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A130624 Binomial transform of A101000.

%I A130624 #25 Jan 01 2024 11:56:53
%S A130624 0,1,5,12,23,43,84,169,341,684,1367,2731,5460,10921,21845,43692,87383,
%T A130624 174763,349524,699049,1398101,2796204,5592407,11184811,22369620,
%U A130624 44739241,89478485,178956972,357913943,715827883,1431655764,2863311529,5726623061,11453246124
%N A130624 Binomial transform of A101000.
%H A130624 Vincenzo Librandi, <a href="/A130624/b130624.txt">Table of n, a(n) for n = 0..1000</a>
%H A130624 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 2).
%F A130624 a(0)=0, a(1)=1, a(2)=5; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
%F A130624 a(0)=0; a(n+1) = 2*a(n) + A119910(n).
%F A130624 G.f.: x*(1 + 2*x)/((1 - 2*x)*(1 - x + x^2)).
%F A130624 a(n) = 2^n + a(n-1) - a(n-2). - _Jon Maiga_, Nov 14 2018
%t A130624 LinearRecurrence[{3,-3,2},{0,1,5},40] (* _Harvey P. Dale_, Mar 05 2013 *)
%t A130624 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(2^n) + a[n-1] - a[n-2]}, a, {n, 50}] (* _Vincenzo Librandi_, Nov 15 2018 *)
%o A130624 (PARI) {m=32; v=concat([0, 1, 5], vector(m-3)); for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} /* _Klaus Brockhaus_, Jun 21 2007 */
%o A130624 (Magma) m:=32; S:=[[0, 1, 3][(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_, Jun 21 2007 */
%o A130624 (Magma) I:=[0,1,5]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Nov 15 2018
%Y A130624 Cf. A101000, A119910, A130625 (first differences), A130626 (second differences).
%K A130624 nonn
%O A130624 0,3
%A A130624 _Paul Curtz_, Jun 18 2007
%E A130624 Edited and extended by _Klaus Brockhaus_, Jun 21 2007