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%I #86 Sep 08 2022 08:45:17
%S 3,1,4,5,9,14,23,37,60,97,157,254,411,665,1076,1741,2817,4558,7375,
%T 11933,19308,31241,50549,81790,132339,214129,346468,560597,907065,
%U 1467662,2374727,3842389,6217116,10059505,16276621,26336126,42612747,68948873,111561620
%N Fibonacci sequence with initial values a(0) = 3 and a(1) = 1.
%C The old name was: The Pibonacci numbers (a Fibonacci-type sequence): each term is the sum of the two previous terms.
%C 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.
%C (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
%D V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
%H G. C. Greubel, <a href="/A104449/b104449.txt">Table of n, a(n) for n = 0..1000</a>
%H John Conway, Alex Ryba, <a href="https://doi.org/10.1007/s00283-015-9582-5">The extra Fibonacci series and the Empire State Building</a>, Math. Intelligencer 38 (2016), no. 1, 41-48. (Uses the name Pibonacci.)
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/">Fibonacci Numbers and the Golden Section </a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H Shaoxiong Yuan, <a href="https://arxiv.org/abs/1907.12459">Generalized Identities of Certain Continued Fractions</a>, arXiv:1907.12459 [math.NT], 2019.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = a(n-1) + a(n-2) with a(0) = 3, a(1) = 1.
%F a(n) = 3*Fibonacci(n-1) + Fibonacci(n). - _Zerinvary Lajos_, Oct 05 2007
%F G.f.: (3-2*x)/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008
%F 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
%F 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
%F a(n) = Fibonacci(n-1) + Lucas(n). - _G. C. Greubel_, May 29 2019
%F 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
%F 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
%p a:=n->3*fibonacci(n-1)+fibonacci(n): seq(a(n), n=0..40); # _Zerinvary Lajos_, Oct 05 2007
%t LinearRecurrence[{1,1},{3,1},40] (* _Harvey P. Dale_, May 23 2014 *)
%o (PARI) a(n)=3*fibonacci(n-1)+fibonacci(n) \\ _Charles R Greathouse IV_, Jun 05 2011
%o (Magma) [Fibonacci(n-1) + Lucas(n): n in [0..40]]; // _G. C. Greubel_, May 29 2019
%o (Sage) ((3-2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 29 2019
%o (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
%Y Cf. Other Fibonacci-type sequences: A000045, A000032, A013655. Other related sequences: A001906, A013655, A089270, A103343, A103344.
%Y Wythoff array: A035513.
%Y Essentially the same as A000285.
%K nonn,easy
%O 0,1
%A _Casey Mongoven_, Mar 08 2005
%E Name changed by _Wolfdieter Lang_, Jun 17 2019
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