OFFSET
0,3
COMMENTS
n^2*(n+1)*(2*n+1)^2*(7*n+1)/36 = Sum(i=1..n, i^2) * Sum(i=n+1..2*n, i^2) = A000330(n)*(A000330(2*n)-A000330(n)) = A000330(n)*n*(2*n+1)*(7*n+1)/6. - Michael Somos, Jul 27 2002
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
K. R. S. Sastry, Problem 533 The College Mathematics Journal, 25, issue 4, 1994, p. 334.
K. R. S. Sastry, Square Products of Sums of Squares The College Mathematics Journal, 26, issue 4, 1995, p. 333.
Index entries for linear recurrences with constant coefficients, signature (1,16,-16,-1,1).
FORMULA
From Michael Somos, Jul 27 2002: (Start)
G.f.: x * (1 + 6*x + x^2) / ((1 - x) * (1 - 16*x^2 + x^4)).
a(n) = 16 * a(n-2) - a(n-4) + 8. (End)
From G. C. Greubel, Feb 10 2020: (Start)
a(2*n) = (4*ChebyshevU(n,8) - 11*ChebyshevU(n-1,8) - 4))/7 = A007751(n).
a(2*n+1) = (11*ChebyshevU(n,8) - 4*ChebyshevU(n-1,8) - 4))/7 = A007752(n+1). (End)
MAPLE
m:=30; S:=series(x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 10 2020
MATHEMATICA
CoefficientList[Series[x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 15 2017 *)
Table[If[EvenQ[n], (4*ChebyshevU[n/2, 8] -11*ChebyshevU[(n-2)/2, 8] -4)/7, (11*ChebyshevU[(n-1)/2, 8] -4*ChebyshevU[(n-3)/2, 8] -4)/7], {n, 0, 30}] (* G. C. Greubel, Feb 10 2020 *)
PROG
(PARI) {a(n) = if( n<0, a(-1-n), if( n<2, n>0, 16 * a(n-2) - a(n-4) + 8))} /* Michael Somos, Jul 27 2002 */
(PARI) {a(n) = local(w); if( n<0, 0, w = 8 + 3*quadgen(28); n = ((n+1)\2) * (-1)^(n%2); imag(w^n) + 4 * (real(w^n) - 1) / 7)} /* Michael Somos, Jul 27 2002 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)) )); // G. C. Greubel, Feb 10 2020
(Sage)
def A007750_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)) ).list()
A007750_list(30) # G. C. Greubel, Feb 10 2020
(GAP) a:=[0, 1, 7, 24, 120];; for n in [6..30] do a[n]:=a[n-1]+16*a[n-2]-16*a[n-3] -a[n-4]+a[n-5]; od; a; # G. C. Greubel, Feb 10 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John C. Hallyburton, Jr. (hallyb(AT)vmsdev.enet.dec.com)
EXTENSIONS
Edited by Michael Somos, Jul 27 2002
STATUS
approved