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A152750 Eight times hexagonal numbers: a(n) = 8*n*(2*n-1).

%I A152750 #41 May 05 2025 01:36:15
%S A152750 0,8,48,120,224,360,528,728,960,1224,1520,1848,2208,2600,3024,3480,
%T A152750 3968,4488,5040,5624,6240,6888,7568,8280,9024,9800,10608,11448,12320,
%U A152750 13224,14160,15128,16128,17160,18224,19320,20448,21608,22800,24024,25280,26568,27888,29240
%N A152750 Eight times hexagonal numbers: a(n) = 8*n*(2*n-1).
%C A152750 Equals Engel expansion of cosh(1/2), except first member (see A067239).
%C A152750 Also sequence found by reading the line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Sep 18 2011
%C A152750 a(n) = the sum of the edges of a rectangular prism having edges 2*(n-1)*n, n^2 - (n-1)^2, and n^2 + (n-1)^2. - _J. M. Bergot_, Apr 24 2014
%H A152750 Ivan Panchenko, <a href="/A152750/b152750.txt">Table of n, a(n) for n = 0..1000</a>
%H A152750 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A152750 a(n) = 16*n^2 - 8*n = 8*A000384(n) = 4*A002939(n) = 2*A085250(n).
%F A152750 a(n) = A067239(n), for n > 0.
%F A152750 a(n) = a(n-1) + 32*n - 24 (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010
%F A152750 From _Colin Barker_, Sep 25 2016: (Start)
%F A152750 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
%F A152750 G.f.: 8*x*(1+3*x)/(1-x)^3. (End)
%F A152750 Sum_{n>=1} 1/a(n) = log(2)/4. - _Vaclav Kotesovec_, Sep 25 2016
%F A152750 E.g.f.: 8*exp(x)*x*(1 + 2*x). - _Elmo R. Oliveira_, Dec 15 2024
%F A152750 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - log(2)/8. - _Amiram Eldar_, May 05 2025
%p A152750 A152750:=n->8*n*(2*n-1); seq(A152750(n), n=0..50); # _Wesley Ivan Hurt_, Jun 09 2014
%t A152750 Table[8*n*(2*n - 1), {n, 0, 50}] (* _Wesley Ivan Hurt_, Jun 09 2014 *)
%o A152750 (Magma) [ 8*n*(2*n-1) : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%o A152750 (PARI) concat(0, Vec(8*x*(1+3*x)/(1-x)^3 + O(x^50))) \\ _Colin Barker_, Sep 25 2016
%Y A152750 Cf. A000384, A002939, A067239, A074377, A085250.
%K A152750 easy,nonn
%O A152750 0,2
%A A152750 _Omar E. Pol_, Dec 12 2008