A292185 One-fifth of the rolling arithmetic mean of the fifth powers of the natural numbers taken five at a time.

177, 488, 1159, 2460, 4781, 8656, 14787, 24068, 37609, 56760, 83135, 118636, 165477, 226208, 303739, 401364, 522785, 672136, 854007, 1073468, 1336093, 1647984, 2015795, 2446756, 2948697, 3530072, 4199983, 4968204, 5845205, 6842176, 7971051, 9244532, 10676113

Offset: 1

Author

Robert G. Burns, Sep 12 2017

Keywords

• nonn
• easy

Comments

This method can be generalized. Replacing all the fives by any odd positive integer m, and taking m at a time, also gives an integer sequence.

If m is 3 then A006527 (from term 3) and A167875 (from term 2) are retrieved.

Examples

			a(1) = (1^5 + 2^5 + 3^5 + 4^5 +5^5)/25 = (1+32+243+1024+3125)/25 = 4425/25 = 177.
a(2) = (2^5 + 3^5 + 4^5 + 5^5 +6^5 )/25 = (32+243+1024+3125+7776)/25 = 12200/25 = 488.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A006527, A167875.

Programs

J

(m(+/ % #) \ (1+i. 44)^(x: m))%m [m=.5 NB. See http://www.jsoftware.com

Mathematica

MovingAverage[Range[40]^5,5]/5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{177,488,1159,2460,4781,8656},40] (* Harvey P. Dale, Aug 03 2024 *)

PARI

Vec(x*(177 - 574*x + 886*x^2 - 714*x^3 + 301*x^4 - 52*x^5) / (1 - x)^6 + O(x^30)) \\ Colin Barker, Sep 18 2017

Formula

a(n) = ((n^5 + (n+1)^5 + (n+2)^5 + (n+3)^5 + (n+4)^5) /5) /5.

From Colin Barker, Sep 18 2017: (Start)

G.f.: x*(177 - 574*x + 886*x^2 - 714*x^3 + 301*x^4 - 52*x^5) / (1 - x)^6.

a(n) = 6*a(n-1)- 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

(End)