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

Author

Philippe Deléham, Dec 01 2008

Keywords

• nonn
• easy

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

Examples

			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
		

Links

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

Crossrefs

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

Programs

Haskell

import Data.List (transpose)
a152271 = a057979 . (+ 2)
a152271_list = concat $ transpose [repeat 1, [1..]]
-- Reinhard Zumkeller, Aug 11 2014

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 *)

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

Python

def A152271(n): return n+1>>1 if n&1 else 1 # Chai Wah Wu, Jan 04 2024

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