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 A135713 #44 Apr 26 2025 18:17:12
%S A135713 0,5,27,78,170,315,525,812,1188,1665,2255,2970,3822,4823,5985,7320,
%T A135713 8840,10557,12483,14630,17010,19635,22517,25668,29100,32825,36855,
%U A135713 41202,45878,50895,56265,62000,68112,74613,81515,88830,96570,104747,113373,122460,132020
%N A135713 a(n) = n*(n+1)*(4*n+1)/2.
%C A135713 This sequence is related to A045944 by a(n) = n*A045944(n)-Sum_{i=0..n-1} A045944(i); this is the case d=6 in the identity n^2*(d*n+d-2)/2 - sum(k*(d*k+d-2)/2, k=0..n-1) = n*(n+1)*(2*d*n+d-3)/6 . - _Bruno Berselli_, Nov 19 2010
%C A135713 Bisection (even part) of A002717. See the Conway and Guy reference. - _Wolfdieter Lang_, Apr 16 2020
%D A135713 J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.
%H A135713 Vincenzo Librandi, <a href="/A135713/b135713.txt">Table of n, a(n) for n = 0..1000</a>
%H A135713 B. 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).
%H A135713 M. E. Larsen, <a href="http://www.jstor.org/stable/2686923">The eternal triangle - a history of a counting problem</a>, College Math. J., 20 (1989), 370-392.
%H A135713 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A135713 O.g.f.: x*(7*x+5)/(x-1)^4. - _R. J. Mathar_, Apr 22 2008.
%F A135713 a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) for n>3. - _Bruno Berselli_, Nov 19 2010
%F A135713 a(-n-1) = -A051895(n). - _Bruno Berselli_, Aug 23 2011
%F A135713 E.g.f.: (1/2)*x*(10 + 17*x + 4*x^2)*exp(x). - _G. C. Greubel_, Oct 29 2016
%F A135713 Sum_{n>=1} 1/a(n) = 2*(5 - 2*Pi/3 - 4*log(2)) = 0.26603235073404654... - _Ilya Gutkovskiy_, Oct 29 2016
%t A135713 LinearRecurrence[{4,-6,4,-1}, {0, 5, 27, 78}, 50] (* _Vincenzo Librandi_, Mar 01 2012 *)
%t A135713 Table[n*(n+1)*(4*n+1)/2,{n,0,25}] (* _G. C. Greubel_, Oct 29 2016 *)
%t A135713 Table[PolygonalNumber[n](4n+1),{n,0,40}] (* _Harvey P. Dale_, Apr 26 2025 *)
%o A135713 (Magma) [n*(n+1)*(4*n+1)/2: n in [0..40]]; // _Bruno Berselli_, Aug 23 2011
%Y A135713 Bisection of A002717 (even part).
%Y A135713 Cf. A045944, A160378; A132124, A006331.
%K A135713 nonn,easy
%O A135713 0,2
%A A135713 _N. J. A. Sloane_, Mar 05 2008