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A218395 If the sum of the squares of 11 consecutive numbers is a square, then a(n) is the square root of this sum. 13
11, 77, 143, 1529, 2849, 30503, 56837, 608531, 1133891, 12140117, 22620983, 242193809, 451285769, 4831736063, 9003094397, 96392527451, 179610602171, 1923018812957, 3583208949023, 38363983731689, 71484568378289, 765356655820823, 1426108158616757 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n)^2 = Sum_{j=0..10} (x(n)+j)^2 = 11*(x(n)+5)^2+110 and b(n) = x(n)+5 give the pell equation a(n)^2-11*b(n)^2 = 110 with the 2 fundamental solutions (11; 1) and (77; 23) and the solution (10; 3) for the unit form.  A198949(n+1) = b(n); A106521(n) = x(n) and x(0) = -4.

General: If the sum of the squares of c neighboring numbers is a square with c=3*k^2-1 and 1<=k, then a(n)^2 = Sum_{j=0..c-1} (x(n)+j)^2 and b(n) = 2*x(n)+c-1 give the pell equation a(n)^2-c*(b(n)/2)^2 = binomial(c+1,3)/2.  a(n) = 2*e1*a(n-k)-a(n-2*k); b(n) = 2*e1*b(n-k) -b(n-2*k); a(n) = e1*a(n-k)+c*e2*b(n-k); b(n) = e2*a(n-k)+e1*b(n-k) with the solution (e1; e2) for the unit form.

LINKS

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

FORMULA

a(n) = 20*a(n-2)-a(n-4); b(n) = 20*b(n-2)-b(n-4);

a(n) = 10*a(n-2)+33*b(n-2); b(n) = 3*a(n-2)+10*b(n-2).

a(n) = a(n-1)+20*a(n-2)-20*a(n-3)-a(n-4)+a(n-5).

G.f.: (11+77*x-77*x^2-11*x^3)/(1-20*x^2+x^4).

with  r=sqrt(11); s=10+3*r; t=10-3*r:

a(2*n) = (11+r)*s^n+(11-r)*t^n.

a(2*n+1) = (77+23*r)+s^n+(77-23*r)*t^n.

EXAMPLE

For n=6 the Sum_{z=17132..17142} z^2 = 3230444569;

a(6) = sqrt(3230444569) = 56837;

b(6) = sqrt((a(6)^2-110)/11) = 17137; x(6) = b(6)-5 = 17132.

MAPLE

s:=0: n:=-1:

for j from -5 to 5 do s:=s+j^2: end do:

for z from -4 to 100000 do

  s:=s-(z-1)^2+(z+10)^2: r:=sqrt(s):

  if (r=floor(r)) then

    n:=n+1: a(n):=r: x(n):=z:

    b(n):=sqrt((s-110)/11):

    print(n, a(n), b(n), x(n)):

  end if:

end do:

CROSSREFS

c=2: A001653(n+1) = a(n); A002315(n) = b(n); A001652(n) = x(n).

Sequence in context: A272395 A305727 A039674 * A208599 A059625 A023010

Adjacent sequences:  A218392 A218393 A218394 * A218396 A218397 A218398

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Oct 28 2012

STATUS

approved

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Last modified March 26 10:18 EDT 2019. Contains 321491 sequences. (Running on oeis4.)