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 A005565 M5087 #77 Jun 24 2025 00:50:42
%S A005565 20,75,189,392,720,1215,1925,2904,4212,5915,8085,10800,14144,18207,
%T A005565 23085,28880,35700,43659,52877,63480,75600,89375,104949,122472,142100,
%U A005565 163995,188325,215264,244992,277695,313565,352800,395604,442187,492765,547560,606800
%N A005565 Number of walks on square lattice.
%D A005565 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005565 Vincenzo Librandi, <a href="/A005565/b005565.txt">Table of n, a(n) for n = 0..1000</a>
%H A005565 R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>
%H A005565 R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%H A005565 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005565 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005565 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A005565 From _Ralf Stephan_, Apr 23 2004: (Start)
%F A005565 a(n) = (1/4)*(n^4+14n^3+69n^2+136n+80).
%F A005565 G.f.: (20-25x+14x^2-3x^3)/(1-x)^5. (End)
%F A005565 a(n) = binomial(n+4,2)^2 - binomial(n+4,1)^2. - _Gary Detlefs_, Nov 22 2011
%F A005565 Using two consecutive triangular numbers t(n) and t(n+1), starting at n=3, compute the determinant of a 2 X 2 matrix with the first row t(n), t(n+1) and the second row t(n+1), 2*t(n+1). This gives (n+1)^2*(n-2)*(n+2)/4 = a(n-3). - _J. M. Bergot_, May 17 2012
%F A005565 E.g.f.: exp(x)*(80 + 220*x + 118*x^2 + 20*x^3 + x^4)/4. - _Stefano Spezia_, Jun 20 2025
%p A005565 seq(add (k^3-n^2, k =0..n), n=4..28 ); # _Zerinvary Lajos_, Aug 26 2007
%p A005565 A005565:=(-20+25*z-14*z**2+3*z**3)/(z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A005565 CoefficientList[Series[(20-25x+14x^2-3x^3)/(1-x)^5,{x,0,40}],x] (* _Vincenzo Librandi_, May 24 2012 *)
%t A005565 LinearRecurrence[{5,-10,10,-5,1},{20,75,189,392,720},40] (* _Harvey P. Dale_, Dec 04 2020 *)
%t A005565 Differences[Table[Sum[x^3 - y^2, {x, 0, g}, {y, x, g}], {g, 3, 30}]] (* _Horst H. Manninger_, Jun 19 2025 *)
%o A005565 (PARI) a(n)=(n^4+14*n^3+69*n^2+136*n)/4+20 \\ _Charles R Greathouse IV_, Nov 22 2011
%o A005565 (Magma) [1/4*(n^4+14*n^3+69*n^2+136*n+80): n in [0..40]]; // _Vincenzo Librandi_, May 24 2012
%K A005565 nonn,walk,easy
%O A005565 0,1
%A A005565 _N. J. A. Sloane_