OFFSET
1,1
COMMENTS
Theorem: if 5^((n-1)/2) = -1 (mod n) then n == 2 or 3 (mod 5) (see Crandall and Pomerance).
Start with 2. The next number, 3, cannot be written as the sum of two of the previous terms. So 3 is in. 4=2+2, 5=2+3, 6=3+3, so these are not in. But you cannot obtain 7, so the next term is 7. And so on. - Fabian Rothelius, Mar 13 2001
Also numbers k such that k^2 == -1 (mod 5). - Vincenzo Librandi, Aug 05 2010
For any (t,s) < n, a(t)*a(s) != a(n) and a(t) - a(s) != a(n). - Anders Hellström, Jul 01 2015
These numbers appear in the product of a Rogers-Ramanujan identity. See A003106 also for references. - Wolfdieter Lang, Oct 29 2016
REFERENCES
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 3.24, p. 154.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = 5*(n-1) - a(n-1) (with a(1)=2). - Vincenzo Librandi, Aug 05 2010
a(n) = (10*n - 3*(-1)^n - 5)/4.
G.f.: x*(2+x+2*x^2)/((1+x)*(1-x)^2).
a(n)^2 = 5*A118015(a(n)) + 4.
a(n) = 5 * (floor(n-1)/2) + 3 - n mod 2. - Reinhard Zumkeller, Nov 27 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/5. - Amiram Eldar, Dec 07 2021
E.g.f.: 2 + ((5*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 23 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1.
Product_{n>=1} (1 + (-1)^n/a(n)) = 1/phi (A094214). (End)
MATHEMATICA
{2, 3}+#&/@(5 Range[0, 30])//Flatten (* Harvey P. Dale, Jan 22 2023 *)
PROG
(Magma) [ n : n in [1..165] | n mod 5 eq 2 or n mod 5 eq 3 ];
(Haskell)
a047221 n = 5 * ((n - 1) `div` 2) + 3 - n `mod` 2
a047221_list = 2 : 3 : map (+ 5) a047221_list
-- Reinhard Zumkeller, Nov 27 2012
(PARI) Vec(x*(2+x+2*x^2)/((1+x)*(1-x)^2) + O(x^80)) \\ Michel Marcus, Jun 30 2015
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Apr 08 2002
Closed formula, g.f. and link added by Bruno Berselli, Nov 28 2010
STATUS
approved