A055267 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=7.

1, 7, 20, 53, 139, 364, 953, 2495, 6532, 17101, 44771, 117212, 306865, 803383, 2103284, 5506469, 14416123, 37741900, 98809577, 258686831, 677250916, 1773065917, 4641946835, 12152774588, 31816376929, 83296356199, 218072691668, 570921718805, 1494692464747

Offset: 0

Author

Barry E. Williams, May 09 2000

Keywords

• easy
• nonn

References

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

Links

• Vincenzo Librandi, 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.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (3,-1).

Crossrefs

Cf. A000032, A000045, A054486, A054492.

Programs

GAP

List([0..30], n-> Fibonacci(2*n+2) +4*Fibonacci(2*n) ); # G. C. Greubel, Jan 17 2020

Magma

[5*Fibonacci(2*n) + Fibonacci(2*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 25 2018

Maple

with(combinat); seq(fibonacci(2*n+2) +4*fibonacci(2*n), n=0..30); # G. C. Greubel, Jan 17 2020

Mathematica

Table[5*Fibonacci[2n] + Fibonacci[2n+1],  {n, 0, 30}]
Table[4*Fibonacci[2n-1] + 3*LucasL[2n-1], {n, 0, 30}] (* Rigoberto Florez, Dec 24 2018 *)
LinearRecurrence[{3,-1}, {1,7}, 30] (* Vincenzo Librandi, Dec 25 2018 *)
nxt[{a_,b_}]:={b,3b-a}; NestList[nxt,{1,7},30][[;;,1]] (* Harvey P. Dale, Mar 23 2025 *)

PARI

Vec((1+4*x)/(1-3*x+x^2) + O(x^40)) \\ Michel Marcus, Sep 06 2017

Sage

[fibonacci(2*n+2) +4*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020

Formula

a(n) = (7*(((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n) - (((3 + sqrt(5))/2)^(n - 1) - ((3 - sqrt(5))/2)^(n - 1)))/sqrt(5).

G.f.: (1 + 4*x)/(1 - 3*x + x^2).

From Rigoberto Florez, Dec 24 2018: (Start)

a(n) = 5*Fibonacci(2*n) + Fibonacci(2*n+1).

a(n) = 4*Fibonacci(2*n - 1) + 3*Lucas(2*n - 1). (End)

E.g.f.: exp(3*t/2)*( cosh(sqrt(5)*t/2) + (11/sqrt(5))*sinh(sqrt(5)*t/2) ). - G. C. Greubel, Jan 17 2020

a(n) = 4*A001906(n) + A001906(n+1). - R. J. Mathar, Mar 06 2022