Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

New record highs reached in A060030.
25

%I #28 Jul 25 2024 19:07:14

%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 196605,262141,393213,524285,786429,1048573,1572861,2097149,3145725,4194301,6291453

%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)

%F E.g.f.: 1 + x + x^2/2 - 3*cosh(x) + 2*cosh(sqrt(2)*x) - 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2). - _Stefano Spezia_, Jul 25 2024

%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