A081199 - OEIS

%I #36 Jul 23 2024 18:55:10

%S 0,1,10,76,520,3376,21280,131776,807040,4907776,29708800,179301376,

%T 1080002560,6496792576,39047864320,234555621376,1408407470080,

%U 8454739787776,50745618595840,304542431051776,1827529464217600,10966276296933376,65802055828111360,394829927154712576

%N 5th binomial transform of (0,1,0,1,...), A000035.

%C Binomial transform of A005059.

%C Conjecture (verified up to a(9)): Number of collinear 4-tuples of points in a 4 X 4 X 4 X ... n-dimensional cubic grid. - _R. H. Hardin_, May 24 2010

%C a(n) is also the total number of words of length n, over an alphabet of six letters, of which one of them appears an odd number of times. See a Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5- and 7-letter case analogs see A131577, A003462, A005059 and A081200, respectively. - _Wolfdieter Lang_, Jul 16 2017

%H Vincenzo Librandi, <a href="/A081199/b081199.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-24).

%F a(n) = 10*a(n-1) - 24*a(n-2) with n>1, a(0)=0, a(1)=1.

%F G.f.: x/((1-4*x)*(1-6*x)).

%F a(n) = 6^n/2 - 4^n/2.

%F E.g.f.: exp(4*x)*(exp(2*x) - 1)/2. - _Stefano Spezia_, Jul 23 2024

%p seq(add(2^(2*n-k)*binomial(n,k)/2,k=1..n),n=0..20); # _Zerinvary Lajos_, Apr 18 2009

%t CoefficientList[Series[x / ((1 - 4 x) (1 - 6 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)

%t LinearRecurrence[{10, -24}, {0, 1}, 21] (* _Michael De Vlieger_, Jul 16 2017 *)

%o (Magma) [6^n/2-4^n/2: n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2013

%Y Cf. A000035, A003462, A005059, A006516, A016149, A081200 (binomial transform of a(n), and 7-letter case), A131577.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Mar 11 2003