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 A348407 a(n) = ((n+1)*3*2^(n+1) + 29*2^n + (-1)^n)/9. 1
 4, 9, 21, 47, 105, 231, 505, 1095, 2361, 5063, 10809, 22983, 48697, 102855, 216633, 455111, 953913, 1995207, 4165177, 8679879, 18058809, 37515719, 77827641, 161247687, 333680185, 689729991, 1424199225, 2937876935, 6054710841, 12467335623, 25650499129, 52732654023, 108328619577 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. 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). LINKS Table of n, a(n) for n=0..32. Index entries for linear recurrences with constant coefficients, signature (3,0,-4). FORMULA a(n) = round(((n+1)*3*2^(n+1) + 29*2^n)/9). a(n) = 2^(n+2) + A113861(n). a(n) = 2^(n+2) + n*2^n - A045883(n) = 2^(n+2) + n*2^n - round(((3*n+1)*2^n)/9). a(n+1) - 2*a(n) = A001045(n+2). a(n) = A034007(n+3) + A045883(n-1) for n > 0. A partial binomial transform in two parts: (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.) A001787(n+2) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2*n-k) = (n+2)*2^(n+1). A052951(n+1) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(a(1+2*n-k) - a(2*n-k)) = (n+2)*2^(n+1) + 2^n. The inverse transform: a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(k+2)*2^(k+1) + Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*((k+2)*2^(k+1) + 2^k). From Stefano Spezia, Oct 20 2021: (Start) G.f.: (4 - 3*x - 6*x^2)/((1 + x)*(1 - 2*x)^2). a(n) = 3*a(n-1) - 4*a(n-3) for n > 2. (End) MATHEMATICA Array[((# + 1)*3*2^(# + 1) + 29*2^# + (-1)^#)/9 &, 33, 0] (* Michael De Vlieger, Oct 19 2021 *) LinearRecurrence[{3, 0, -4}, {4, 9, 21}, 40] (* Harvey P. Dale, Aug 12 2023 *) CROSSREFS Cf. A001045, A001787, A034007, A045883, A052951, A113861, A348405. Sequence in context: A144527 A117880 A027973 * A103040 A084861 A122498 Adjacent sequences: A348404 A348405 A348406 * A348408 A348409 A348410 KEYWORD nonn,easy AUTHOR Paul Curtz and Thomas Scheuerle, Oct 17 2021 STATUS approved

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Last modified August 4 13:44 EDT 2024. Contains 374923 sequences. (Running on oeis4.)