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%I #24 Jan 14 2023 12:41:51
%S 1,2,3,5,9,13,21,29,45,61,93,125,189,253,381,509,765,1021,1533,2045,
%T 3069,4093,6141,8189,12285,16381,24573,32765,49149,65533,98301,131069,
%U 196605,262141,393213,524285,786429,1048573,1572861,2097149,3145725
%N New record highs reached in A060030.
%H Harry J. Smith, <a href="/A060482/b060482.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).
%F a(n) = a(n-1) + 2^((n-1)/2) = 2*a(n-2) + 3 = a(n-1) + A016116(n-1) = A027383(n-1) - 1 = 2*A027383(n-3) + 1 = 4*A052955(n-4) + 1. a(2n) = 2^(n+1) - 3; a(2n+1) = 3*2^n - 3.
%F From _Colin Barker_, Jan 12 2013: (Start)
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n > 5.
%F G.f.: x*(2*x^4-x^2+x+1) / ((x-1)*(2*x^2-1)). (End)
%t LinearRecurrence[{1,2,-2},{1,2,3,5,9},50] (* _Harvey P. Dale_, Sep 11 2016 *)
%o (PARI) { for (n=1, 1000, if (n%2==0, m=n/2; a=2^(m + 1) - 3, m=(n - 1)/2; a=3*2^m - 3); if (n<3, a=n); write("b060482.txt", n, " ", a); ) } \\ _Harry J. Smith_, Jul 05 2009
%Y Cf. A060013, A060030.
%Y The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - _N. J. A. Sloane_, Jul 14 2022
%K nonn,easy
%O 1,2
%A _Henry Bottomley_, Mar 19 2001
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