A065261 -id:A065261

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A065262

The nonpositive side (-1, -2, -3, ...) of the site swap sequence A065261. The bisection of odd terms of A065261.

+20
2

1, 1, 5, 2, 9, 3, 13, 4, 17, 5, 21, 6, 25, 7, 29, 8, 33, 9, 37, 10, 41, 11, 45, 12, 49, 13, 53, 14, 57, 15, 61, 16, 65, 17, 69, 18, 73, 19, 77, 20, 81, 21, 85, 22, 89, 23, 93, 24, 97, 25, 101, 26, 105, 27, 109, 28, 113, 29, 117, 30, 121, 31, 125, 32, 129, 33, 133, 34, 137, 35

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

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

FORMULA

G.f.: x*(3*x^2+x+1) / ((x-1)^2*(x+1)^2). [Colin Barker, Feb 18 2013]

From Wesley Ivan Hurt, Dec 06 2015: (Start)

a(n) = 2*a(n-2)-a(n-4) for n>4.

a(n) = n - Sum_{i=1..n} ceiling( (-1)^i*(2*n-3+i)/2 ). (End)

MAPLE

a:=proc(n) option remember; if n=1 then 1 elif n=2 then 1 elif n=3 then 5 elif n=4 then 2 else 2*a(n-2)-a(n-4); fi; end: seq(a(n), n=1..100); # Wesley Ivan Hurt, Dec 06 2015

MATHEMATICA

CoefficientList[Series[(3*x^2 + x + 1)/(x^2 - 1)^2, {x, 0, 100}], x] (* or *) LinearRecurrence[{0, 2, 0, -1}, {1, 1, 5, 2}, 100] (* Wesley Ivan Hurt, Dec 06 2015 *)

PROG

(PARI) Vec(x*(3*x^2+x+1) / ((x-1)^2*(x+1)^2) + O(x^100)) \\ Michel Marcus, Dec 06 2015

(PARI) vector(100, n, n - sum(i=1, n, ceil((-1)^i*(2*n-3+i)/2 ))) \\ Altug Alkan, Dec 06 2015

(Magma) I:=[1, 1, 5, 2]; [n le 4 select I[n] else 2*Self(n-2)-Self(n-4) : n in [1..100]]; // Wesley Ivan Hurt, Dec 06 2015

CROSSREFS

Cf. A065261.

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Oct 28 2001

STATUS

approved

A065260

A057115 conjugated with A059893, inverse of A065259.

+10
6

2, 4, 1, 8, 6, 12, 3, 16, 10, 20, 5, 24, 14, 28, 7, 32, 18, 36, 9, 40, 22, 44, 11, 48, 26, 52, 13, 56, 30, 60, 15, 64, 34, 68, 17, 72, 38, 76, 19, 80, 42, 84, 21, 88, 46, 92, 23, 96, 50, 100, 25, 104, 54, 108, 27, 112, 58, 116, 29, 120, 62, 124, 31, 128, 66, 132, 33, 136, 70

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This permutation of N induces also such permutation of Z, that p(i)-i >= 0 for all i.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.

Index entries for sequences that are permutations of the natural numbers

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

FORMULA

a(n) = A059893(A057115(A059893(n))).

a(2*k+2) = 4*k+4, a(4*k+1) = 4*k+2, a(4*k+3) = 2*k+1. - Ralf Stephan, Jun 10 2005

G.f.: x*(x^6+4*x^5+2*x^4+8*x^3+x^2+4*x+2) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, Feb 18 2013

a(n) = 2*a(n-4) - a(n-8) for n>8. - Colin Barker, Oct 29 2016

a(n) = (11*n+1+(5*n-1)*(-1)^n-(n+3)*(1-(-1)^n)*(-1)^((2*n+3+(-1)^n)/4))/8. - Luce ETIENNE, Oct 20 2016

EXAMPLE

G.f. = 2*x + 4*x^2 + x^3 + 8*x^4 + 6*x^5 + 12*x^6 + 3*x^7 + 16*x^8 + ...

PROG

(PARI) Vec(x*(2+4*x+x^2+8*x^3+2*x^4+4*x^5+x^6)/((1-x)^2*(1+x)^2*(1+x^2)^2) + O(x^100)) \\ Colin Barker, Oct 29 2016

(PARI) {a(n) = if( n%2==0, n*2, n%4==1, n+1, n\2)}; /* Michael Somos, Nov 06 2016 */