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 A115567 #25 Sep 08 2022 08:45:24
%S A115567 0,1,3,7,15,31,63,126,246,465,847,1485,2509,4095,6475,9948,14892,
%T A115567 21777,31179,43795,60459,82159,110055,145498,190050,245505,313911,
%U A115567 397593,499177,621615,768211,942648,1149016,1391841,1676115,2007327,2391495
%N A115567 a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).
%C A115567 a(n) = n + T(n) + Tet(n) + Ptop(n) + 5-Simplex(n) + 6-Simplex(n), where T(n) = n-th triangular number A000217(n), Tet(n) = n-th tetrahedral number A000292(n), Ptop(n) = n-th pentatope number A000332(n), 5-Simplex(n) = n-th 5-simplex number A000389(n), 6-Simplex(n) = n-th 6-simplex number A000579(n).
%C A115567 By analogy to A004006, A055795 and A057703, I presume that a(n) = Answer to the question: if you have a tall building and 6 plates and you need to find the highest story, a plate thrown from which does not break, what is the number of stories you can handle given n tries?
%H A115567 G. C. Greubel, <a href="/A115567/b115567.txt">Table of n, a(n) for n = 0..1000</a>
%H A115567 Michael Boardman, <a href="http://www.jstor.org/stable/3219201">The Egg-Drop Numbers</a>, Mathematics Magazine, 77 (2004), 368-372. [_Parthasarathy Nambi_, Sep 30 2009]
%H A115567 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A115567 a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).
%F A115567 a(n) = A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.
%F A115567 a(n) = A000579(n) + A057703(n).
%F A115567 G.f.: x*(1-x+x^2)*(1-3*x+3*x^2)/(1-x)^7. - _Colin Barker_, Mar 16 2012
%F A115567 From _G. C. Greubel_, Nov 25 2017: (Start)
%F A115567 a(n) = n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720.
%F A115567 E.g.f.: x*(720 + 360*x + 120*x^2 + 30*x^3 + 6*x^4 + x^5)*exp(x)/720. (End)
%p A115567 seq(sum(binomial(n,k),k=1..6),n=0..36); # _Zerinvary Lajos_, Dec 13 2007
%t A115567 Table[n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720, {n,0,30}] (* _G. C. Greubel_, Nov 25 2017 *)
%o A115567 (Sage) [binomial(n,2)+binomial(n,4)+binomial(n,6) for n in range(1, 38)] # _Zerinvary Lajos_, May 17 2009
%o A115567 (Sage) [binomial(n,1)+binomial(n,3)+binomial(n,5)+binomial(n,2)+binomial(n,4)+binomial(n,6) for n in range(0, 37)] # _Zerinvary Lajos_, May 17 2009
%o A115567 (PARI) for(n=0,30, print1(n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720, ", ")) \\ _G. C. Greubel_, Nov 25 2017
%o A115567 (Magma) [n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720: n in [0..30]]; // _G. C. Greubel_, Nov 25 2017
%Y A115567 Cf. A000217, A000332, A000389, A000292, A000579, A004006, A055795, A057703.
%K A115567 easy,nonn
%O A115567 0,3
%A A115567 _Jonathan Vos Post_, Mar 12 2006