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%I #52 Apr 22 2023 21:48:35
%S 1,5,0,3,0,0,0,6,0,0,0,0,0,0,0,6,0,0,3,0,0,2,3,0,0,6,0,0,0,1,0,0,0,0,
%T 0,0,1,0,2,6,0,0,3,0,0,0,2,0,8,3,0,0,6,0,0,0,0,3,0,8,0,0,0,0,0,0,3,1,
%U 0,4,0,4,6,0,0,3,0,0,3,6,7,0,5,0,0,3,0,0,6,0,0,5,5,0,6,0,6,0,4,0,0,0,0,0,0,6
%N a(n) is the n-th decimal digit of p(n)/q(n) where p(n) = A002260(n) and q(n) = A004736(n).
%C Decimal digit positions are numbered 1 for the units, 2 for immediately after the decimal point, and so on.
%C This sequence can be interpreted as the decimal digits of a constant 1.5030006...
%C This constant shares an infinite number of decimal digits with any given rational r. This is since p,q go through all pairs of integers >= 1 and so p(n)/q(n) = r for infinitely many n.
%C This constant is irrational.
%e p(1) = 1, q(1) = 1, p/q = 1/1 = 1, a(1) = 1.
%e p(2) = 1, q(2) = 2, p/q = 1/2 = 0.5, a(2) = 5.
%e p(3) = 2, q(3) = 1, p/q = 2/1 = 2.00, a(3) = 0.
%e p(4) = 1, q(4) = 3, p/q = 1/3 = 0.333..., a(4) = 3.
%o (PARI) p(n) = n-binomial(floor(1/2+sqrt(2*n)), 2); \\ A002260
%o q(n) = binomial( floor(3/2 + sqrt(2*n)), 2) - n + 1; \\ A004736
%o a(n) = my(r = p(n)/q(n)); floor(r*10^(n-1)) % 10; \\ _Michel Marcus_, Apr 05 2023
%Y Cf. A002260, A004736, A061480.
%K base,cons,easy,nonn
%O 1,2
%A _Jesiah Darnell_, Apr 01 2023