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A263449
Permutation of the natural numbers: [4k+1, 4k+4, 4k+3, 4k+2, ...].
3
1, 4, 3, 2, 5, 8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 17, 20, 19, 18, 21, 24, 23, 22, 25, 28, 27, 26, 29, 32, 31, 30, 33, 36, 35, 34, 37, 40, 39, 38, 41, 44, 43, 42, 45, 48, 47, 46, 49, 52, 51, 50, 53, 56, 55, 54, 57, 60, 59, 58, 61, 64, 63, 62, 65, 68, 67
OFFSET
0,2
COMMENTS
From Franklin T. Adams-Watters, Jul 13 2017: (Start)
For this to be a permutation, it should have offset one, not zero.
With offset 1, a(n) is the smallest positive integer == n (mod 2) with a(n) != a(n-1) + 1. (End)
FORMULA
G.f.: (1+2*x-3*x^2+2*x^3)/((x-1)^2*(1+x^2)).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>3.
a(n) = n+1+(1-(-1)^n)*(-1)^(n*(n-1)/2).
a(2n) = A005408(n), a(2n+1) = 2*A014681(n+1).
a(n) = n+1+i*((-i)^n-i^n), where i=sqrt(-1). - Colin Barker, Oct 27 2015
a(n) = 4*ceiling(n/4) - (n mod 4) + 1. - Wesley Ivan Hurt, Nov 07 2015
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023
MAPLE
A263449:=n->n+1+(1-(-1)^n)*(-1)^(n*(n-1)/2): seq(A263449(n), n=0..100);
MATHEMATICA
Table[n + 1 + (1 - (-1)^n) (-1)^(n (n - 1)/2), {n, 0, 100}] (* or *) LinearRecurrence[{2, -2, 2, -1}, {1, 4, 3, 2}, 70]
PROG
(Magma) [n+1+(1-(-1)^n)*(-1)^(n*(n-1) div 4) : n in [0..100]];
(Magma) /* By definition: */ &cat[[4*k+1, 4*k+4, 4*k+3, 4*k+2]: k in [0..20]]; // Bruno Berselli, Oct 19 2015
(PARI) Vec((1+2*x-3*x^2+2*x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ Altug Alkan, Oct 19 2015
(PARI) a(n) = n+1+I*((-I)^n-I^n) \\ Colin Barker, Oct 27 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 18 2015
STATUS
approved