A084634 Binomial transform of 1, 1, 1, 2, 2, 2, 2, 2, ...
1, 2, 4, 9, 21, 48, 106, 227, 475, 978, 1992, 4029, 8113, 16292, 32662, 65415, 130935, 261990, 524116, 1048385, 2096941, 4194072, 8388354, 16776939, 33554131, 67108538, 134217376, 268435077, 536870505, 1073741388, 2147483182, 4294966799, 8589934063
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).
Programs
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Magma
[2^(n+1)-1-Binomial(n+1,2): n in [0..50]]; // G. C. Greubel, Mar 18 2023
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Maple
A084634:=n->2^(n+1) - (n^2 +n +2)/2; seq(A084634(n), n=0..50); # Wesley Ivan Hurt, Jan 31 2014
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Mathematica
LinearRecurrence[{5,-9,7,-2}, {1,2,4,9}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *) -
Sage
[2^(n+1)-1-binomial(n+1,2) for n in range(52)] # Zerinvary Lajos, May 29 2009
Formula
a(n) = 2^(n+1) - (n^2 + n + 2)/2.
a(n) = 1 + n + n*(n-1)/2 + 2*Sum_{k=3..n} C(n, k).
O.g.f.: (1-3*x+3*x^2)/((1-2*x)*(1-x)^3). - R. J. Mathar, Apr 07 2008
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4). - R. J. Mathar, Apr 07 2008
a(n) = Sum_{i=0..n} (2^i - i). - Ctibor O. Zizka, Oct 15 2010
a(n) = A000225(n+1) - binomial(n+1,2). - G. C. Greubel, Mar 18 2023
Comments