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A348407 a(n) = ((n+1)*3*2^(n+1) + 29*2^n + (-1)^n)/9.

%I A348407 #72 Aug 12 2023 15:08:38
%S A348407 4,9,21,47,105,231,505,1095,2361,5063,10809,22983,48697,102855,216633,
%T A348407 455111,953913,1995207,4165177,8679879,18058809,37515719,77827641,
%U A348407 161247687,333680185,689729991,1424199225,2937876935,6054710841,12467335623,25650499129,52732654023,108328619577
%N A348407 a(n) = ((n+1)*3*2^(n+1) + 29*2^n + (-1)^n)/9.
%C A348407 The ratio (count of ones)/(count of zeros) in the binary expansion of a(n) is > 1/2 and <= 5 for all n > 0, this is because the division by 9 adds a repeating pattern 111000... after some binary digits.
%C A348407 This sequence has in its "partial binomial transform" (see formula section) no other constants than 2 and 1 despite of its more complicated looking closed form expression. This transform has a deep connection to the Grünwald-Letnikov fractional derivative if we replace the order of the derivative with the variable x: D^x*f(x).
%H A348407 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4).
%F A348407 a(n) = round(((n+1)*3*2^(n+1) + 29*2^n)/9).
%F A348407 a(n) = 2^(n+2) + A113861(n).
%F A348407 a(n) = 2^(n+2) + n*2^n - A045883(n) = 2^(n+2) + n*2^n - round(((3*n+1)*2^n)/9).
%F A348407 a(n+1) - 2*a(n) = A001045(n+2).
%F A348407 a(n) = A034007(n+3) + A045883(n-1) for n > 0.
%F A348407 A partial binomial transform in two parts:
%F A348407 (Partial means a diagonal in a difference table a(0), a(2)-a(1), ... . This is partial because one diagonal alone is no invertible transform.)
%F A348407 A001787(n+2) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2*n-k)
%F A348407              = (n+2)*2^(n+1).
%F A348407 A052951(n+1) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(a(1+2*n-k) - a(2*n-k))
%F A348407              = (n+2)*2^(n+1) + 2^n.
%F A348407 The inverse transform:
%F A348407 a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(k+2)*2^(k+1)
%F A348407        + Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*((k+2)*2^(k+1) + 2^k).
%F A348407 From _Stefano Spezia_, Oct 20 2021: (Start)
%F A348407 G.f.: (4 - 3*x - 6*x^2)/((1 + x)*(1 - 2*x)^2).
%F A348407 a(n) = 3*a(n-1) - 4*a(n-3) for n > 2. (End)
%t A348407 Array[((# + 1)*3*2^(# + 1) + 29*2^# + (-1)^#)/9 &, 33, 0] (* _Michael De Vlieger_, Oct 19 2021 *)
%t A348407 LinearRecurrence[{3,0,-4},{4,9,21},40] (* _Harvey P. Dale_, Aug 12 2023 *)
%Y A348407 Cf. A001045, A001787, A034007, A045883, A052951, A113861, A348405.
%K A348407 nonn,easy
%O A348407 0,1
%A A348407 _Paul Curtz_ and _Thomas Scheuerle_, Oct 17 2021