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A152271
a(n)=1 for even n and (n+1)/2 for odd n.
15
1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 1, 43, 1, 44
OFFSET
0,4
COMMENTS
A000012 and A000027 interleaved. - Omar E. Pol, Mar 12 2012
Run lengths in A128218. - Reinhard Zumkeller, Jun 20 2015
a(n+1) is the number of reversible binary strings of length n+1 with Hamming weight 1 or 2 such that the 1's are separated by an even number of 0's. - Christian Barrientos, Jan 28 2019
Simple continued fraction of -1 + BesselJ(1,2)/BesselJ(2,2) = 1/(1 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + 1/(4 + 1/(1 + ... ))))))))). - Peter Bala, Oct 06 2023
FORMULA
a(n) = 2*a(n-2) - a(n-4) with a(0)=a(1)=a(2)=1 and a(3)=2.
a(n) = (a(n-2) + a(n-3))/a(n-1).
G.f.: (1 + x - x^2)/(1 - 2*x^2 + x^4).
a(n) = A057979(n+2).
a(n)*a(n+1) = floor((n+2)/2) = A008619(n). - Paul Barry, Feb 27 2009
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*0^floor((n-2k)/2). - Paul Barry, Feb 27 2009
a(n) = gcd(floor((n+1)/2), (n+1)). - Enrique Pérez Herrero, Mar 13 2012
G.f.: U(0) where U(k) = 1 + x*(k+1)/(1 - x/(x + (k+1)/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 04 2012
E.g.f.: ((2 + x)*cosh(x) + sinh(x))/2. - Stefano Spezia, Mar 26 2022
a(n) = (-1)^n * a(-2-n) for all n in Z. - Michael Somos, Mar 26 2022
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 3*x^5 + x^6 + 4*x^7 + x^8 + ... - Michael Somos, Mar 26 2022
MATHEMATICA
Table[If[EvenQ[n], 1, (n+1)/2], {n, 0, 120}] (* or *) LinearRecurrence[{0, 2, 0, -1}, {1, 1, 1, 2}, 120] (* or *) Riffle[Range[60], 1, {1, -1, 2}] (* Harvey P. Dale, Jan 20 2018 *)
PROG
(PARI) Vec((1+x-x^2)/(1-2*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012
(PARI) a(n)=gcd(n+1, (n+1)\2) \\ Charles R Greathouse IV, Mar 13 2012
(Haskell)
import Data.List (transpose)
a152271 = a057979 . (+ 2)
a152271_list = concat $ transpose [repeat 1, [1..]]
-- Reinhard Zumkeller, Aug 11 2014
(Python)
def A152271(n): return n+1>>1 if n&1 else 1 # Chai Wah Wu, Jan 04 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Dec 01 2008
STATUS
approved