A045502 Numbers k such that 2*k+1 and 3*k+1 are squares.

0, 40, 3960, 388080, 38027920, 3726348120, 365144087880, 35780394264160, 3506113493799840, 343563341998120200, 33665701402321979800, 3298895174085555900240, 323258061358982156243760, 31675991118006165755988280, 3103923871503245261930607720

Offset

0,2

Comments

Problem 1 for the 3rd grade of the 38th Mathematics Competition of the Republic of Slovenia (1998) was to prove that if k is a natural number such that 2*k+1 and 3*k+1 are perfect squares, then k is divisible by 40 (see link with solution Crux Mathematicorum and formula Mar 25 2021). - Bernard Schott, Mar 25 2021

Links

Colin Barker, Table of n, a(n) for n = 0..500.

John Albert, Pell Equations, Putnam practice, November 17, 2004 (1-2).

R. S. Luthar, Problem E2606, Amer. Math. Monthly, 84 (1977), 823-824.

R. E. Woodrow, Problem 1 for the third grade of the 38th Mathematics competition of the Republic of Slovenia (1998), Crux Mathematicorum, The Olympiad Corner, p. 208, April 1999, Vol. 25, No. 4.

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Index to sequences related to Olympiads and other Mathematical Competitions.

Formula

From Colin Barker, Mar 23 2017: (Start)

O.g.f.: 40*x / ((1 - x)*(1 - 98*x + x^2)).

a(n) = 99*a(n-1)- 99*a(n-2) + a(n-3) for n>2.

a(n) = (-10 + (5 - 2*sqrt(6))*(49 + 20*sqrt(6))^(-n) + (5 + 2*sqrt(6))*(49 + 20*sqrt(6))^n)/24. (End)

From G. C. Greubel, Jan 13 2020: (Start)

a(n) = 5*(ChebyshevT(n, 49) + 48*ChebyshevU(n-1, 48) - 1)/12.

a(n) = 4*ChebyshevU(n-1, 5)*ChebyshevU(n, 5). (End)

a(n) = 40*A278620(n). - Bernard Schott, Mar 25 2021

Maple

seq(coeff(series(40*x/((1-x)*(x^2-98*x+1)), x, n+1), x, n), n=0..15); # Muniru A Asiru, Jul 17 2018

Mathematica

f[0]=0; f[1]=2; f[n_]:= f[n]= 10*f[n-1] -f[n-2]; a[n_]:= f[n]*f[n+1];
CoefficientList[Series[40x/((1-x)(1-98x+x^2)), {x, 0, 15}], x] (* Michael De Vlieger, Jul 20 2018 *)
Table[5*(ChebyshevT[n, 49] +48*ChebyshevU[n-1, 49] -1)/12, {n, 0, 15}] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{99, -99, 1}, {0, 40, 3960}, 20] (* Harvey P. Dale, Dec 02 2023 *)

Prog

(PARI) concat(0, Vec(40*x/((1-x)*(1-98*x+x^2))+O(x^20))) \\ Colin Barker, Mar 23 2017
(GAP) a:=[0, 40, 3960];; for n in [4..15] do a[n]:=99*a[n-1]-99*a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Jul 17 2018
(Magma) I:=[0, 40, 3960]; [n le 3 select I[n] else 99*Self(n-1) -99*Self(n-2) + Self(n-3): n in [1..15]]; // G. C. Greubel, Jan 13 2020
(Sage) [4*chebyshev_U(n-1, 5)*chebyshev_U(n, 5) for n in (0..15)] # G. C. Greubel, Jan 13 2020

Crossrefs

Cf. A001078, A245031, A278620.

Sequence in context: A178721 A059948 A229604 * A347854 A378234 A259461

Adjacent sequences: A045499 A045500 A045501 * A045503 A045504 A045505

Keyword

nonn,easy

Author

Fred Schwab (fschwab(AT)nrao.edu)

Status

approved