|
%I #53 May 04 2022 07:00:19
%S 0,1,14,50,120,235,406,644,960,1365,1870,2486,3224,4095,5110,6280,
%T 7616,9129,10830,12730,14840,17171,19734,22540,25600,28925,32526,
%U 36414,40600,45095,49910,55056,60544,66385,72590,79170,86136,93499,101270
%N Partial sums of A051865.
%C This sequence is related to A180223 by 2*a(n) = n*A180223(n) - Sum_{i=0..n-1} A180223(i). Also, 13-gonal (or tridecagonal) pyramidal numbers. - _Bruno Berselli_, Dec 14 2010
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189-196.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%H Bruno Berselli, <a href="/A050441/b050441.txt">Table of n, a(n) for n = 0..1000</a>
%H Bruno Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian), 2008.
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = n*(n+1)*(11*n-8)/6.
%F G.f.: x*(1+10*x)/(1-x)^4. - _Bruno Berselli_, Aug 19 2010
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Bruno Berselli_, Aug 19 2010
%F a(n) = Sum_{i=0..n-1} (n-i)*(11*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014
%F E.g.f.: exp(x)*x*(6 + 36*x + 11*x^2)/6. - _Stefano Spezia_, May 04 2022
%e After 0, the sequence is provided by the row sums of the triangle (see above, fourth formula):
%e 1;
%e 2, 12;
%e 3, 24, 23;
%e 4, 36, 46, 34;
%e 5, 48, 69, 68, 45; ... - _Vincenzo Librandi_, Feb 12 2014
%p seq(n*(n+1)*(11*n-8)/6, n=0..40); # _G. C. Greubel_, Aug 30 2019
%t Accumulate[Table[n (11n-9)/2,{n,0,40}]] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,1,14,50},40] (* _Harvey P. Dale_, Nov 14 2011 *)
%t CoefficientList[Series[x (1 + 10 x)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%o (Magma) I:=[0,1,14,50]; [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)*(11*n-8)/6 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Sage) [n*(n+1)*(11*n-8)/6 for n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o (GAP) List([0..40], n-> n*(n+1)*(11*n-8)/6); # _G. C. Greubel_, Aug 30 2019
%Y Cf. A051865, A180223.
%Y Similar sequences are listed in A237616.
%K nonn,easy,nice
%O 0,3
%A _Barry E. Williams_, Dec 23 1999
|
