This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A172117 #57 Apr 17 2022 11:19:21
%S A172117 0,1,23,86,210,415,721,1148,1716,2445,3355,4466,5798,7371,9205,11320,
%T A172117 13736,16473,19551,22990,26810,31031,35673,40756,46300,52325,58851,
%U A172117 65898,73486,81635,90365,99696,109648,120241,131495,143430,156066
%N A172117 a(n) = n*(n+1)*(20*n-17)/6.
%C A172117 Generated by the formula n*(n+1)*(2*d*n-2*d+3)/6 for d=10.
%C A172117 This sequence is related to A051624 by a(n) = n*A051624(n) - Sum_{i=0..n-1} A051624(i) = n*(n+1)*(20*n-17)/2; in fact, this is the case d=10 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - _Bruno Berselli_, Aug 26 2010
%C A172117 Also, a(n) = n*A190816(n) - Sum_{i=0..n-1} A190816(i) for n>0. - _Bruno Berselli_, Dec 18 2013
%C A172117 Starting with offset 1, the sequence is the binomial transform of (1, 22, 41, 20, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 31 2015
%D A172117 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. [_Bruno Berselli_, Feb 13 2014]
%H A172117 Vincenzo Librandi, <a href="/A172117/b172117.txt">Table of n, a(n) for n = 0..1000</a>
%H A172117 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 A172117 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A172117 G.f.: x*(1+19*x)/(1-x)^4. - _Bruno Berselli_, Aug 26 2010
%F A172117 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - _Harvey P. Dale_, May 15 2011
%F A172117 a(n) = Sum_{i=0..n-1} (n-i)*(20*i+1), with a(0)=0. - _Bruno Berselli_, Feb 11 2014
%F A172117 E.g.f.: (1/6)*x*(6 + 63*x + 20*x^2)*exp(x). - _G. C. Greubel_, Apr 15 2022
%t A172117 Table[(20n^3+3n^2-17n)/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,23,86},40] (* _Harvey P. Dale_, May 15 2011 *)
%o A172117 (PARI) a(n)=n*(20*n^2+3*n-17)/6 \\ _Charles R Greathouse IV_, Jan 11 2012
%o A172117 (Magma) [n*(n+1)*(20*n-17)/6: n in [0..50]]; // _Vincenzo Librandi_, Aug 01 2015
%o A172117 (SageMath) [sum( (-1)^j*(20-j)*binomial(n+2-j, 3-j) for j in (0..1)) for n in (0..50)] # _G. C. Greubel_, Apr 15 2022
%Y A172117 Cf. A051624.
%Y A172117 Cf. similar sequences listed in A237616.
%K A172117 nonn,easy
%O A172117 0,3
%A A172117 _Vincenzo Librandi_, Jan 26 2010