A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.

2, 3, 0, 1, 4, 5, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 18, 19, 16, 17, 20, 21, 22, 23, 26, 27, 24, 25, 28, 29, 30, 31, 34, 35, 32, 33, 36, 37, 38, 39, 42, 43, 40, 41, 44, 45, 46, 47, 50, 51, 48, 49, 52, 53, 54, 55, 58, 59, 56, 57, 60, 61, 62, 63, 66, 67, 64

Offset

0,1

Comments

Write n in base 8, then apply the following substitution to the rightmost digit: '0'->'2, '1'->'3', and vice versa. Convert back to decimal.

A self-inverse permutation: a(a(n)) = n.

Array whose columns are, in this order, A047463, A047621, A047451 and A047522, read by rows.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

OEIS wiki, Lodumo transform.

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

Index entries for sequences that are permutations of the natural numbers.

Formula

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - a(n-8), n > 7.

a(n) = (4*(floor(((2*n + 4) mod 8)/4) - floor(((n + 2) mod 8)/4)) + 2*n)/2.

a(n) = lod_4(A047247(n+1)).

a(4*n) = A047463(n+1).

a(4*n+1) = A047621(n+1).

a(4*n+2) = A047451(n+1).

a(4*n+3) = A047522(n+1).

a(A042948(n)) = A047596(n+1).

a(A042964(n+1)) = A047551(n+1).

G.f.: (x^7 + x^5 + 3*x^3 - 2*x^2 - x + 2)/((x-1)^2 * (x^2+1) * (x^4+1)).

E.g.f.: x*exp(x) + cos(x) + sin(x) + cos(x/sqrt(2))*cosh(x/sqrt(2)) + (sqrt(2)*cos(x/sqrt(2)) - sin(x/sqrt(2)))*sinh(x/sqrt(2)).

a(n+8) = a(n) + 8 . - Philippe Deléham, Mar 09 2023

Sum_{n>=3} (-1)^(n+1)/a(n) = 1/6 + log(2). - Amiram Eldar, Mar 12 2023

Example

a(25) = a('3'1') = '3'3' = 27.
a(26) = a('3'2') = '3'0' = 24.
a(27) = a('3'3') = '3'1' = 25.
a(28) = a('3'4') = '3'4' = 28.
a(29) = a('3'5') = '3'5' = 29.
The sequence as array read by rows:
  A047463, A047621, A047451, A047522;
        2,       3,       0,       1;
        4,       5,       6,       7;
       10,      11,       8,       9;
       12,      13,      14,      15;
       18,      19,      16,      17;
       20,      21,      22,      23;
       26,      27,      24,      25;
       28,      29,      30,      31;
  ...

Mathematica

Table[(4*(Floor[1/4 Mod[2*n + 4, 8]] - Floor[1/4 Mod[n + 2, 8]]) + 2*n)/2, {n, 0, 100}]
f[n_] := Block[{id = IntegerDigits[n, 8]}, FromDigits[ Join[Most@ id /. {{} -> {0}}, {id[[-1]] /. {0 -> 2, 1 -> 3, 2 -> 0, 3 -> 1}}], 8]]; Array[f, 67, 0] (* or *)
CoefficientList[ Series[(x^7 + x^5 + 3x^3 - 2x^2 - x + 2)/((x - 1)^2 (x^6 + x^4 + x^2 + 1)), {x, 0, 70}], x] (* or *)
LinearRecurrence[{2, -2, 2, -2, 2, -2, 2, -1}, {2, 3, 0, 1, 4, 5, 6, 7}, 70] (* Robert G. Wilson v, Aug 01 2018 *)

Prog

(Maxima) makelist((4*(floor(mod(2*n + 4, 8)/4) - floor(mod(n + 2, 8)/4)) + 2*n)/2, n, 0, 100);
(PARI) my(x='x+O('x^100)); Vec((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1))) \\ G. C. Greubel, Sep 25 2018
(Magma) m:=100; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1)))); // G. C. Greubel, Sep 25 2018

Crossrefs

Cf. A047225, A047243, A047257, A064429, A080412, A159959, A026185.

Sequence in context: A355889 A275736 A276074 * A292628 A376586 A163465

Adjacent sequences: A317610 A317611 A317612 * A317614 A317615 A317616

Keyword

nonn,easy,base

Author

Franck Maminirina Ramaharo, Aug 01 2018

Status

approved