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A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.

%I A317613 #33 Mar 12 2023 04:19:45
%S A317613 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,
%T A317613 24,25,28,29,30,31,34,35,32,33,36,37,38,39,42,43,40,41,44,45,46,47,50,
%U A317613 51,48,49,52,53,54,55,58,59,56,57,60,61,62,63,66,67,64
%N A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.
%C A317613 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.
%C A317613 A self-inverse permutation: a(a(n)) = n.
%C A317613 Array whose columns are, in this order, A047463, A047621, A047451 and A047522, read by rows.
%H A317613 G. C. Greubel, <a href="/A317613/b317613.txt">Table of n, a(n) for n = 0..10000</a>
%H A317613 OEIS wiki, <a href="https://oeis.org/wiki/Lodumo_transform">Lodumo transform</a>.
%H A317613 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-2,2,-1).
%H A317613 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A317613 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.
%F A317613 a(n) = (4*(floor(((2*n + 4) mod 8)/4) - floor(((n + 2) mod 8)/4)) + 2*n)/2.
%F A317613 a(n) = lod_4(A047247(n+1)).
%F A317613 a(4*n) = A047463(n+1).
%F A317613 a(4*n+1) = A047621(n+1).
%F A317613 a(4*n+2) = A047451(n+1).
%F A317613 a(4*n+3) = A047522(n+1).
%F A317613 a(A042948(n)) = A047596(n+1).
%F A317613 a(A042964(n+1)) = A047551(n+1).
%F A317613 G.f.: (x^7 + x^5 + 3*x^3 - 2*x^2 - x + 2)/((x-1)^2 * (x^2+1) * (x^4+1)).
%F A317613 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)).
%F A317613 a(n+8) = a(n) + 8 . - _Philippe Deléham_, Mar 09 2023
%F A317613 Sum_{n>=3} (-1)^(n+1)/a(n) = 1/6 + log(2). - _Amiram Eldar_, Mar 12 2023
%e A317613 a(25) = a('3'1') = '3'3' = 27.
%e A317613 a(26) = a('3'2') = '3'0' = 24.
%e A317613 a(27) = a('3'3') = '3'1' = 25.
%e A317613 a(28) = a('3'4') = '3'4' = 28.
%e A317613 a(29) = a('3'5') = '3'5' = 29.
%e A317613 The sequence as array read by rows:
%e A317613   A047463, A047621, A047451, A047522;
%e A317613         2,       3,       0,       1;
%e A317613         4,       5,       6,       7;
%e A317613        10,      11,       8,       9;
%e A317613        12,      13,      14,      15;
%e A317613        18,      19,      16,      17;
%e A317613        20,      21,      22,      23;
%e A317613        26,      27,      24,      25;
%e A317613        28,      29,      30,      31;
%e A317613   ...
%t A317613 Table[(4*(Floor[1/4 Mod[2*n + 4, 8]] - Floor[1/4 Mod[n + 2, 8]]) + 2*n)/2, {n, 0, 100}]
%t A317613 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 *)
%t A317613 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 *)
%t A317613 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 *)
%o A317613 (Maxima) makelist((4*(floor(mod(2*n + 4, 8)/4) - floor(mod(n + 2, 8)/4)) + 2*n)/2, n, 0, 100);
%o A317613 (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
%o A317613 (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
%Y A317613 Cf. A047225, A047243, A047257, A064429, A080412, A159959, A026185.
%K A317613 nonn,easy,base
%O A317613 0,1
%A A317613 _Franck Maminirina Ramaharo_, Aug 01 2018