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A045502
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Numbers k such that 2*k+1 and 3*k+1 are squares.
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5
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0, 40, 3960, 388080, 38027920, 3726348120, 365144087880, 35780394264160, 3506113493799840, 343563341998120200, 33665701402321979800, 3298895174085555900240, 323258061358982156243760, 31675991118006165755988280, 3103923871503245261930607720
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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John Albert, Pell Equations, Putnam practice, November 17, 2004 (1-2).
R. S. Luthar, Problem E2606, Amer. Math. Monthly, 84 (1977), 823-824.
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FORMULA
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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)
a(n) = 5*(ChebyshevT(n, 49) + 48*ChebyshevU(n-1, 48) - 1)/12.
a(n) = 4*ChebyshevU(n-1, 5)*ChebyshevU(n, 5). (End)
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Fred Schwab (fschwab(AT)nrao.edu)
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STATUS
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approved
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