A081201 - OEIS

%I #43 Nov 10 2024 02:29:47

%S 0,1,14,148,1400,12496,107744,908608,7548800,62070016,506637824,

%T 4113568768,33271347200,268347559936,2159841173504,17357093552128,

%U 139326933401600,1117436577120256,8956419276406784,71752914167922688,574632673083392000,4600717543107198976

%N 7th binomial transform of (0,1,0,1,0,1,....), A000035.

%C Binomial transform of A081200.

%C Conjecture (verified up to a(8)): Number of collinear 6-tuples of points in a 6 X 6 X 6 X... n-dimensional cubic grid. [_R. H. Hardin_, May 23 2010]

%C From _Wolfdieter Lang_, Jul 17 2017: (Start)

%C For a combinatorial interpretation of a(n) with special 8-letter words of length n see the comment in A081200 on the 7-letter analog.

%C The binomial transform of {a(n)}_{n >= 0} is A081202, the 9-letter analog.

%C (End)

%H Vincenzo Librandi, <a href="/A081201/b081201.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 (14,-48).

%F a(n) = 14*a(n-1) - 48*a(n-2) with n>1, a(0)=0, a(1)=1.

%F G.f.: x/((1-6*x)*(1-8*x)).

%F a(n) = (1/2)*(8^n - 6^n).

%F E.g.f.: exp(7*x)*sinh(x). - _G. C. Greubel_, Nov 10 2024

%t CoefficientList[Series[x/((1 - 6 x) (1 - 8 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)

%t LinearRecurrence[{14,-48},{0,1},30] (* _Harvey P. Dale_, Oct 24 2022 *)

%o (Magma) [(8^n-6^n)/2: n in [0..30]]; // _Vincenzo Librandi_, Aug 07 2013

%o (SageMath)

%o A081201=BinaryRecurrenceSequence(14,-48,0,1)

%o [A081201(n) for n in range(41)] # _G. C. Greubel_, Nov 10 2024

%Y Cf. A000035, A016169, A016170, A081200, A081202.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Mar 11 2003

%E Name clarified by _Pontus von Brömssen_, Nov 11 2020