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
Examples
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;
...
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.
Programs
-
Magma
m:=100; R
:=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 -
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 *) -
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
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).
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
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