A106527 Values of x in x^2 - 289 = 2*y^2.

19, 33, 51, 81, 179, 289, 467, 1041, 1683, 2721, 6067, 9809, 15859, 35361, 57171, 92433, 206099, 333217, 538739, 1201233, 1942131, 3140001, 7001299, 11319569, 18301267, 40806561, 65975283, 106667601, 237838067, 384532129, 621704339

Offset

1,1

Comments

The sums of the following consecutive integer sequences are squares for any integer i and n >=1: if mod(n,3) = 0, 17*i + 1, ..., a(n)*i + (A001541(n/3)-1)/2 if mod(n,3) = 1 or 2, 17*i + 9, ..., a(n)*i + (a(n) - 1)/2.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Formula

a(3n) = 17*A001541(n).

a(3n+1) = A001541(n+2) - A001541(n+1) - A001541(n) + 2*A001541(n-1).

a(3n+2) = 2*A001541(n+2) - A001541(n+1) - A001541(n) + A001541(n-1).

From Colin Barker, Mar 29 2012: (Start)

a(n) = 6*a(n-3) - a(n-6).

G.f.: x*(19 +33*x +51*x^2 -33*x^3 -19*x^4 -17*x^5)/(1 -6*x^3 +x^6). (End)

Example

a(6) = 289, 17*A001541(2) = 17*17 = 289;
a(7) = 467, A001541(4) -A001541(3) -A001541(2) +2*A001541(1) = 577 -99 -17 + 2*3 = 467;
a(8) = 1041, 2*A001541(4) -A001541(3) -A001541(2) +A001541(1) = 2*577 -99 -17 +3 = 1041;
Also, a(8)^2 - 289 = 2*A106528(8)^2 : 1041^2 - 289 = 2*736^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {19, 33, 51, 81, 179, 289}, 40] (* G. C. Greubel, Aug 18 2021 *)

Prog

(Magma) I:=[19, 33, 51, 81, 179, 289]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // G. C. Greubel, Aug 18 2021
(Sage)
def A106527_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(19 +33*x +51*x^2 -33*x^3 -19*x^4 -17*x^5)/(1 -6*x^3 +x^6) ).list()
a=A106527_list(41); a[1:] # G. C. Greubel, Aug 18 2021

Crossrefs

Cf. A001541, A106528.

Sequence in context: A152088 A372427 A362410 * A223608 A146438 A146571

Adjacent sequences: A106524 A106525 A106526 * A106528 A106529 A106530

Keyword

nonn

Author

Andras Erszegi (erszegi.andras(AT)chello.hu), May 09 2005

Status

approved