A152271 a(n)=1 for even n and (n+1)/2 for odd n.

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

Links

Table of n, a(n) for n=0..87.

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

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

Cf. A000012, A000027, A008619, A057979, A128218.

Sequence in context: A177815 A007879 A057979 * A133622 A158416 A318225

Adjacent sequences: A152268 A152269 A152270 * A152272 A152273 A152274

Keyword

nonn,easy

Author

Philippe Deléham, Dec 01 2008

Status

approved