A275788 - OEIS

%I #29 Oct 09 2016 04:13:44

%S 0,1,3,7,13,25,49,99,199,399,797,1593,3185,6371,12743,25487,50973,

%T 101945,203889,407779,815559,1631119,3262237,6524473,13048945,

%U 26097891,52195783,104391567,208783133,417566265,835132529,1670265059,3340530119,6681060239

%N a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

%C a(n) and its successive differences:

%C 0,  1,  3,  7, 13, 25, 49, ...

%C 1,  2,  4,  6, 12, 24, 50, 100, ...

%C 1,  2,  2,  6, 12, 26, 50, 100, 198, ...

%C 1,  0,  4,  6, 14, 24, 50,  98, 200, 398, ...

%C -1, 4,  2,  8, 10, 26, 48, 102, 198, 400, 794, ...

%C 5, -2,  6,  2, 16, 22, 54,  96, 202, 394, 800, 1590, ...

%C -7, 8, -4, 14,  6, 32, 42, 106, 192, 406, 790, 1600, 3178, ...

%C ... .

%C Each row has the recurrence a(n) + a(n+3) = 7*2^n.

%C Main diagonal: 2*A001045(n).

%C Upper diagonals: A084214(n+1), 3*2^n, ... .

%C Subdiagonals: 2^n, A078008(n), A084214(n+1), -2^n, ... .

%C a(-n) = 0, 1/2, 3/4, 7/8, -1/16, -17/32, -49/64, 15/128, ... .

%C b(n), numerators of a(-n), and first differences:

%C 0, 1, 3,  7,  -1, -17, -49,  15, 143,  399,  -113, -1137, ...

%C 1, 2, 4, -8, -16, -32,  64, 128, 256, -512, -1024, ... = A000079(n)*A130151(n), not in the OEIS.

%H Colin Barker, <a href="/A275788/b275788.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,2).

%F From _Colin Barker_, Aug 09 2016: (Start)

%F a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.

%F G.f.: x*(1 + x + x^2) / ((1+x)*(1-2*x)*(1-x+x^2)).

%F (End)

%F a(n+3) = 7*2^n - a(n), a(0)=0, a(1)=1, a(2)=3.

%e a(1)=2*0+1=1, a(2)=2*1+1=3, a(2)=2*3+1=7, a(3)=2*7-1=13, a(4)=2*13-1=25, ... .

%t CoefficientList[Series[x (1 + x + x^2)/((1 + x) (1 - 2 x) (1 - x + x^2)), {x, 0, 33}], x] (* _Michael De Vlieger_, Aug 11 2016 *)

%t LinearRecurrence[{2,0,-1,2}, {0, 1, 3, 7}, 25] (* _G. C. Greubel_, Aug 16 2016 *)

%o (PARI) concat(0, Vec(x*(1+x+x^2)/((1+x)*(1-2*x)*(1-x+x^2)) + O(x^40))) \\ _Colin Barker_, Aug 10 2016

%Y Cf. A000079, A001045, A002264, A005009, A007283, A078008, A084214, A113405, A130151, A274817.

%K nonn

%O 0,3

%A _Paul Curtz_, Aug 09 2016

%E More terms from _Colin Barker_, Aug 10 2016