A073536 -id:A073536 - cp's OEIS frontend

A077468 Greedy powers of (2/3): Sum_{n>=1} (2/3)^a(n) = 1.

1, 3, 9, 12, 15, 17, 27, 34, 39, 46, 49, 52, 54, 66, 70, 73, 81, 84, 90, 95, 102, 106, 110, 116, 119, 124, 132, 140, 143, 149, 153, 158, 161, 165, 171, 177, 180, 183, 186, 189, 194, 198, 209, 215, 221, 224, 226, 233, 235, 241, 244, 248, 251, 255, 259, 262, 272

Offset: 1

Paul D. Hanna, Nov 06 2002

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series Sum_{k=1..n} x^a(k) to exceed unity.

A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n >= m} log(1 + x^n)/log(x) = 4.9298413943..., where x=2/3 and m=floor(log(1-x)/log(x))=2. - Paul D. Hanna, Nov 16 2002

			a(3)=9 since (2/3) +(2/3)^3 +(2/3)^9 < 1 and (2/3) +(2/3)^3 +(2/3)^8 > 1; since the power 8 makes the sum > 1, then 9 is the 3rd greedy power of (2/3).

Michael Somos, Non-integer Radix Expansions and Modular Functions (1993)

Cf. A077469, A077470, A077471, A077472, A077473, A077474, A077475.

s = 0; a = {}; Do[ If[s + (2/3)^n < 1, s = s + (2/3)^n; a = Append[a, n]], {n, 1, 278}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[2/3], 20]

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) at x= 2/3 and frac(y) = y - floor(y).

It appears that, for n>1, a(n) = A073536(n-1) - Benoit Cloitre, Jun 04 2004

Extended by John W. Layman, Robert G. Wilson v and Benoit Cloitre, Nov 07 2002

A094384 Determinant of n X n partial Hadamard matrix with coefficient m(i,j) 1<=i,j<=n (see comment).

Original entry on oeis.org

1, -2, 4, 16, -32, -128, -512, 4096, -8192, -32768, -131072, 1048576, 4194304, -33554432, 268435456, 4294967296, -8589934592, -34359738368, -137438953472, 1099511627776, 4398046511104, -35184372088832, 281474976710656

Offset: 1

Benoit Cloitre, Jun 03 2004

sign

Let M(infinity) be the infinite matrix with coefficient m(i,j) i>=1, j>=1 defined as follows : M(0)=1 and M(k) is the 2^k X 2^k matrix following the recursion : +M(k-1)-M(k-1) M(k)= -M(k-1)-M(k-1)

			M(2)=/1,-1/-1,-1/ then a(2)=detM(2)=-2

Cf. A000788, A073536.

from sympy import Matrix
def A094384(n):
    m = Matrix([1])
    for i in range((n-1).bit_length()):
        m = Matrix([[m, -m],[-m, -m]])
    return m[:n,:n].det() # Chai Wah Wu, Nov 12 2024

It appears that abs(a(n))=2^A000788(n). What is the rule for signs? Does sum(k=1, n, a(k+1)/a(k))=0 iff n is in A073536 ?

Conjecture: a(n) = (-2)^A000788(n-1). - Chai Wah Wu, Nov 12 2024

cp's OEIS Frontend

A077468 Greedy powers of (2/3): Sum_{n>=1} (2/3)^a(n) = 1.

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