A115567 a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

0, 1, 3, 7, 15, 31, 63, 126, 246, 465, 847, 1485, 2509, 4095, 6475, 9948, 14892, 21777, 31179, 43795, 60459, 82159, 110055, 145498, 190050, 245505, 313911, 397593, 499177, 621615, 768211, 942648, 1149016, 1391841, 1676115, 2007327, 2391495

Offset: 0

Author

Jonathan Vos Post, Mar 12 2006

Keywords

• easy
• nonn

Comments

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).

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?

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [_Parthasarathy Nambi_, Sep 30 2009]
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000217, A000332, A000389, A000292, A000579, A004006, A055795, A057703.

Programs

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

Maple

seq(sum(binomial(n,k),k=1..6),n=0..36); # Zerinvary Lajos, Dec 13 2007

Mathematica

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 *)

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

Sage

[binomial(n,2)+binomial(n,4)+binomial(n,6) for n in range(1, 38)] # Zerinvary Lajos, May 17 2009

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

Formula

a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

a(n) = A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.

a(n) = A000579(n) + A057703(n).

G.f.: x*(1-x+x^2)*(1-3*x+3*x^2)/(1-x)^7. - Colin Barker, Mar 16 2012

From G. C. Greubel, Nov 25 2017: (Start)

a(n) = n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720.

E.g.f.: x*(720 + 360*x + 120*x^2 + 30*x^3 + 6*x^4 + x^5)*exp(x)/720. (End)