| Displaying 1-4 of 4 results found.
page
1
5, 7, 23, 37, 133, 215, 775, 1253, 4517, 7303, 26327, 42565, 153445, 248087, 894343, 1445957, 5212613, 8427655, 30381335, 49119973, 177075397, 286292183, 1032071047, 1668633125, 6015350885, 9725506567, 35060034263, 56684406277, 204344854693, 330380931095
COMMENTS
a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077241(n). Because there is only one primitive Pythagorean triangle with sum of the legs L = 17 (see also A120681), namely (5,12,13), all positive solutions (x(n), y(n)) = (a(n), 2* A077241(n)) of the (generalized) Pell equation x^2 - 2*y^2 = +17 satisfy x(n) < 2*y(n), for n >= 1, only 5 = x(0) > 2*y(0) = 4. The proof runs along the same line as the one given in a comment on the L=7 case in A077443. - Wolfdieter Lang, Feb 05 2015
FORMULA
G.f.: (1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4).
a(n) = ((6-5*sqrt(2))*(1-sqrt(2))^n - (-1-sqrt(2))^n*(-4+sqrt(2)) + 4*(-1+sqrt(2))^n + sqrt(2)*(-1+sqrt(2))^n + 6*(1+sqrt(2))^n + 5*sqrt(2)*(1+sqrt(2))^n)/4. - Colin Barker, Mar 27 2016
EXAMPLE
23 = a(2) = sqrt(8* A077241(2)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.
MATHEMATICA
A077239 = Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 12}]; A077240 = Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 12}]; Riffle[ A077240, A077239] (* Jean-François Alcover, Dec 19 2013 *)
CoefficientList[Series[(1 - x) (5 + 12 x + 5 x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 18 2014 *)
PROG
(Magma) I:=[5, 7, 23, 37]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014 (PARI) Vec((1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4) + O(x^50)) \\ Colin Barker, Mar 27 2016
Expansion of (1+2*x)/(1-6*x+x^2).
+10
10
1, 8, 47, 274, 1597, 9308, 54251, 316198, 1842937, 10741424, 62605607, 364892218, 2126747701, 12395593988, 72246816227, 421085303374, 2454265004017, 14304504720728, 83372763320351, 485932075201378, 2832219687887917
COMMENTS
Bisection (even part) of Chebyshev sequence with Diophantine property.
b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.
FORMULA
a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=8.
a(n) = ((3 + 2*sqrt(2))^(n+1) - (3 - 2*sqrt(2))^(n+1) + 2*((3 + 2*sqrt(2))^n - (3 - 2*sqrt(2))^n))/(4*sqrt(2)).
a(n) = S(n, 6) + 2*S(n-1, 6), with S(n, x) Chebyshev's polynomials of the second kind, A049310. S(n, 6) = A001109(n+1). a(n) = (Pell(2*n+2) + 2*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + Pell(2*n))/2. - G. C. Greubel, Jan 19 2020 E.g.f.: (1/4)*exp(3*x)*(4*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x)). - Stefano Spezia, Jan 27 2020
EXAMPLE
8 = a(1) = sqrt(( A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) = 8.
MAPLE
a[0]:=1: a[1]:=8: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006
MATHEMATICA
LinearRecurrence[{6, -1}, {1, 8}, 30] (* Harvey P. Dale, Oct 09 2017 *) Table[(LucasL[2*n+1, 2] + Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jan 19 2020 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1+2*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jan 19 2020 (PARI) apply( { A054488(n)=[1, 8]*([0, -1; 1, 6]^n)[, 1]}, [0..30]) \\ M. F. Hasler, Feb 27 2020 (Magma) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2020 (Magma) R<x>:=PowerSeriesRing(Integers(), 21); Coefficients(R!( (1+2*x)/(1-6*x+x^2))); // Marius A. Burtea, Jan 20 2020 (Sage) [(lucas_number2(2*n+1, 2, -1) + lucas_number1(2*n, 2, -1))/2 for n in (0..30)] # G. C. Greubel, Jan 19 2020 (GAP) a:=[1, 8];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020
Bisection (odd part) of Chebyshev sequence with Diophantine property.
+10
6
2, 13, 76, 443, 2582, 15049, 87712, 511223, 2979626, 17366533, 101219572, 589950899, 3438485822, 20040964033, 116807298376, 680802826223, 3968009658962, 23127255127549, 134795521106332, 785645871510443, 4579079707956326, 26688832376227513, 155553914549408752
COMMENTS
-8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n) = A077239(n).
FORMULA
a(n) = 6*a(n-1) - a(n-2), a(-1)=-1, a(0)=2.
a(n) = 2*S(n, 6)+S(n-1, 6), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 6) = A001109(n+1). G.f.: (2+x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-7+4*sqrt(2))+(3+2*sqrt(2))^n*(7+4*sqrt(2))))/(4*sqrt(2)). - Colin Barker, Oct 12 2015
EXAMPLE
8*a(1)^2 + 17 = 8*13^2+17 = 1369 = 37^2 = A077239(1)^2. G.f. = 2 + 13*x + 76*x^2 + 443*x^3 + 2582*x^4 + ... - Michael Somos, Jul 30 2024
MATHEMATICA
LinearRecurrence[{6, -1}, {2, 13}, 30] (* or *) CoefficientList[Series[ (2+x)/(1-6*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Jan 18 2018 *) a[ n_] := 2*ChebyshevU[n, 3] + ChebyshevU[n-1, 3]; (* Michael Somos, Jul 30 2024 *)
PROG
(PARI) Vec((2+x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Jun 16 2015 (PARI) {a(n) = 2*polchebyshev(n, 2, 3) + polchebyshev(n-1, 2, 3)}; /* Michael Somos, Jul 30 2024 */ (PARI) {a(n) = my(w = 3 + quadgen(32)); imag(w^n + 2*w^(n+1))}; /* Michael Somos, Jul 30 2024 */ (Magma) I:=[2, 13]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018
Bisection (odd part) of Chebyshev sequence with Diophantine property.
+10
5
7, 37, 215, 1253, 7303, 42565, 248087, 1445957, 8427655, 49119973, 286292183, 1668633125, 9725506567, 56684406277, 330380931095, 1925601180293, 11223226150663, 65413755723685, 381259308191447, 2222142093424997, 12951593252358535, 75487417420726213
COMMENTS
a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077413(n).
FORMULA
a(n) = 6*a(n-1) - a(n-2), a(-1) := 5, a(0)=7.
a(n) = 2*T(n+1, 3)+T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n). G.f.: (7-5*x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-8+7*sqrt(2))+(3+2*sqrt(2))^n*(8+7*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015
EXAMPLE
37 = a(1) = sqrt(8* A077413(1)^2 +17) = sqrt(8*13^2 + 17)= sqrt(1369) = 37.
PROG
(PARI) Vec((7-5*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015
Search completed in 0.005 seconds
|