A133622 a(n) = 1 if n is odd, a(n) = n/2+1 if n is even.

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, 1

Offset: 1

Author

Hieronymus Fischer, Sep 30 2007

Keywords

• nonn
• easy

Comments

a(n) is the count of terms a(n+1) present so far in the sequence, with a(n+1) included in the count; example: a(1) = 1 "says" that there is 1 term "2" so far in the sequence; a(2) = 2 "says" that there are 2 terms "1" so far in the sequence... etc. This comment was inspired by A039617. - Eric Angelini, Mar 03 2020

Links

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

Crossrefs

Cf. A133620, A133621, A133623, A133624, A133625, A133630, A038509, A133634-A133636, A133872, A133882, A133880, A133890, A133900, A133910, A165157 (partial sums).

Other related sequences: A000217, A007879, A057979.

Programs

Haskell

import Data.List (transpose)
a133622 n = (1 - m) * n' + 1 where (n', m) = divMod n 2
a133622_list = concat $ transpose [[1, 1 ..], [2 ..]]
-- Reinhard Zumkeller, Feb 20 2015

Maple

seq([1,n][],n=2..100); # Robert Israel, May 27 2016

Mathematica

Riffle[Range[2,50],1,{1,-1,2}] (* Harvey P. Dale, Jan 19 2013 *)

PARI

a(n)=if(n%2,1,n/2+1) \\ Charles R Greathouse IV, Sep 02 2015

Formula

a(n)=1+(binomial(n+1,2)mod n)=1+(binomial(n+1,n-1)mod n).

a(n)=binomial(n+2,2) mod n = binomial(n+2,n) mod n for n>2.

a(n)=1+(1+(-1)^n)*n/4.

a(n)=1+(A000217(n) mod n).

a(n)=a(n-2)+1, if n is even, a(n)=a(n-2) if n is odd.

a(n)=a(n-2)+1-(n mod 2)=a(n-2)+(1+(-1)^n)/2 for n>2.

a(n)=(a(n-3)+a(n-2))/a(n-1) for n>3.

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

G.f.: (Q(0)-1-x)/x^2, where Q(k)= 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 23 2013

a(n) = 2*a(n-2)-a(n-4) for n > 4. - Chai Wah Wu, May 26 2016

E.g.f.: exp(x) - 1 + x*sinh(x)/2. - Robert Israel, May 27 2016