A006004 a(n) = C(n+2,3) + C(n,3) + C(n-1,3).

1, 4, 11, 25, 49, 86, 139, 211, 305, 424, 571, 749, 961, 1210, 1499, 1831, 2209, 2636, 3115, 3649, 4241, 4894, 5611, 6395, 7249, 8176, 9179, 10261, 11425, 12674, 14011, 15439, 16961, 18580, 20299, 22121, 24049, 26086, 28235, 30499, 32881, 35384, 38011, 40765

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Equals binomial transform of [1, 3, 4, 3, 0, 0, 0, ...]. Example: a(4) = 25 = (1, 3, 3, 1) dot (1, 3, 4, 3) = (1 + 9 + 12 + 3). - Gary W. Adamson, Jul 25 2008

References

• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
• Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Programs

Maple

A006004:=n->(n^3 - 2*n^2 + 5*n - 2)/2; seq(A006004(n), n=1..50); # Wesley Ivan Hurt, Feb 09 2014

Mathematica

Table[Binomial[n+2,3]+Binomial[n,3]+Binomial[n-1,3],{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,11,25},50] (* Harvey P. Dale, Jun 15 2011 *)

PARI

a(n) = (n^3 - 2*n^2 + 5*n - 2)/2 \\ Charles R Greathouse IV, Feb 10 2017

Formula

a(n) = (n^3 - 2n^2 + 5n - 2)/2.

G.f.: (x^3+x^2+1)/(x-1)^4. - Harvey P. Dale, Jun 15 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=1, a(1)=4, a(2)=11, a(3)=25. - Harvey P. Dale, Jun 15 2011

Extensions

Terms added by Wesley Ivan Hurt, Feb 09 2014