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%I #48 Sep 08 2022 08:46:02
%S 1,4,7,36,65,340,615,3220,5825,30500,55175,288900,522625,2736500,
%T 4950375,25920500,46890625,245522500,444154375,2325622500,4207090625,
%U 22028612500,39850134375,208658012500,377465890625,1976437062500,3575408234375
%N a(n) is the difference between multiples of 5 with even and odd digit sum in base 4 in interval [0,4^n).
%C Let the term "transform" mean the operation of summing the products of the numbers in the n-th row of an m-nomial triangle (m-nomial T(n,k)) and the ascending numbers of a sequence. And let T(0,0) be the top entry (0th row, 0th column) in an m-nomial triangle. Then starting with a(1)=1, the bisection of this sequence (1,7,65,615,5825...) is the quadrinomial (4-nomial) transform of A000045 (Fibonacci sequence, F(j)), starting at T(0,0)=1, F(1)=1. - _Bob Selcoe_, May 24 2014
%H Vincenzo Librandi, <a href="/A212500/b212500.txt">Table of n, a(n) for n = 1..1000</a>
%H V. Shevelev, <a href="http://arxiv.org/abs/0710.3177">On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m</a>, arXiv:0710.3177 [math.NT], 2007.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-5).
%F For n>=5, a(n) = 10*a(n-2)-5*a(n-4).
%F a(n) = 0.4*((5+2*sqrt(5))^(n/2)+ (5-2*sqrt(5))^(n/2)) , if n is even, and
%F a(n) = 0.1*((5+2*sqrt(5))^((n-1)/2)*sqrt(30+10*sqrt(5))+(5-2*sqrt(5))^((n-1)/2)*sqrt(30-10*sqrt(5))), if n is odd.
%F a(2n+1) = Sum_{j=0..3n+1} fibonacci(j+1)*A008287(n,j). - _Bob Selcoe_, May 28 2014
%F G.f.: -x*(-1-4*x+3*x^2+4*x^3) / ( 1-10*x^2+5*x^4 ). - _R. J. Mathar_, Jun 16 2014
%e Let n=3. In interval [0,4^3) we have 13 multiples of 5,from which in base 4 only three (namely, 35,50,55) have odd digit sums. Thus a(3)=(13-3)-3=7.
%e From _Bob Selcoe_, May 28 2014: (Start)
%e n=2: a(5)=65 because T(2,k) {k=0..6} is {1,2,3,4,3,2,1} and {j=1..7} is {1,1,2,3,5,8,13}: 1*1 + 2*1 + 3*2 + 4*3 + 3*5 + 2*8 + 1*13 = 65.
%e n=3: a(7)=615 because T(3,k) {k=0..9} is {1,3,6,10,12,12,10,6,3,1} and {j=1..10} is {1,1,2,3,5,8,13,21,34,55}: 1*1 + 3*1 + 6*2 + 10*3 + 12*5 + 12*8 + 10*13 + 6*21 + 3*34 + 1*55 = 615. (End)
%t CoefficientList[Series[-(-1 - 4 x + 3 x^2 + 4 x^3)/(1 - 10 x^2 + 5 x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 17 2014 *)
%t LinearRecurrence[{0,10,0,-5},{1,4,7,36},30] (* _Harvey P. Dale_, Apr 07 2019 *)
%o (Magma) I:=[1,4,7,36]; [n le 4 select I[n] else 10*Self(n-2)-5*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jun 17 2014
%o (PARI) Vec(-x*(-1-4*x+3*x^2+4*x^3)/(1-10*x^2+5*x^4) + O(x^30)) \\ _Michel Marcus_, Feb 06 2016
%Y Cf. A038754, A084990, A189334 (bisection).
%K nonn,base,easy
%O 1,2
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, May 19 2012
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