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%I #30 May 07 2023 05:40:31
%S 2,6,11,18,29,48,83,150,281,540,1055,2082,4133,8232,16427,32814,65585,
%T 131124,262199,524346,1048637,2097216,4194371,8388678,16777289,
%U 33554508,67108943,134217810,268435541,536871000,1073741915,2147483742,4294967393,8589934692,17179869287
%N a(n) = 2^n + 3n + 1.
%C Inverse binomial transform of this sequence: 2,4,1,1 (1 continued).
%H Bruno Berselli, <a href="/A194455/b194455.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F G.f.: (2 - 2*x - 3*x^2)/((1 - 2*x)*(1 - x)^2).
%F a(n) = A086653(n) - 1 for n > 0.
%F Sum_{i=0..n} a(i) = A115067(n+1) + 2^(n+1).
%F a(n) = 3*a(n-1) - 2*a(n-2) - 3 for n > 1.
%F a(n)^2 = 2^(n+1)*(a(n-1) + 3) + (3*n + 1)^2 for n > 2.
%F E.g.f.: exp(x)*(1 + exp(x) + 3*x). - _Stefano Spezia_, May 06 2023
%t Table[2^n + 3 n + 1, {n, 0, 40}] (* _Vincenzo Librandi_, Mar 26 2013 *)
%t LinearRecurrence[{4,-5,2},{2,6,11},40] (* _Harvey P. Dale_, Oct 01 2014 *)
%o (Magma) [2^n+3*n+1: n in [0..31]];
%o (PARI) for(n=0, 31, print1(2^n+3*n+1", "));
%Y Cf. A000051, A005126, A176691; A120845.
%Y Cf. A062709 (first differences), A000079 (second and successive differences).
%Y Cf. A146529 (differences between alternate terms, for n>2).
%Y Cf. A086653, A115067.
%K nonn,easy
%O 0,1
%A _Bruno Berselli_, Sep 01 2011
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