A104449 Fibonacci sequence with initial values a(0) = 3 and a(1) = 1.

3, 1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11933, 19308, 31241, 50549, 81790, 132339, 214129, 346468, 560597, 907065, 1467662, 2374727, 3842389, 6217116, 10059505, 16276621, 26336126, 42612747, 68948873, 111561620

Offset: 0

Author

Casey Mongoven, Mar 08 2005

Keywords

• nonn
• easy

Comments

The old name was: The Pibonacci numbers (a Fibonacci-type sequence): each term is the sum of the two previous terms.

The 6th row in the Wythoff array begins with the 6th term of the sequence (14, 23, 37, 60, 97, 157, ...). a(n) = f(n-3) + f(n+2) for the Fibonacci numbers f(n) = f(n-1) + f(n-2); f(0) = 0, f(1) = 1.

(a(2*k), a(2*k+1)) give for k >= 0 the proper positive solutions of one of two families (or classes) of solutions (x, y) of the indefinite binary quadratic form x^2 + x*y - y^2 of discriminant 5 representing 11. The other family of such solutions is given by (x2, y2) = (b(2*k), b(2*k+1)) with b = A013655. See the formula in terms of Chebyshev S polynomials S(n, 3) = A001906(n+1) below, which follows from the fundamental solution (3, 1) by applying positive powers of the automorphic matrix given in a comment in A013655. See also A089270 with the Alfred Brousseau link with D = 11. - Wolfdieter Lang, May 28 2019

References

• V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• John Conway, Alex Ryba, The extra Fibonacci series and the Empire State Building, Math. Intelligencer 38 (2016), no. 1, 41-48. (Uses the name Pibonacci.)
• Tanya Khovanova, Recursive Sequences
• Ron Knott, Fibonacci Numbers and the Golden Section .
• Eric Weisstein's World of Mathematics, Fibonacci Number
• Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
• Index entries for linear recurrences with constant coefficients, signature (1,1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. Other Fibonacci-type sequences: A000045, A000032, A013655. Other related sequences: A001906, A013655, A089270, A103343, A103344.

Wythoff array: A035513.

Essentially the same as A000285.

Programs

GAP

a:=[3,1];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # G. C. Greubel, May 29 2019

Magma

[Fibonacci(n-1) + Lucas(n): n in [0..40]]; // G. C. Greubel, May 29 2019

Maple

a:=n->3*fibonacci(n-1)+fibonacci(n): seq(a(n), n=0..40); # Zerinvary Lajos, Oct 05 2007

Mathematica

LinearRecurrence[{1,1},{3,1},40] (* Harvey P. Dale, May 23 2014 *)

PARI

a(n)=3*fibonacci(n-1)+fibonacci(n) \\ Charles R Greathouse IV, Jun 05 2011

Sage

((3-2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 29 2019

Formula

a(n) = a(n-1) + a(n-2) with a(0) = 3, a(1) = 1.

a(n) = 3*Fibonacci(n-1) + Fibonacci(n). - Zerinvary Lajos, Oct 05 2007

G.f.: (3-2*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = ( (3*sqrt(5)-1)*((1+sqrt(5))/2)^n + (3*sqrt(5)+1)*((1-sqrt(5) )/2)^n )/(2*sqrt(5)). - Bogart B. Strauss, Jul 19 2013

Bisection: a(2*k) = 4*S(k-1, 3) - 3*S(k-2, 3), a(2*k+1) = 2*S(k-1, 3) + S(k, 3) for k >= 0, with the Chebyshev S(n, 3) polynomials from A001906(n+1) for n >= -1. - Wolfdieter Lang, May 28 2019

a(n) = Fibonacci(n-1) + Lucas(n). - G. C. Greubel, May 29 2019

a(3n + 4)/a(3n + 1) = continued fraction 4,4,4,...,4,9 (that's n 4's followed by a single 9). - Greg Dresden and Shaoxiong Yuan, Jul 16 2019

E.g.f.: (exp((1/2)*(1 - sqrt(5))*x)*(1 + 3*sqrt(5) + (- 1 + 3*sqrt(5))*exp(sqrt(5)*x)))/(2*sqrt(5)). - Stefano Spezia, Jul 18 2019

Extensions

Name changed by Wolfdieter Lang, Jun 17 2019