cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A369580 a(n) := f(n, n), where f(0,0) = 1/3, f(0,k) = 0 and f(k,0) = 3^(k-1) if k > 0, and f(n, m) = f(n, m-1) + f(n-1, m) + 3*f(n-1, m-1) otherwise.

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%I A369580 #33 Nov 18 2024 16:39:29
%S A369580 2,16,138,1216,10802,96336,861114,7708416,69072354,619380496,
%T A369580 5557080938,49879087296,447852531986,4022246329936,36132550233498,
%U A369580 324645166734336,2917340834679234,26219438520320016,235672871308226634,2118552629658530496,19046140604787232242,171241206828437556816
%N A369580 a(n) := f(n, n), where f(0,0) = 1/3, f(0,k) = 0 and f(k,0) = 3^(k-1) if k > 0, and f(n, m) = f(n, m-1) + f(n-1, m) + 3*f(n-1, m-1) otherwise.
%C A369580 Take turns flipping a fair coin. The first to n heads wins. Sequence gives numerator of probability of first player winning. The denominator is .3^(2n-1).
%C A369580 It appears that a(n) for any n is divisible by 2^(A001511(n)).
%H A369580 Tadayoshi Kamegai, <a href="/A369580/b369580.txt">Table of n, a(n) for n = 1..100</a>
%H A369580 Bartosz Sobolewski and Lukas Spiegelhofer, <a href="https://arxiv.org/abs/2411.07779">Decomposing the sum-of-digits correlation measure</a>, arXiv:2411.07779 [math.NT], 2024. See p. 16.
%F A369580 Limit_{n->oo} a(n)/3^(2n-1) = 1/2.
%F A369580 a(n) = Sum_{i>=n} Sum_{j=0..n-1} binomial(i-1,n-1)*binomial(i-1,j)*3^(2n-1)/2^(2i-1).
%F A369580 9*a(n) - a(n+1) = 2*A162326(n) (conjectured).
%F A369580 a(n) = 3^(2n-1)*A(n, n) where A(0, k) = 0 for k > 0, A(k, 0) = 1 for k >= 0 and A(n, m) = (A(n-1, m) + A(n, m-1) + A(n-1, m-1))/3.
%o A369580 (Python)
%o A369580 def lis(n):
%o A369580     table = [[0]*(n+1) for _ in range(n+1)]
%o A369580     table[1][1] = 2
%o A369580     for i in range(1, n+1) :
%o A369580         table[i][0] = 3**(i-1)
%o A369580     for i in range(1, n+1) :
%o A369580         for j in range(1, n+1) :
%o A369580             if (i == 1 and j == 1) :
%o A369580                 continue
%o A369580             table[i][j] = table[i][j-1] + table[i-1][j] + 3*table[i-1][j-1]
%o A369580     return [int(table[i][i]) for i in range(1, n+1)]
%Y A369580 Cf. A344576, A050231, A001850.
%Y A369580 Cf. A001511 (see comments), A162326 (see formula).
%K A369580 nonn
%O A369580 1,1
%A A369580 _Tadayoshi Kamegai_, Jan 26 2024