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A262592 a(n) = (3^(n+1) - 2n^2 + 4n + 5) / 8.

%I A262592 #25 Jul 18 2025 13:22:10
%S A262592 1,2,4,10,29,88,268,812,2449,7366,22124,66406,199261,597836,1793572,
%T A262592 5380792,16142465,48427498,145282612,435847970,1307544061,3922632352,
%U A262592 11767897244,35303691940,105911076049,317733228398,953199685468,2859599056702,8578797170429,25736391511636
%N A262592 a(n) =  (3^(n+1) - 2n^2 + 4n + 5) / 8.
%H A262592 Colin Barker, <a href="/A262592/b262592.txt">Table of n, a(n) for n = 0..1000</a>
%H A262592 K. Satyanarayana, <a href="/A262592/a262592.pdf">Sequences whose kth differences form a geometrical progression</a>, Math. Student, 12 (1944), page 109. [Annotated scanned copy. This sequence was formerly A2752 but has now been renumbered]
%H A262592 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,10,-3).
%F A262592 G.f.: (1-2*x)^2/((1-x)^3*(1-3*x)).
%F A262592 a(n) = 6*a(n-1)-12*a(n-2)+10*a(n-3)-3*a(n-4) for n>3. - _Colin Barker_, Oct 23 2015
%p A262592 f1:=(a,b)->(1-a*x)^a/((1-x)^b*(1-b*x));
%p A262592 f2:=(a,b)->seriestolist(series(f1(a,b),x,40));
%p A262592 f2(2,3);
%t A262592 Table[3^(n + 1)/8 + 5/8 - n^2/4 + n/2, {n, 0, 29}] (* _Michael De Vlieger_, Oct 23 2015 *)
%t A262592 LinearRecurrence[{6,-12,10,-3},{1,2,4,10},30] (* _Harvey P. Dale_, Jul 18 2025 *)
%o A262592 (PARI) a(n) = 3^(n+1)/8+5/8-n^2/4+n/2 \\ _Colin Barker_, Oct 23 2015
%o A262592 (PARI) Vec((1-2*x)^2/((1-x)^3*(1-3*x)) + O(x^40)) \\ _Colin Barker_, Oct 23 2015
%Y A262592 Other sequences with generating functions like this: A000340, A052161, A262593, A262594.
%K A262592 nonn,easy
%O A262592 0,2
%A A262592 _N. J. A. Sloane_, Oct 21 2015