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%I #58 Feb 11 2024 04:42:22
%S 1,2,4,9,20,45,102,231,520,1161,2570,5643,12300,26637,57358,122895,
%T 262160,557073,1179666,2490387,5242900,11010069,23068694,48234519,
%U 100663320,209715225,436207642,905969691,1879048220,3892314141,8053063710,16642998303
%N a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.
%C The inverse binomial transform of A176328/A176591 (see Comments field in A228827) begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, ... Consider these values without sign and the fractions rounded to the nearest integer, the sequence lists the resulting numbers.
%C Differences table of a(n):
%C 1, 2, 4, 9, 20, 45, 102, 231, 520, 1161, ...
%C 1, 2, 5, 11, 25, 57, 129, 289, 641, 1409, ... After 2: 2^m*(m+4)+1.
%C 1, 3, 6, 14, 32, 72, 160, 352, 768, 1664, ... A078836 (after 3).
%C 2, 3, 8, 18, 40, 88, 192, 416, 896, 1920, ... A129955.
%C 1, 5, 10, 22, 48, 104, 224, 480, 1024, 2176, ... A079861 (after 5).
%C 4, 5, 12, 26, 56, 120, 256, 544, 1152, 2432, ... After 5: 2^m*(m+12).
%C 1, 7, 14, 30, 64, 136, 288, 608, 1280, 2688, ... After 7: 2^m*(m+14).
%C 6, 7, 16, 34, 72, 152, 320, 672, 1408, 2944, ..., etc.
%C (n-1)*a(n)-n*a(n-1) = A001788(n-1) for n>1. [_Bruno Berselli_, Oct 11 2013]
%H Bruno Berselli, <a href="/A227978/b227978.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).
%F a(2n+2) = A229135(n+1); a(2n-1) = -A228767(n) for n>0.
%F a(n) = 6*a(n-1) -13*a(n-2) +12*a(n-3) -4*a(n-4) for n>5.
%F G.f.: (1-4*x+5*x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1-2*x)^2). - _Colin Barker_, Oct 09 2013
%t Join[{1, 2}, Table[n (2^n + 4)/4, {n, 2, 40}]] (* _Bruno Berselli_, Oct 11 2013 *)
%o (Magma) [1,2] cat [n*(2^n+4)/4: n in [2..40]]; // _Bruno Berselli_, Oct 11 2013
%o (PARI) a(n) = if (n == 0, 1, if (n == 1, 2, n*(2^n+4)/4)); \\ _Michel Marcus_, Oct 11 2013
%Y Cf. A001788, A079862, A079863.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Oct 07 2013
%E More terms from _Colin Barker_, Oct 09 2013
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