A116082 a(n) = C(n,7) + 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, 127, 254, 501, 967, 1815, 3301, 5811, 9907, 16383, 26332, 41225, 63003, 94183, 137979, 198439, 280599, 390655, 536154, 726205, 971711, 1285623, 1683217, 2182395, 2804011, 3572223, 4514872, 5663889, 7055731, 8731847

Offset: 0

Author

Jonathan Vos Post, Mar 13 2006

Keywords

• easy
• nonn

Comments

Number of compositions with at most three parts distinct from 1 and with a sum at most n. - Beimar Naranjo, Mar 12 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. See Table 2 on page 370. [_Parthasarathy Nambi_, Jun 18 2009]
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A000217, A000225, A004006, A055795, A057703, A115567, A116082. [Parthasarathy Nambi, Jun 18 2009]

Programs

Magma

[n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: n in [0..40]]; // Vincenzo Librandi, Jun 21 2011

Maple

a:=n->n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: seq(a(n),n=0..35); # Emeric Deutsch, Apr 14 2006
seq(sum(binomial(n,k),k=1..7),n=0..35); # Zerinvary Lajos, Dec 14 2007

Mathematica

Table[Total[Binomial[n,Range[7]]],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,1,3,7,15,31,63,127},41](* Harvey P. Dale, Aug 05 2011 *)

PARI

for(n=0,30, print1(n*(n^6 -14*n^5 +112*n^4 -350*n^3 +1099*n^2 +364*n +3828)/5040, ", ")) \\ G. C. Greubel, Nov 25 2017

Formula

a(n) = A000580(n) + A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.

a(n) = A000580(n) + A115567(n).

a(n) = n*(n^6 - 14*n^5 + 112*n^4 - 350*n^3 + 1099*n^2 + 364*n + 3828)/5040. - Emeric Deutsch, Apr 14 2006

G.f.: x*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)/(1-x)^8. - R. J. Mathar, Jun 20 2011

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31, a(6)=63, a(7)=127. - Harvey P. Dale, Aug 05 2011