A007750 - OEIS

A007750

Nonnegative integers n such that n^2*(n+1)*(2*n+1)^2*(7*n+1)/36 is a square.

0, 1, 7, 24, 120, 391, 1921, 6240, 30624, 99457, 488071, 1585080, 7778520, 25261831, 123968257, 402604224, 1975713600, 6416405761, 31487449351, 102259887960, 501823476024, 1629741801607, 7997688167041, 25973608937760

(list; graph; refs; listen; history; text; internal format)

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

(SageMath)

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

Cf. A007751, A007752, A077412.

Sequence in context: A283457 A129797 A188120 * A153577 A009643 A108095

Adjacent sequences: A007747 A007748 A007749 * A007751 A007752 A007753

KEYWORD