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Fixed points of A001175 (Pisano periods).
4

%I #25 Nov 10 2024 08:58:02

%S 1,24,120,600,3000,15000,75000,375000,1875000,9375000,46875000,

%T 234375000,1171875000,5859375000,29296875000,146484375000,

%U 732421875000,3662109375000,18310546875000,91552734375000,457763671875000,2288818359375000,11444091796875000

%N Fixed points of A001175 (Pisano periods).

%H Michael De Vlieger, <a href="/A235702/b235702.txt">Table of n, a(n) for n = 1..1430</a>

%H Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, <a href="https://arxiv.org/abs/2411.02897">Patterns in Multi-dimensional Permutations</a>, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 26.

%H J. D. Fulton and W. L. Morris, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf">On arithmetical functions related to the Fibonacci numbers</a>, Acta Arithmetica, 16 (1969), 105-110.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pisano_period">Pisano period</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (5).

%F A001175(a(n)) = a(n); A001178(a(n)) = 0.

%F From _Colin Barker_, Jan 16 2014: (Start)

%F a(n) = 24*5^(n-2) for n > 1.

%F a(n) = 5*a(n-1) for n > 2.

%F G.f.: -x*(19*x+1) / (5*x-1). (End)

%F E.g.f.: (24*(exp(5*x) - 1) - 95*x)/25. - _Stefano Spezia_, Nov 09 2024

%t LinearRecurrence[{5},{1,24},30] (* or *) Join[{1},NestList[5#&,24,30]] (* _Harvey P. Dale_, May 07 2017 *)

%o (Haskell)

%o a235702 n = if n == 1 then 1 else 24 * 5 ^ (n - 2)

%o a235702_list = 1 : iterate (* 5) 24

%o (PARI)

%o Vec(-x*(19*x+1)/(5*x-1) + O(x^100)) \\ _Colin Barker_, Jan 16 2014

%Y Cf. A001175, A001178, A008606, A000351.

%K nonn,easy,changed

%O 1,2

%A _Reinhard Zumkeller_, Jan 15 2014