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%I #21 Oct 24 2015 12:04:21
%S 1,2,4,10,29,88,268,812,2449,7366,22124,66406,199261,597836,1793572,
%T 5380792,16142465,48427498,145282612,435847970,1307544061,3922632352,
%U 11767897244,35303691940,105911076049,317733228398,953199685468,2859599056702,8578797170429,25736391511636
%N a(n) = (3^(n+1) - 2n^2 + 4n + 5) / 8..
%H Colin Barker, <a href="/A262592/b262592.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,10,-3).
%F G.f.: (1-2*x)^2/((1-x)^3*(1-3*x)).
%F 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 f1:=(a,b)->(1-a*x)^a/((1-x)^b*(1-b*x));
%p f2:=(a,b)->seriestolist(series(f1(a,b),x,40));
%p f2(2,3);
%t Table[3^(n + 1)/8 + 5/8 - n^2/4 + n/2, {n, 0, 29}] (* _Michael De Vlieger_, Oct 23 2015 *)
%o (PARI) a(n) = 3^(n+1)/8+5/8-n^2/4+n/2 \\ _Colin Barker_, Oct 23 2015
%o (PARI) Vec((1-2*x)^2/((1-x)^3*(1-3*x)) + O(x^40)) \\ _Colin Barker_, Oct 23 2015
%Y Other sequences with generating functions like this: A000340, A052161, A262593, A262594.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Oct 21 2015
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