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%I #65 May 14 2023 02:10:37
%S 1,3,5,9,13,21,29,45,61,93,125,189,253,381,509,765,1021,1533,2045,
%T 3069,4093,6141,8189,12285,16381,24573,32765,49149,65533,98301,131069,
%U 196605,262141,393213,524285,786429,1048573,1572861,2097149,3145725,4194301,6291453,8388605
%N a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
%C For n >= 2, number of n X n arrays with values that are squares of integers, having all 2 X 2 subblocks summing to 4. - _R. H. Hardin_, Apr 03 2009
%C Number of moves required in 4-peg Tower of Hanoi solution using a (suboptimal) recursive algorithm: Move (n-2) disks, move bottom 2 disks, move (n-2) disks. Cf. A007664. - _Toby Gottfried_, Nov 29 2010
%H G. C. Greubel, <a href="/A136252/b136252.txt">Table of n, a(n) for n = 0..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) = 2^((1/2)*n-1)*(4 + 4(-1)^n + 3*sqrt(2)*(1-(-1)^n)) - 3. - _Emeric Deutsch_, Mar 31 2008
%F G.f.: (1+2*x)/((1-x)*(1-2*x^2)). - _Jaume Oliver Lafont_, Aug 30 2009
%F a(n) = 2*a(n-2) + 3; first differences are powers of 2, occurring in pairs. - _Toby Gottfried_, Nov 29 2010
%F a(n) = A027383(n+1) - 1. - _Jason Kimberley_, Nov 01 2011
%F a(2n+1) = (a(2n) + a(2n+2))/2. - _Richard R. Forberg_, Nov 30 2013
%F E.g.f.: 4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) - 3*cosh(x) - 3*sinh(x). - _Stefano Spezia_, May 13 2023
%p a:=proc(n) options operator,arrow: 2^((1/2)*n-1)*(4+4*(-1)^n+3*sqrt(2)*(1-(-1)^n))-3 end proc: seq(a(n),n=0..40); # _Emeric Deutsch_, Mar 31 2008
%t LinearRecurrence[{1, 2, -2}, {1, 3, 5}, 100] (* _G. C. Greubel_, Feb 18 2017 *)
%o (PARI) x='x+O('x^50); Vec((1+2*x)/((1-x)*(1-2*x^2))) \\ _G. C. Greubel_, Feb 18 2017
%Y Same recurrence as in A135530.
%Y Partial sums of A163403.
%Y A060482 without the term 2.
%Y Cf. A007664 (Optimal 4-peg Tower of Hanoi).
%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 0,2
%A _Paul Curtz_, Mar 17 2008
%E Edited by _N. J. A. Sloane_, Apr 18 2008
%E More terms from _Emeric Deutsch_, Mar 31 2008
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