A075667 Sum of next n 6th powers.

1, 793, 66377, 1911234, 28504515, 271739011, 1874885963, 10136389172, 45311985069, 173957200405, 589679082421, 1802148522758, 5045944649967, 13108508706879, 31915866810295, 73427944186856, 160710828298553, 336507487921137, 677266380588289, 1315464522556810

Offset: 1

Author

Zak Seidov, Sep 24 2002

Keywords

• nonn
• easy

Examples

			a(1) = 1^6 = 1; a(2) = 2^6 + 3^6 = 793; a(3) = 4^6 + 5^6 + 6^6 = 66377; a(4) = 7^6 + 8^6 + 9^6 + 10^6 = 1911234.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Crossrefs

Cf. A001014 (6th powers).

Cf. A006003, A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).

Programs

Mathematica

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=6; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^6,Range[nn]]] (* or *) LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,793,66377,1911234,28504515,271739011,1874885963,10136389172,45311985069,173957200405,589679082421,1802148522758,5045944649967,13108508706879},20] (* Harvey P. Dale, Mar 29 2022 *)

Formula

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^6.

a(n) = (21n^13 + 231n^11 + 693n^9 + 549n^7 - 126n^5 - 56n^3 + 32n)/1344. - Charles R Greathouse IV, Sep 17 2009

G.f.: x*(x^12 +779*x^11 +55366*x^10 +1053755*x^9 +7499895*x^8 [phone]*x^7 [phone]*x^6 [phone]*x^5 +7499895*x^4 +1053755*x^3 +55366*x^2 +779*x +1)/(x-1)^14. - Colin Barker, Jul 22 2012