A133283 Numbers k such that 30*k^2 + 6 is a square.

1, 23, 505, 11087, 243409, 5343911, 117322633, 2575754015, 56549265697, 1241508091319, 27256628743321, 598404324261743, 13137638505015025, 288429642786068807, 6332314502788498729, 139022489418560903231, 3052162452705551372353, 67008551470103569288535

Offset: 1

Author

Richard Choulet, Oct 16 2007

Keywords

• nonn
• easy

Comments

From Klaus Purath, Apr 19 2025: (Start)

Nonnegative solutions to the Diophantine equation 5*a(n)^2 - 6*b(n)^2 = -1. The corresponding b(n) are A157014(n). Note that (a(n+1)^2 - a(n)*a(n+2))/4 = 6 and (b(n)*b(n+2) - b(n+1)^2)/4 = 5.

(a(n) + b(n))/2 = (a(n+1) - b(n+1))/2 = A077421(n-1) = Lucas U(22,1). Also a(n)*b(n+1) - a(n+1)*b(n) = -2.

a(n+1) = (t(i+2*n+1) - t(i))/(t(i+n+1) - t(i+n)) as long as t(i+n+1) - t(i+n) != 0 for integer i and n >= 0 where (t) is a sequence satisfying t(i+3) = 23*t(i+2) - 23*t(i+1) + t(i) or t(i+2) = 22*t(i+1) - t(i), regardless of the initial values and including this sequence itself. (End)

Links

• Colin Barker, Table of n, a(n) for n = 1..700
• Index entries for linear recurrences with constant coefficients, signature (22,-1).

Crossrefs

Cf. A221874.

Programs

GAP

a:=[1,23];; for n in [3..20] do a[n]:=22*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2020

Magma

I:=[1,23]; [n le 2 select I[n] else 22*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Jan 13 2020

Maple

a[1]:=1: a[2]:=23: for n to 14 do a[n+2]:=22*a[n+1]-a[n] end do: seq(a[n],n= 1..16); # Emeric Deutsch, Oct 24 2007

Mathematica

Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && 30*n^2+6 == k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 1, 20}] // Flatten // Union (* Jean-François Alcover, Dec 19 2013 *)
Rest@ CoefficientList[Series[x(1+x)/(1-22x+x^2), {x,0,20}], x] (* Michael De Vlieger, Jul 14 2016 *)
LinearRecurrence[{22,-1},{1,23},20] (* Harvey P. Dale, Sep 22 2017 *)
Table[ChebyshevU[n-1, 11] + ChebyshevU[n-2, 11], {n, 20}] (* G. C. Greubel, Jan 13 2020 *)

PARI

Vec(x*(1+x)/(1-22*x+x^2) + O(x^20)) \\ Colin Barker, Jul 14 2016

PARI

vector(20, n, polchebyshev(n-1,2,11) + polchebyshev(n-2,2,11) ) \\ G. C. Greubel, Jan 13 2020

Sage

[chebyshev_U(n-1,11) + chebyshev_U(n-2,11) for n in (1..20)] # G. C. Greubel, Jan 13 2020

Formula

a(n+2) = 22*a(n+1) - a(n); a(n+1) = 11*a(n) + 2*sqrt(30*a(n)^2 + 6).

a(n) = (sqrt(30)/10 - 1/2)*(11 + 2*sqrt(30))^n - (sqrt(30)/10 + 1/2) * (11 - 2*sqrt(30))^n. - Emeric Deutsch, Oct 24 2007

G.f.: x*(1+x)/(1-22*x+x^2). - R. J. Mathar, Nov 14 2007

a(n) = A077421(n) + A077421(n-1). - R. J. Mathar, Feb 19 2016

a(n) = Chebyshev(n-1, 11) + Chebyshev(n-2, 11). - G. C. Greubel, Jan 13 2020

Extensions

More terms from Emeric Deutsch, Oct 24 2007