A215149 - OEIS

%I #26 Sep 08 2022 08:46:03

%S 0,2,6,15,36,85,198,455,1032,2313,5130,11275,24588,53261,114702,

%T 245775,524304,1114129,2359314,4980755,10485780,22020117,46137366,

%U 96469015,201326616,419430425,872415258,1811939355,3758096412

%N a(n) = n * (1 + 2^(n-1)).

%C Related to Bernoulli numbers.

%C Essentially the same as A135854.

%H G. C. Greubel, <a href="/A215149/b215149.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).

%F Discovered by a(n) = (A157809(n) - A164555(n)) / A027642(n).

%F a(n) = n (the nonnegative integers A001477(n)) + n*2^(n-1) (their binomial transform A001787(n)).

%F a(n+1) - a(n) = 2,4,9,21,... = A001792(n) + 1.

%F a(n+1) - 2*a(n) = 2 before A132045(n+1).

%F a(n) is the binomial transform of b(n) = 0,2,2,3,4,5,... = A001477(n) with 2 instead of 1. b(n) = (A164558(n) - A027641(n))/A027642(n)?

%F G.f. x*(2-6*x+5*x^2) / ( (2*x-1)^2*(x-1)^2 ). - _R. J. Mathar_, Aug 06 2012

%t Table[n(1+2^(n-1)),{n,0,30}] (* or *) LinearRecurrence[{6,-13,12,-4},{0,2,6,15},30] (* _Harvey P. Dale_, Oct 18 2013 *)

%o (PARI) a(n) = n*(1+2^(n-1)) \\ _Michel Marcus_, Mar 10 2013

%o (Magma) [n*(1 + 2^(n-1)): n in [0..30]]; // _G. C. Greubel_, Apr 19 2018

%K nonn

%O 0,2

%A _Paul Curtz_, Aug 04 2012