A119617 Integers of the form c(n)/b(n) where c(n+1)=c(n)+(n+1)^4 with c(0)=1 and b(n+1)=b(n)+(n+1)^2 with b(0)=1.

1, 7, 25, 43, 79, 109, 163, 205, 277, 331, 421, 487, 595, 673, 799, 889, 1033, 1135, 1297, 1411, 1591, 1717, 1915, 2053, 2269, 2419, 2653, 2815, 3067, 3241, 3511, 3697, 3985, 4183, 4489, 4699, 5023, 5245, 5587, 5821, 6181, 6427, 6805, 7063, 7459, 7729

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 06 2006

Keywords

• easy
• nonn

Comments

The sequence is the union of A134153 and A134154 (without the first term of A134154): A134153(0)=1, A134154(1)=7, A134153(1)=25, A134154(2)=43, A134153(2)=79 and so on.

Examples

			c(0)/b(0) = 1/1 =1.
c(3)/b(3) = (1+2^4+3^4)/(1+2^2+3^2)= (1+16+81)/(1+4+9) = 98/14 = 7.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000538, A000330.

Programs

Magma

[(30*n*(n-1)-3*(2*n-1)*(-1)^n+5)/8: n in [1..46]]; // Bruno Berselli, Jun 27 2011

Maple

P:=proc(n) local f,i,j,nu,de; nu:=0; de:=0; for i from 1 by 1 to n do nu:=nu+i^4; de:=de+i^2; f:=nu/de; if trunc(f)=f then print(f); fi; od; end: P(1000);

Mathematica

LinearRecurrence[{1,2,-2,-1,1},{1,7,25,43,79},50] (* Harvey P. Dale, Jan 21 2017 *)

Maxima

makelist((30*n*(n-1)-3*(2*n-1)*(-1)^n+5)/8,n,1,46); /* Bruno Berselli, Jun 27 2011 */

PARI

for(n=1,46, print1((30*n*(n-1)-3*(2*n-1)*(-1)^n+5)/8", ")); \\ Bruno Berselli, Jun 27 2011

Formula

From Bruno Berselli, Jun 27 2011: (Start)

G.f.: x*(1+6*x+16*x^2+6*x^3+x^4)/((1+x)^2*(1-x)^3).

a(n) = (30*n*(n-1)-3*(2*n-1)*(-1)^n+5)/8. (End)