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%I #31 Sep 08 2022 08:45:38
%S 3,11,45,197,903,4271,20625,100937,498123,2470931,12295605,61300877,
%T 305972943,1528270391,7636568985,38168496017,190799433363,
%U 953868026651,4768952712765,23843601302357,119214519727383,596062138283711,2980279310358945,14901302408615897
%N Number of n-digit numbers with each digit odd where the digits 1 and 3 occur an even number of times.
%C Let M = [3,1,1,0; 1,3,0,1; 1,0,3,1; 0,1,1,3] be a 4 x 4 matrix. Then a(n) = [M^n]_ (0,1); n = 1,2,3,.... - _Philippe Deléham_, Aug 24 2020
%C With a(0) = 1, binomial transform of the sequence 1,2,6,20,72,272, ... (see A063376). - _Philippe Deléham_, Aug 24 2020
%H Vincenzo Librandi, <a href="/A146086/b146086.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-23,15).
%F a(n) = (5^n+2*3^n+1)/4.
%F From _Colin Barker_, Dec 31 2013: (Start)
%F a(n) = 9*a(n-1)-23*a(n-2)+15*a(n-3).
%F G.f.: -x*(15*x^2-16*x+3) / ((x-1)*(3*x-1)*(5*x-1)). (End)
%F E.g.f.: exp(3*x)*(cosh(x))^2 - 1. - _G. C. Greubel_, Jan 31 2016
%e For n=2 the a(2)=11 numbers are 11, 33, 55, 57, 59, 75, 77, 79, 95, 97, 99.
%t Table[(5^n + 2 3^n + 1)/4, {n, 1, 30}] (* _Vincenzo Librandi_, Dec 31 2013 *)
%t LinearRecurrence[{9,-23,15},{3,11,45},30] (* _Harvey P. Dale_, Dec 15 2014 *)
%o (PARI) a(n)=(5^n+2*3^n+1)/4; \\ _Michel Marcus_, Aug 22 2013
%o (Magma) [(5^n+2*3^n+1)/4: n in [1..30]]; // _Vincenzo Librandi_, Dec 31 2013
%Y Cf. A063376, A122983.
%K base,easy,nonn
%O 1,1
%A _Jake Foster_, Oct 27 2008
%E More terms from _Colin Barker_, Dec 31 2013
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