A106526 Values of y in x^2 - 49 = 2*y^2.

4, 6, 14, 30, 40, 84, 176, 234, 490, 1026, 1364, 2856, 5980, 7950, 16646, 34854, 46336, 97020, 203144, 270066, 565474, 1184010, 1574060, 3295824, 6900916, 9174294, 19209470, 40221486, 53471704, 111960996, 234428000, 311655930, 652556506

Offset

1,1

Comments

The expression 2*n^2 + c with c = 49 yields more squares than any other value of c in the range 1 < c < 100 and n < 5*10^4. - K. D. Bajpai, Nov 04 2013

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(n) = 6*a(n-3) - a(n-6), with initial terms 4, 6, 14, 30, 40, 84. - T. D. Noe, Nov 04 2013

From G. C. Greubel, Aug 12 2021: (Start)

a(n) = 2*A276600(n+1).

G.f.: (2*x)*(2 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/(1 - 6*x^3 + x^6). (End)

Example

a(12) = 2856; as 12 mod 3 = 0, a(12) = 14*A001109(12/3) = 204*14 = 2856; also 2*2856^2 = 4039^2 - 49, i.e., A106525(12)^2 - 49;
a(13) = 5980; as 13 mod 3 = 1, a(13) = A001109(4+2) - A001109(4+1) + A001109(4) + A001109(4-1) = 6930 - 1189 + 204 + 35 = 5980; also 2*5980^2 = 8457^2 - 49, i.e., A106525(13)^2 - 49;
a(14) = 7950; as 14 mod 3 = 2, a(14) = A001109(4+2) + A001109(4+1) - A001109(4) + A001109(4-1) = 6930 + 1189 - 204 + 35 = 7950; also 2*7950^2 = 11243^2 - 49, i.e., A106525(14)^2 - 49.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {4, 6, 14, 30, 40, 84}, 40] (* T. D. Noe, Nov 04 2013 *)

Prog

(Magma) I:=[4, 6, 14, 30, 40, 84]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // G. C. Greubel, Aug 12 2021
(Sage)
def A106526_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (2*x)*(2 +3*x +7*x^2 +3*x^3 +2*x^4)/(1 -6*x^3 +x^6) ).list()
a=A106526_list(41); a[1:] # G. C. Greubel, Aug 12 2021

Crossrefs

Cf. A001109, A106525, A276600.

Sequence in context: A103419 A005202 A342602 * A319766 A373567 A108516

Adjacent sequences: A106523 A106524 A106525 * A106527 A106528 A106529

Keyword

nonn

Author

Andras Erszegi (erszegi.andras(AT)chello.hu), May 07 2005

Status

approved