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A353094 a(1) = 2; for n > 1, a(n) = 3*a(n-1) + 3 - n.

%I A353094 #26 May 28 2023 22:21:29
%S A353094 2,7,21,62,184,549,1643,4924,14766,44291,132865,398586,1195748,
%T A353094 3587233,10761687,32285048,96855130,290565375,871696109,2615088310,
%U A353094 7845264912,23535794717,70607384131,211822152372,635466457094,1906399371259,5719198113753,17157594341234
%N A353094 a(1) = 2; for n > 1, a(n) = 3*a(n-1) + 3 - n.
%H A353094 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,3).
%F A353094 G.f.: x * (2 - 3*x)/((1 - x)^2 * (1 - 3*x)).
%F A353094 a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
%F A353094 a(n) = A000340(n-1) + n.
%F A353094 a(n) = (3^(n+1) + 2*n - 3)/4.
%F A353094 a(n) = Sum_{k=0..n-1} (3 - n + k) * 3^k.
%F A353094 E.g.f.: exp(x)*(3*exp(2*x) + 2*x - 3)/4. - _Stefano Spezia_, May 28 2023
%t A353094 LinearRecurrence[{5, -7, 3}, {2, 7, 21}, 28] (* _Amiram Eldar_, Apr 23 2022 *)
%o A353094 (PARI) my(N=30, x='x+O('x^N)); Vec(x*(2-3*x)/((1-x)^2*(1-3*x)))
%o A353094 (PARI) a(n) = (3^(n+1)+2*n-3)/4;
%o A353094 (PARI) b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
%o A353094 a(n) = b(n, 3);
%Y A353094 Cf. A064617, A353095, A353096, A353097, A353098, A353099, A353100.
%Y A353094 Cf. A000340.
%K A353094 nonn,easy
%O A353094 1,1
%A A353094 _Seiichi Manyama_, Apr 23 2022