A133095 - OEIS

%I #16 Sep 08 2022 08:45:31

%S 1,4,9,20,43,90,185,376,759,1526,3061,6132,12275,24562,49137,98288,

%T 196591,393198,786413,1572844,3145707,6291434,12582889,25165800,

%U 50331623,100663270,201326565,402653156,805306339,1610612706,3221225441,6442450912,12884901855,25769803742,51539607517

%N Row sums of triangle A133094.

%H G. C. Greubel, <a href="/A133095/b133095.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 (4,-5,2).

%F Binomial transform of [1, 3, 2, 4, 2, 4, 2, 4, ...].

%F From _G. C. Greubel_, Oct 21 2017: (Start)

%F a(n) = 3*2^(n-1) - n, for n >= 2.

%F a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).

%F G.f.: x*(1 - 2*x^2 + 2*x^3)/((1-2*x)*(1-x)^2).

%F E.g.f.: (3*exp(2*x) - 2*x*exp(x) - 2*x - 3)/2. (End)

%F a(n) = A123720(n) for n >= 2. - _Georg Fischer_, Nov 02 2018

%e a(4) = 20 = sum of row 4 terms of triangle A133094: (7 + 7 + 5 + 1).

%e a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 2, 4) = (1 + 9 + 6 + 4).

%t Join[{1}, Table[ 3*2^(n - 1) - n, {n, 2, 50}]] (* _G. C. Greubel_, Oct 21 2017 *)

%t LinearRecurrence[{4,-5,2},{1,4,9,20},50] (* _Harvey P. Dale_, Aug 09 2022 *)

%o (PARI) concat(1, for(n=1,50, print1(3*2^(n - 1) - n, ", "))) \\ _G. C. Greubel_, Oct 21 2017

%o (Magma) [1] cat [3*2^(n - 1) - n: n in [2..50]]; // _G. C. Greubel_, Oct 21 2017

%Y Cf. A123720, A133094.

%K nonn

%O 1,2

%A _Gary W. Adamson_, Sep 09 2007

%E Terms a(11) onward added by _G. C. Greubel_, Oct 21 2017