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%I #46 Sep 08 2022 08:45:09
%S 0,1,12,109,888,6841,51012,372709,2687088,19200241,136354812,
%T 964249309,6798573288,47834153641,336059778612,2358521965909,
%U 16540171339488,115933787267041,812299450322412,5689910849522509,39848449432985688
%N 6th binomial transform of (0,1,0,1,0,1,...), A000035.
%C Binomial transform of A081199.
%C Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X ... n-dimensional cubic grid. - _Ron Hardin_, May 24 2010
%C a(n) is also the total number of words of length n, over an alphabet of seven letters, of which one of them appears an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5-, 6- and 8-letter case analogs see A131577, A003462, A005059, A081199, A081201 respectively. - _Wolfdieter Lang_, Jul 17 2017
%H Vincenzo Librandi, <a href="/A081200/b081200.txt">Table of n, a(n) for n = 0..200</a>
%H Takao Komatsu, <a href="https://doi.org/10.22436/jnsa.012.12.05">Some recurrence relations of poly-Cauchy numbers</a>, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-35).
%F a(n) = 12*a(n-1) - 35*a(n-2), a(0) = 0, a(1) = 1.
%F G.f.: x/((1-5*x)*(1-7*x)).
%F a(n) = 7^n/2 - 5^n/2.
%F a(n) = Sum_{k=0..n-1} 7^k * 5^(n-k-1), with a(0)=0. - _Reinhard Zumkeller_, Aug 01 2010
%F a(n) = A121213(n)/2. - _Reinhard Zumkeller_, Aug 01 2010
%F E.g.f.: exp(5*x)*(exp(2*x) - 1)/2. - _Stefano Spezia_, Jun 19 2021
%e The a(2) = 12 words of length 2 over {A ,B, C, D, E, F, G} with say, A, appearing an odd number of times (that is once) are: AB, AC, AD, AE, AF, AG; BA, CA, DA, EA, FA, GA. - _Wolfdieter Lang_, Jul 17 2017
%t CoefficientList[Series[x / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)
%t LinearRecurrence[{12,-35},{0,1},30] (* _Harvey P. Dale_, Feb 07 2014 *)
%o (Sage) [lucas_number1(n,12,35) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 27 2009
%o (Magma) [7^n/2-5^n/2: n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2013
%Y Cf. A000035, A003462, A005059, A006516, A016161, A081199, A081201 (binomial transform, and 8-letter analog), A121213, A131577.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Mar 11 2003
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