A048397 Sum of consecutive non-fourth-powers.

0, 119, 3104, 29319, 162104, 643535, 2040744, 5502959, 13129424, 28468359, 57167120, 107793719, 192849864, 329995679, 543506264, 865980255, 1340320544, 2022007319, 2981683584, 4308073319, 6111252440, 8526292719, 11717298824

Offset: 0

Author

Patrick De Geest, Mar 15 1999

Keywords

• nonn
• nice
• easy

Comments

Relationship with tetrahedral numbers: a(4) = (first term + last term)*(6*Tetra_n + n^3) = (82+255)*(6*10+27) = (337)*(87) = 29319.

Examples

			Between 3^4 and 4^4 we have 82+83+...+254+255 which is 29319 or a(4).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Crossrefs

Cf. A048395, A048396, A000292.

Programs

Maple

A048397:=n->4*n^7 + 14*n^6 + 28*n^5 + 34*n^4 + 26*n^3 + 11*n^2 + 2*n; seq(A048397(n), n=0..40); # Wesley Ivan Hurt, Feb 10 2014

Mathematica

Table[Total[Range[n^4+1,(n+1)^4-1]],{n,0,40}] (* or *) Table[4n^7+ 14n^6+28n^5+34n^4+26n^3+11n^2+2n,{n,0,40}] (* Harvey P. Dale, Apr 23 2011 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,119,3104,29319,162104,643535,2040744,5502959},40] (* Harvey P. Dale, Jul 31 2021 *)

PARI

a(n)=n*(2*n^2 + 3*n + 2)*(2*n^4 + 4*n^3 + 6*n^2 + 4*n + 1) \\ Charles R Greathouse IV, Jan 24 2022

Python

def A048397(n): return n*(n*((n<<1)+3)+2)*(n*(n*(n*((n+2)<<1)+6)+4)+1) # Chai Wah Wu, Oct 19 2024

Formula

a(n) = 4*n^7 + 14*n^6 + 28*n^5 + 34*n^4 + 26*n^3 + 11*n^2 + 2*n.

G.f.: (119*x +2152*x^2 +7819*x^3 +7800*x^4 +2141*x^5 +128*x^6 +x^7)/(x-1)^8. - Harvey P. Dale, Apr 23 2011