A097065 Interleave n+1 and n-1.

1, -1, 2, 0, 3, 1, 4, 2, 5, 3, 6, 4, 7, 5, 8, 6, 9, 7, 10, 8, 11, 9, 12, 10, 13, 11, 14, 12, 15, 13, 16, 14, 17, 15, 18, 16, 19, 17, 20, 18, 21, 19, 22, 20, 23, 21, 24, 22, 25, 23, 26, 24, 27, 25, 28, 26, 29, 27, 30, 28, 31, 29, 32, 30, 33, 31, 34, 32, 35, 33, 36, 34, 37, 35, 38

Offset

0,3

Comments

Pairwise sums are abs(A023443), or n - 1 + 2*0^n. The partial sums of this sequence is A000124, with extra leading 1. Partial sums are A097066. Binomial transform is A097067.

Links

Table of n, a(n) for n=0..74.

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

Formula

G.f.: (1 - 2*x + 2*x^2)/((1 + x)*(1 - x)^2).

a(n) = (2*n - 1)/4 + 5*(-1)^n/4.

a(n) = floor((n+2)/2) - 2 * (n mod 2). - Reinhard Zumkeller, Apr 06 2015

a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2. - Wesley Ivan Hurt, Jan 10 2017

E.g.f.: ((2 + x)*cosh(x) - (3 - x)*sinh(x))/2. - Stefano Spezia, Jul 01 2023

Maple

A097065:=n->(2*n-1)/4 + 5*(-1)^n/4: seq(A097065(n), n=0..150); # Wesley Ivan Hurt, Jan 10 2017

Mathematica

Table[(2n - 1)/4 + 5(-1)^n/4, {n, 0, 75}] (* Or *) Flatten[ Table[{n + 1, n - 1}, {n, 0, 37}]] (* Or *) CoefficientList[Series[(1 - 2x + 2x^2)/((1 + x)(1 - x)^2), {x, 0, 75}], x] (* Robert G. Wilson v, Jul 24 2004 *)

Prog

(Haskell)
import Data.List (transpose)
a097065 n = n' - 2 * m where (n', m) = divMod (n + 2) 2
a097065_list = concat $ transpose [[1 ..], [-1 ..]]
(PARI) a(n)=n\2+1-n%2*2 \\ Charles R Greathouse IV, Sep 02 2015
(Magma) [(2*n-1)/4 + 5*(-1)^n/4 : n in [0..100]]; // Wesley Ivan Hurt, Jan 10 2017

Crossrefs

Essentially the same as A084964.

Cf. A000124, A023443, A097066, A097067.

Sequence in context: A195826 A331478 A378500 * A084964 A267182 A008720

Adjacent sequences: A097062 A097063 A097064 * A097066 A097067 A097068

Keyword

easy,sign

Author

Paul Barry, Jul 22 2004

Status

approved