A007586 - OEIS

%I M4835 #53 Sep 08 2022 08:44:35

%S 0,1,12,42,100,195,336,532,792,1125,1540,2046,2652,3367,4200,5160,

%T 6256,7497,8892,10450,12180,14091,16192,18492,21000,23725,26676,29862,

%U 33292,36975,40920,45136,49632,54417,59500,64890,70596,76627,82992,89700,96760,104181

%N 11-gonal (or hendecagonal) pyramidal numbers: n*(n+1)*(3*n-2)/2.

%C Starting with 1 equals binomial transform of [1, 11, 19, 9, 0, 0, 0, ...]. - _Gary W. Adamson_, Nov 02 2007

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A007586/b007586.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: x*(1+8*x)/(1-x)^4.

%F a(0)=0, a(1)=1, a(2)=12, a(3)=42; for n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Apr 09 2012

%F a(n) = Sum_{i=0..n-1}  (n-i)*(9*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014

%F From _Amiram Eldar_, Jun 28 2020: (Start)

%F Sum_{n>=1} 1/a(n) = (9*log(3) + sqrt(3)*Pi - 4)/10.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)*Pi + 2 - 4*log(2))/5. (End)

%e From _Vincenzo Librandi_, Feb 12 2014: (Start)

%e After 0, the sequence is provided by the row sums of the triangle (see above, third formula):

%e   1;

%e   2, 10;

%e   3, 20, 19;

%e   4, 30, 38, 28;

%e   5, 40, 57, 56, 37;

%e   6, 50, 76, 84, 74, 46; etc. (End)

%p seq(n*(n+1)*(3*n-2)/2, n=0..45); # _G. C. Greubel_, Aug 30 2019

%t 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 CoefficientList[Series[x(1+8x)/(1-x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)

%o (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 (PARI) a(n)=n*(n+1)*(3*n-2)/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Sage) [n*(n+1)*(3*n-2)/2 for n in (0..45)] # _G. C. Greubel_, Aug 30 2019

%o (GAP) List([0..45], n-> n*(n+1)*(3*n-2)/2); # _G. C. Greubel_, Aug 30 2019

%Y Cf. A051682.

%Y Cf. A093644 ((9, 1) Pascal, column m=3).

%Y Cf. similar sequences listed in A237616.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_, _R. K. Guy_

%E More terms from _Vincenzo Librandi_, Feb 12 2014