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A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n*(n+1)*(3*n-2)/2.

%I A007586 M4835 #57 Aug 04 2025 08:57:14
%S A007586 0,1,12,42,100,195,336,532,792,1125,1540,2046,2652,3367,4200,5160,
%T A007586 6256,7497,8892,10450,12180,14091,16192,18492,21000,23725,26676,29862,
%U A007586 33292,36975,40920,45136,49632,54417,59500,64890,70596,76627,82992,89700,96760,104181
%N A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n*(n+1)*(3*n-2)/2.
%C A007586 Starting with 1 equals binomial transform of [1, 11, 19, 9, 0, 0, 0, ...]. - _Gary W. Adamson_, Nov 02 2007
%D A007586 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D A007586 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%D A007586 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007586 Vincenzo Librandi, <a href="/A007586/b007586.txt">Table of n, a(n) for n = 0..1000</a>
%H A007586 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A007586 G.f.: x*(1+8*x)/(1-x)^4.
%F A007586 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, a(0)=0, a(1)=1, a(2)=12, a(3)=42. - _Harvey P. Dale_, Apr 09 2012
%F A007586 a(n) = Sum_{i=0..n-1} (n-i)*(9*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014
%F A007586 From _Amiram Eldar_, Jun 28 2020: (Start)
%F A007586 Sum_{n>=1} 1/a(n) = (9*log(3) + sqrt(3)*Pi - 4)/10.
%F A007586 Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)*Pi + 2 - 4*log(2))/5. (End)
%F A007586 E.g.f.: exp(x)*x*(2 + 10*x + 3*x^2)/2. - _Elmo R. Oliveira_, Aug 03 2025
%e A007586 From _Vincenzo Librandi_, Feb 12 2014: (Start)
%e A007586 After 0, the sequence is provided by the row sums of the triangle (see above, third formula):
%e A007586   1;
%e A007586   2, 10;
%e A007586   3, 20, 19;
%e A007586   4, 30, 38, 28;
%e A007586   5, 40, 57, 56, 37;
%e A007586   6, 50, 76, 84, 74, 46; etc. (End)
%p A007586 seq(n*(n+1)*(3*n-2)/2, n=0..45); # _G. C. Greubel_, Aug 30 2019
%t A007586 Table[n(n+1)(3n-2)/2,{n,0,45}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,12,42}, 45] (* _Harvey P. Dale_, Apr 09 2012 *)
%t A007586 CoefficientList[Series[x(1+8x)/(1-x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%o A007586 (Magma) I:=[0,1,12,42]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4) : n in [1..50]]; // _Vincenzo Librandi_, Feb 12 2014
%o A007586 (PARI) a(n)=n*(n+1)*(3*n-2)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A007586 (Sage) [n*(n+1)*(3*n-2)/2 for n in (0..45)] # _G. C. Greubel_, Aug 30 2019
%o A007586 (GAP) List([0..45], n-> n*(n+1)*(3*n-2)/2); # _G. C. Greubel_, Aug 30 2019
%Y A007586 Cf. A051682.
%Y A007586 Cf. A093644 ((9,1) Pascal, column m=3).
%Y A007586 Cf. similar sequences listed in A237616.
%K A007586 nonn,easy,nice
%O A007586 0,3
%A A007586 _N. J. A. Sloane_, _R. K. Guy_
%E A007586 More terms from _Vincenzo Librandi_, Feb 12 2014