A055269 a(n) = 4*a(n-1) - a(n-2) + 3 with a(0)=1, a(1)=7.

1, 7, 30, 116, 437, 1635, 6106, 22792, 85065, 317471, 1184822, 4421820, 16502461, 61588027, 229849650, 857810576, 3201392657, 11947760055, 44589647566, 166410830212, 621053673285, 2317803862931, 8650161778442, 32282843250840, 120481211224921, 449642001648847

Offset: 0

Author

Barry E. Williams, May 10 2000

Keywords

• easy
• nonn

Comments

Also partial sums of A054491.

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
• E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
• Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Crossrefs

Cf. A001834, A054491.

Programs

Magma

I:=[1,7,30]; [n le 3 select I[n] else 5*Self(n-1) - 5*Self(n-2) + Self(n-3): n in [1..40]]; // G. C. Greubel, Mar 16 2020

Maple

A055269:= n-> simplify((5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2); seq( A055269(n), n=0..40); # G. C. Greubel, Mar 16 2020

Mathematica

LinearRecurrence[{5,-5,1},{1,7,30},40] (* or *) CoefficientList[ Series[ (1+2*x)/(1-5*x+5*x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Dec 01 2013 *)
Table[(5*ChebyshevU[n, 2] -3*ChebyshevU[n-1, 2] - 3)/2, {n,0,40}] (* G. C. Greubel, Mar 16 2020 *)

Sage

[(5*chebyshev_U(n, 2) - 3*chebyshev_U(n-1, 2) - 3)/2 for n in (0..40)] # G. C. Greubel, Mar 16 2020

Formula

G.f.: (1+2*x)/((1-x)*(1-4*x+x^2)).

a(n) = ( ( (17 - 5*(2-sqrt(3)))*(2+sqrt(3))^n + (5*(2+sqrt(3))-17)*(2-sqrt(3))^n )/(4*sqrt(3)) ) - 3/2.

a(n) = (5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2. - G. C. Greubel, Mar 16 2020

Extensions

Corrected by T. D. Noe, Nov 07 2006