A068236 First differences of (n+1)^5-n^5.

30, 180, 570, 1320, 2550, 4380, 6930, 10320, 14670, 20100, 26730, 34680, 44070, 55020, 67650, 82080, 98430, 116820, 137370, 160200, 185430, 213180, 243570, 276720, 312750, 351780, 393930, 439320, 488070, 540300, 596130, 655680, 719070, 786420, 857850, 933480

Offset: 0

Author

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Mar 25 2002

Keywords

• nonn
• easy

Comments

For n>=0, a(n) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n+2} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A022521 ((n+1)^5-n^5), A000584 (5th powers), A005900 (octahedral numbers).

Programs

Mathematica

Table[20*n^3 + 10*n, {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Differences[#[[2]]-#[[1]]&/@Partition[Range[0,40]^5,2,1]] (* or *) LinearRecurrence[{4,-6,4,-1},{30,180,570,1320},40] (* Harvey P. Dale, Jun 05 2019 *)

PARI

Vec(30*(x+1)^2 / (x-1)^4 + O(x^100)) \\ Colin Barker, Dec 13 2014

Formula

a(n) = (n+2)^5-2*(n+1)^5+n^5.

a(n) = 30*A005900(n+1). - R. J. Mathar, Sep 02 2008

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Colin Barker, Dec 13 2014

G.f.: 30*(x+1)^2 / (x-1)^4. - Colin Barker, Dec 13 2014