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A233656 a(n) = 3*a(n-1) - 2*(n-1), with a(0) = 1.

%I A233656 #21 Mar 24 2022 04:21:47
%S A233656 1,3,7,17,45,127,371,1101,3289,9851,29535,88585,265733,797175,2391499,
%T A233656 7174469,21523377,64570099,193710263,581130753,1743392221,5230176623,
%U A233656 15690529827,47071589437,141214768265,423644304747,1270932914191,3812798742521,11438396227509,34315188682471
%N A233656 a(n) = 3*a(n-1) - 2*(n-1), with a(0) = 1.
%C A233656 Inverse binomial transform of this sequence is A090129.
%C A233656 Conjecture: Last digit of a(n) has repeating pattern of 20 digits as follows: {1, 3, 7, 7, 5, 7, 1, 1, 9, 1, 5, 5, 3, 5, 9, 9, 7, 9, 3, 3}, with an equal frequency of the five odd digits.
%H A233656 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,3).
%F A233656 a(n) = 3*a(n-1) - 2*(n-1), with a(0) = 1.
%F A233656 (a(n+1) - a(n))/2 = A007051(n).
%F A233656 From _Colin Barker_, Dec 17 2013: (Start)
%F A233656 a(n) = (1 + 3^n + 2*n)/2.
%F A233656 a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
%F A233656 G.f.: (x^2+2*x-1) / ((x-1)^2*(3*x-1)). (End)
%F A233656 E.g.f.: exp(x)*(1 + exp(2*x) + 2*x)/2. - _Stefano Spezia_, Mar 20 2022
%e A233656 a(1) = 3*1 - 2*0 = 3;
%e A233656 a(2) = 3*3 - 2*1 = 7.
%t A233656 LinearRecurrence[{5,-7,3},{1,3,7},30] (* _Harvey P. Dale_, Feb 13 2015 *)
%o A233656 (PARI) Vec((x^2+2*x-1)/((x-1)^2*(3*x-1)) + O(x^100)) \\ _Colin Barker_, Dec 17 2013
%Y A233656 Cf. A007051, A090129.
%K A233656 nonn,easy
%O A233656 0,2
%A A233656 _Richard R. Forberg_, Dec 14 2013