A064551 Ado [Simone Caramel]'s Fibonacci function: define the Fibonacci sequence by f(0) = 1, f(1) = 1, f(n) = f(n-1)+f(n-2); then a(0) = 1, a(n) = a(n-1) + 2*(f(n)-n), n > 0.

1, 1, 1, 1, 3, 9, 23, 51, 103, 195, 353, 619, 1061, 1789, 2981, 4925, 8087, 13221, 21547, 35039, 56891, 92271, 149541, 242231, 392233, 634969, 1027753, 1663321, 2691723, 4355745, 7048223, 11404779, 18453871, 29859579, 48314441, 78175075, 126490637, 204666901, 331158797

Offset

0,5

Comments

A Pickover sequence with properties analogous to the primes.

References

Ado [Simone Caramel], Postings in egroups and newsgroups.

Links

T. D. Noe, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Formula

From T. D. Noe, Oct 12 2007: (Start)

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

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5). (End)

a(n) = (1/5)*2^(-n)*(-15*2^n + (10-4*sqrt(5))*(1-sqrt(5))^n + (1+sqrt(5))^n*(10+4*sqrt(5))) - n - n^2. - Jean-François Alcover, May 28 2013

a(n) = a(n-1) - 2 * A065220(n), n > 0. - Reinhard Zumkeller, Sep 13 2013

a(n) = 2*F(n+3) - n^2 - n - 3 = 1 + 2*Sum_{k=1..n} F(k+1) - k = 1 + 2*Sum_{k=1..n} A001924(k-3), F=A000045. - Ehren Metcalfe, Dec 27 2018

E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x)*(3 + x*(2 + x)). - Stefano Spezia, Oct 16 2023

Maple

a := proc(n) option remember: f := n->fibonacci(n+1): if n=0 then RETURN(1) fi: a(n-1)+2*(f(n)-n): end: for n from 1 to 60 do printf(`%d, `, a(n)) od:

Mathematica

a[0] = f[0] = f[1] = 1; f[n_] := f[n] = f[n - 1] + f[n - 2]; a[n_] := a[n] = a[n - 1] + 2*(f[n] - n); Table[ a[n], {n, 0, 40} ]
LinearRecurrence[{4, -5, 1, 2, -1}, {1, 1, 1, 1, 3}, 50] (* Harvey P. Dale, Sep 27 2011 *)

Prog

(ARIBAS): function a064551(maxarg: integer); var n, r, rm, q, qm1, qm2: integer; begin qm2 := 0; qm1 := 0; rm := 0; for n := 0 to maxarg do if n < 2 then q := 1; else q := qm1 + qm2; end; qm2 := qm1; qm1 := q; if n = 0 then r := 1; else r := rm + 2*(q - n); end; rm := r; write(r, " "); end; end; a064551(35).
(Haskell)
a064551 n = a064551_list !! n
a064551_list = 1 : zipWith (+) a064551_list
                   (map (* 2) $ zipWith (-) (drop 2 a000045_list) [1..])
-- Reinhard Zumkeller, Sep 13 2013

Crossrefs

Cf. A064550, A064552, A065220, A000045, A001924.

Sequence in context: A230119 A146693 A227259 * A005783 A263330 A146440

Adjacent sequences: A064548 A064549 A064550 * A064552 A064553 A064554

Keyword

nonn,nice,easy

Author

Roger L. Bagula, Oct 08 2001

Status

approved