A133095 Row sums of triangle A133094.
1, 4, 9, 20, 43, 90, 185, 376, 759, 1526, 3061, 6132, 12275, 24562, 49137, 98288, 196591, 393198, 786413, 1572844, 3145707, 6291434, 12582889, 25165800, 50331623, 100663270, 201326565, 402653156, 805306339, 1610612706, 3221225441, 6442450912, 12884901855, 25769803742, 51539607517
Offset: 1
Keywords
Examples
a(4) = 20 = sum of row 4 terms of triangle A133094: (7 + 7 + 5 + 1). a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 2, 4) = (1 + 9 + 6 + 4).
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
Programs
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Magma
[1] cat [3*2^(n - 1) - n: n in [2..50]]; // G. C. Greubel, Oct 21 2017
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Mathematica
Join[{1}, Table[ 3*2^(n - 1) - n, {n, 2, 50}]] (* G. C. Greubel, Oct 21 2017 *) LinearRecurrence[{4,-5,2},{1,4,9,20},50] (* Harvey P. Dale, Aug 09 2022 *) -
PARI
concat(1, for(n=1,50, print1(3*2^(n - 1) - n, ", "))) \\ G. C. Greubel, Oct 21 2017
Formula
Binomial transform of [1, 3, 2, 4, 2, 4, 2, 4, ...].
From G. C. Greubel, Oct 21 2017: (Start)
a(n) = 3*2^(n-1) - n, for n >= 2.
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: x*(1 - 2*x^2 + 2*x^3)/((1-2*x)*(1-x)^2).
E.g.f.: (3*exp(2*x) - 2*x*exp(x) - 2*x - 3)/2. (End)
a(n) = A123720(n) for n >= 2. - Georg Fischer, Nov 02 2018
Extensions
Terms a(11) onward added by G. C. Greubel, Oct 21 2017