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%I #54 Feb 16 2022 23:58:09
%S 0,5,29,86,190,355,595,924,1356,1905,2585,3410,4394,5551,6895,8440,
%T 10200,12189,14421,16910,19670,22715,26059,29716,33700,38025,42705,
%U 47754,53186,59015,65255,71920,79024,86581,94605,103110,112110,121619
%N Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.
%H Vincenzo Librandi, <a href="/A050409/b050409.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%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)*(14*n+1)/6.
%F a(n) = A132121(n,4) for n>3. - _Reinhard Zumkeller_, Aug 12 2007
%F From _Bruno Berselli_, Feb 11 2011: (Start)
%F G.f.: x*(5+9*x)/(1-x)^4.
%F a(n) = A129371(2*n). (End)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 22 2012
%F E.g.f.: x*(30 + 57*x + 14*x^2)*exp(x)/6. - _G. C. Greubel_, Oct 30 2019
%p seq(add((n+k)^2,k=0..n),n=0..40); # _Zerinvary Lajos_, Dec 01 2006
%t LinearRecurrence[{4,-6,4,-1},{0,5,29,86},40] (* _Vincenzo Librandi_, Jun 22 2012 *)
%t Table[(n(n+1)(14n+1))/6,{n,0,40}] (* _Harvey P. Dale_, Mar 08 2020 *)
%o (Magma) [&+[k^2: k in [n..2*n]]: n in [0..40]]; // _Bruno Berselli_, Feb 11 2011
%o (Magma) I:=[0, 5, 29, 86]; [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..40]]; // _Vincenzo Librandi_, Jun 22 2012
%o (PARI) a(n)=sum(k=n,n+n,k^2)
%o (PARI) vector(40, n, n*(n-1)*(14*n-13)/6) \\ _G. C. Greubel_, Oct 30 2019
%o (Sage) [n*(n+1)*(14*n+1)/6 for n in (0..40)] # _G. C. Greubel_, Oct 30 2019
%o (GAP) List([0..40], n-> n*(n+1)*(14*n+1)/6); # _G. C. Greubel_, Oct 30 2019
%Y Cf. A000330, A033994, A129371, A132112, A132121, A132124.
%Y Cf. A225144. - _Bruno Berselli_, Jun 06 2013
%Y Cf. A045943: Sum_{k = n..2*n} k.
%Y Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.
%K nonn,easy,nice
%O 0,2
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999
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