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%I #25 Sep 08 2022 08:45:30
%S 3,4,7,15,32,65,129,256,511,1023,2048,4097,8193,16384,32767,65535,
%T 131072,262145,524289,1048576,2097151,4194303,8388608,16777217,
%U 33554433,67108864,134217727,268435455,536870912,1073741825,2147483649
%N Binomial transform of periodic sequence (3, 1, 2).
%C The third sequence of "less twisted numbers"; this sequence, A130750 and A130752 form a "suite en trio" (cf. reference, p. 130).
%C First differences of A130752, second differences of A130750.
%C Sequence equals its third differences:
%C 3 4 7 15 32 65 129 256 511 1023
%C 1 3 8 17 33 64 127 255 512
%C 2 5 9 16 31 63 128 257
%C 3 4 7 15 32 65 129
%D P. Curtz, Exercise Book, manuscript, 1995.
%H G. C. Greubel, <a href="/A130755/b130755.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Vincenzo Librandi)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2).
%F G.f.: (3-5*x+4*x^2)/((1-2*x)*(1-x+x^2)).
%F a(0) = 3; a(1) = 4; a(2) = 7; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
%F a(n) = 2^(n+1) + A128834(n+2).
%F a(0) = 3; for n > 0, a(n) = 2*a(n-1) + A057079(n+3).
%t CoefficientList[Series[(3-5*x+4*x^2)/((1-2*x)*(1-x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{3,-3,2}, {3,4,7}, 30] (* _G. C. Greubel_, Jan 15 2018 *)
%o (Magma) m:=31; S:=[ [3, 1, 2][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // _Klaus Brockhaus_, Aug 03 2007
%o (Magma) I:=[3,4,7]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018
%o (PARI) {m=31; v=vector(m); v[1]=3; v[2]=4; v[3]=7; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ _Klaus Brockhaus_, Aug 03 2007
%o (PARI) {for(n=0, 30, print1(2^(n+1)+[1, 0, -1, -1, 0, 1][n%6+1], ","))} \\ _Klaus Brockhaus_, Aug 03 2007
%Y Cf. A010882 (periodic (1, 2, 3)), A128834 (periodic (0, 1, 1, 0, -1, -1)), A057079 (periodic (1, 2, 1, -1, -2, -1)), A130750 (first differences), A130752 (second differences).
%K nonn,easy
%O 0,1
%A _Paul Curtz_, Jul 13 2007
%E Edited and extended by _Klaus Brockhaus_, Aug 03 2007