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 A141001 #16 Mar 26 2023 12:06:27
%S A141001 1,1,1,2,3,4,7,11,16,21,26,32,38,44,51,59,67,75,84,94,104,114,125,137,
%T A141001 149,161,174,188,202,216,231,247,263,279,296,314,332,350,369,389,409,
%U A141001 429,450,472,494,516,539,563,587,611,636,662,688,714,741,769,797
%N A141001 a(n) = number of different landings of a grasshopper after n hops.
%C A141001 Consider a grasshopper (cf. A141000) that starts at x=0 at time 0, then makes successive hops of sizes 1, 2, 3, ..., n, subject to the constraint that it must always land on a point x >= 0; sequence gives number of different places x where it can land after the n-th jump.
%C A141001 Here, unlike A141000, there is no restriction on how large x can be (of course x <= n(n+1)/2).
%H A141001 Colin Barker, <a href="/A141001/b141001.txt">Table of n, a(n) for n = 0..1000</a>
%H A141001 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F A141001 a(n) = floor(n * (n+1) / 4 - 1) for n >= 9. The induction proof for A141000 shows that for n >= 21, you hit all numbers of the right parity except n(n+1)/2-2 and n(n+1)/2-4. The floor expression handles the various parity cases. - _David Applegate_.
%F A141001 G.f.: (1 - 2*x + 2*x^2 - x^3 + x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 - x^12 + x^13) / ((1 - x)^3*(1 + x^2)). - _Colin Barker_, May 21 2013
%F A141001 From _Colin Barker_, Aug 06 2017: (Start)
%F A141001 a(n) = (1/8+i/8) * ((-5+5*i) + (-i)^(1+n) + i^n + (1-i)*n + (1-i)*n^2) for n>8 where i=sqrt(-1).
%F A141001 a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>9.
%F A141001 (End)
%e A141001 For example, for n=3 the grasshopper can hit 0=1+2-3 or 6=1+2+3; for n=4 it can hit 2=1+2+3-4, 4=1+2-3+4, or 10=1+2+3+4.
%t A141001 LinearRecurrence[{3,-4,4,-3,1},{1,1,1,2,3,4,7,11,16,21,26,32,38,44},60] (* _Harvey P. Dale_, Mar 26 2023 *)
%o A141001 (PARI) Vec((1 - 2*x + 2*x^2 - x^3 + x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 - x^12 + x^13) / ((1 - x)^3*(1 + x^2)) + O(x^60)) \\ _Colin Barker_, Aug 06 2017
%Y A141001 Cf. A141000, A141002.
%K A141001 nonn,easy
%O A141001 0,4
%A A141001 _David Applegate_ and _N. J. A. Sloane_, Jul 21 2008