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

Author

Fred Schwab (fschwab(AT)nrao.edu)

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A001078, A245031, A278620.

Programs

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

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 *)

PARI

concat(0, Vec(40*x/((1-x)*(1-98*x+x^2))+O(x^20))) \\ Colin Barker, Mar 23 2017

Sage

[4*chebyshev_U(n-1,5)*chebyshev_U(n,5) for n in (0..15)] # G. C. Greubel, Jan 13 2020

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