn,easy
%O A195145 0,3
%A A195145 _Omar E. Pol_, Sep 17 2011