A225055 -id:A225055 - cp's OEIS frontend

A246416 A permutation of essentially the duplicate nonnegative numbers: a(4n) = n + 1/2 - (-1)^n/2, a(2n+1) = a(4n+2) = 2n+1.

0, 1, 1, 3, 2, 5, 3, 7, 2, 9, 5, 11, 4, 13, 7, 15, 4, 17, 9, 19, 6, 21, 11, 23, 6, 25, 13, 27, 8, 29, 15, 31, 8, 33, 17, 35, 10, 37, 19, 39, 10, 41, 21, 43, 12, 45, 23, 47, 12, 49, 25, 51, 14, 53, 27, 55, 14, 57, 29, 59, 16, 61, 31, 63, 16

Offset: 0

Paul Curtz, Sep 14 2014

A permutation of A004526 (n > 0).
0 is at its own place. Distance between the two (2*k+1)'s: 2*k+1 terms. 0 is in position 0, the first 1 in position 1, the second 1 in position 2, the first 2 in position 4, the second 2 in position 8. Hence, r(n) = 0, 1, 2, 4, 8, 3, 6, 12, 16, 5, 10, 20, 24, ..., a permutation of A001477. See A225055. The recurrence r(n) = r(n-4) + r(n-8) - r(n-12) is the same as for a(n).
A061037(n+3) is divisible by a(n+5) (= 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 3, ...). Hence a link, via A212831 and A214282, between the Catalan numbers A000108 and the Balmer series.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,1,0,0,0,-1).

Cf. A000108, A004526, A052928, A061037, A176895, A212831, A214282, A226279, A225055, A247617.

I:=[0,1,1,3,2,5,3,7,2,9,5,11,4,13,7,15,4,17,9,19,6,21,11,23]; [n le 24 select I[n] else 3*Self(n-8)-3*Self(n-16)+Self(n-24): n in [1..80]]; // Vincenzo Librandi, Oct 15 2014

A246416:=n->n*(1+floor((2-n)/4)+floor((n-2)/4))/2+n*(1+floor((1-n)/2)+floor((n-1)/2))+(-n-2+2*(-1)^(n/4))*(ceil(n/4)-floor(n/4)-1)/4: seq(A246416(n), n=0..50); # Wesley Ivan Hurt, Sep 14 2014

Table[n (1 + Floor[(2 - n)/4] + Floor[(n - 2)/4])/2 + n (1 + Floor[(1 - n)/2] + Floor[(n - 1)/2]) + (-n - 2 + 2 (-1)^(n/4)) (Ceiling[n/4] - Floor[n/4] - 1)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 14 2014 *)
a[n_] := Switch[Mod[n, 4], 0, n/4-(-1)^(n/4)/2+1/2, 1|3, n, 2, n/2]; Table[a[n], {n, 0, 64}] (* Jean-François Alcover, Oct 09 2014 *)
LinearRecurrence[{0,0,0,1,0,0,0,1,0,0,0,-1},{0,1,1,3,2,5,3,7,2,9,5,11},70] (* Harvey P. Dale, Mar 23 2015 *)

a(n)=if(n%4,n/(2-n%2),if(n%8,1,0)+n/4) \\ Charles R Greathouse IV, Sep 14 2014

a(n) = 3*a(n-8) - 3*a(n-16) + a(n-24).
a(n+4) = a(n) + period 8: repeat [2, 4, 2, 4, 0, 4, 2, 4].
a(n+8) = a(n) + period 4: repeat [2, 8, 4, 8] (= 2 * A176895).
a(2n) = A212831(n).
a(n) = n*(1+floor((2-n)/4)+floor((n-2)/4))/2+n*(1+floor((1-n)/2)+floor((n-1)/2))+(-n-2+2*(-1)^(n/4))*(ceiling(n/4)-floor(n/4)-1)/4. - Wesley Ivan Hurt, Sep 14 2014
a(n) = a(n-4) + a(n-8) - a(n-12). - Charles R Greathouse IV, Sep 14 2014
G.f.: x*(x^10+x^9+3*x^8+4*x^6+2*x^5+4*x^4+2*x^3+3*x^2+x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)). - Colin Barker, Sep 15 2014

A247555 A permutation of the nonnegative numbers: a(4n) = 8n, a(4n+1) = 2n + 1, a(4n+2) = 4n + 2, a(4n+3) = 8n + 4.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 6, 12, 16, 5, 10, 20, 24, 7, 14, 28, 32, 9, 18, 36, 40, 11, 22, 44, 48, 13, 26, 52, 56, 15, 30, 60, 64, 17, 34, 68, 72, 19, 38, 76, 80, 21, 42, 84, 88, 23, 46, 92, 96, 25, 50, 100, 104, 27, 54, 108, 112, 29, 58, 116, 120

Offset: 0

Paul Curtz, Sep 19 2014

A permutation of the nonnegative integers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A005408, A008590, A016825, A017113, A225055, A001477, A111284.

&cat[[4*(i-1),i,2*i,4*i]: i in [1..50 by 2]]; // Bruno Berselli, Sep 19 2014

a[n_]:=Switch[Mod[n,4],0,2 n,1,(n+1)/2,2,n,3,2 n-2]; Table[a[n],{n,0,60}] (* Jean-François Alcover, Oct 09 2014 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1}, {0,1,2,4,8,3,6,12}, 50] (* G. C. Greubel, May 01 2018 *)

Vec(x*(4*x^6+2*x^5+x^4+8*x^3+4*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100)) \\ Colin Barker, Sep 19 2014

a(n) = a(n-4) + a(n-8) - a(n-12).
a(n) = 2*a(n-4) - a(n-8). - Colin Barker, Sep 19 2014
G.f.: x*(4*x^6 + 2*x^5 + x^4 + 8*x^3 + 4*x^2 + 2*x + 1) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, Sep 19 2014
a(n) = (11*n-3+(n+3)*(-1)^n+(4*n-1+(-1)^n)*cos(n*Pi/2)+2*(9-3*n+4(-1)^n)* sin(n*Pi/2))/8. - Wesley Ivan Hurt, May 07 2021

cp's OEIS Frontend

A246416 A permutation of essentially the duplicate nonnegative numbers: a(4n) = n + 1/2 - (-1)^n/2, a(2n+1) = a(4n+2) = 2n+1.

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