cp's OEIS Frontend

Binomial transform of A000918: (-1, 0, 2, 6, 14, 30, ...). - Gary W. Adamson, Mar 23 2012

This sequence demonstrates 2^n as a loose lower bound for g(n) in Waring's problem. Since 3^n > 2(2^n) for all n > 2, the number 2^(n + 1) - 1 requires 2^n n-th powers for its representation since 3^n is not available for use in the sum: the gulf between the relevant powers of 2 and 3 widens considerably as n gets progressively larger. - Alonso del Arte, Feb 01 2013

a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1).

a(n,1) = a(n-1,1) + a(n-1,3) for n > 0; a(n,1) = A001047(n) = 2^(2n) - A083324(n); a(n,2) = A000244(n) = 2^(2n) - A005061(n); a(n,3) = 2a(n-1,4) for n > 0; a(n,3) = A027649(n); a(n,4) = A083313(n+1); a(n,5) = A084171(n+1).

Sum[a(n-k,k), {k,0,n}] = A098703(n+1), antidiagonal sums.

Let R, S and T be binary relations on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x, xSy if x is a subset of y and xTy if x is a proper subset of y. Then a(n,3) = |R|, a(n,2) = |S| and a(n,1) = |T|. Note that a binary relation W on P(A) can be defined also such that for every element x, y of P(A) xWy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. A090802(n,1) = |W|. Also, a(n,0) = |P(A)|.

Binomial transform of A083313. - Paul Barry, Apr 25 2003

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x and x and y are disjoint, 1) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 2) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009

A biquam or biquanimous number (A064544) is a number whose digits can be split into two groups with equal sum.

This is the same as A083313 (apart from the initial term). Proof: Let sum(w) denote the sum of the digits of w. There are 2*3^(n-1) n-digit base-3 numbers: w = (w_1,w_2,...,w_n) with w_i in {0,1,2} for all i and w_1 != 0. Partition them into 4 classes: (i) sum(w) is odd, (ii) sum(w) is even, w contains no 1's and has an odd number of 2s, (iii) sum(w) is even, w contains no 1's and has an even number of 2s and (iv) sum(w) is even and w contains some 1's. Clearly, no biquams occur in cases (i) and (ii), case (iii) consists entirely of biquams and, we claim, so does case (iv). For case (iv) forces an even number, say 2k, of 1's. An even number of 2s clearly gives a biquam and an odd number 2m+1 of 2s does too because {m 2s, (k+1) 1's} and {(m+1) 2s, (k-1) 1's} is a biquam split. There are 3^(n-1) w's in case (i) and 2^(n-2) w's in case (ii) and hence 2*3^(n-1) - (3^(n-1) + 2^(n-2)) = 3^(n-1) - 2^(n-2) (A083313) biquams among n-digit base-3 numbers. - David Callan, Sep 15 2004

a(n) % 100 = 23 for n = 4*k-1, k>=1; a(n) % 100 = 27 for n = 4*k+1, k>=1. - Alex Ratushnyak, Jul 03 2012

The fraction of biquams for any base approaches 1/2 as the number of digits grows but only if you count leading zeros. Without counting leading zeros, the fraction appears to converge to (b-1)/2b where b is the base used. For base 3 this is 1/3 which fits the data in this sequence (see paper cited below for proofs and the OEIS data collated as fractions). - Timothy Varghese, Aug 08 2021

Binomial transform of A083313

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

For n>=2, a(n) is the number of vertices of the Sierpinski Gasket Rhombus graph SR(n) (see Theorem 2.1 in the D. Antony Xavier et al. reference).

The round (nearest) function uses the convention that values halfway between two integers are rounded up, so a(4) = 37 = round(36 + 1/2).

T(m, n) is the number of saturated Cp^m*q^n-transfer systems where Cp^m*q^n is the cyclic group of order p^m*q^n, for m, n >= 0, p and q primes. See Hafeez et al. link page 1.