cp's OEIS Frontend

The a(n) formulas given below correspond to the first ten rows of the ED1 array A167546.

The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A003148 and A007318.

The GF(z) formulas given below correspond to the first ten rows of the ED1 array A167546. The polynomials in their numerators lead to the triangle given above.

We discovered that the numbers that appear in the lower left triangle of the ED1 array A167546 (m <= n) behave in a regular way, see the formula below. This rather simple regularity doesn't show up in the upper right triangle of the ED1 array (m > n).

The first six terms coincide with the known first six terms of A006373.

Equals the triangular numbers convolved with [1, 3, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009

From Ant King, Jun 15 2012: (Start)

a(n) == 1 (mod 5) for all n.

The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 6, 7, 4, 6, 4, 7, 6, 1.

The units' digits of the a(n) form a purely periodic palindromic 4-cycle 1, 6, 6, 1.

(End)

Binomial transform of (1, 5, 5, 0, 0, 0, ...) and second partial sum of (1, 4, 5, 5, 5, ...). - Gary W. Adamson, Sep 09 2015

a(n) = a(-1-n) for all n in Z. - Michael Somos, Jan 25 2019

On the plane start with a single regular pentagon, and repeat the following procedure, "For each edge of any pentagon not already connected to an existing pentagon create a mirror image such that the mirror image does not overlap with an existing pentagon." a(n) is the number of pentagons occupying the plane after n repetitions. - Torlach Rush, Sep 14 2022

Denominators in series for sqrt(Pi/4)*erf(x): sqrt(Pi/4)*erf(x)= x/1 - x^3/3 + x^5/10 - x^7/42 + x^9/216 -+ ...

Appears to be the BinomialMean transform of A000354 (after truncating the first term of A000354). (See A075271 for the definition of BinomialMean.) - John W. Layman, Apr 16 2003

Number of permutations p of {1,2,...,n+2} such that max|p(i)-i|=n+1. Example: a(1)=3 since only the permutations 312,231 and 321 of {1,2,3} satisfy the given condition. - Emeric Deutsch, Jun 04 2003

Stirling transform of A000670(n+1) = [3, 13, 75, 541, ...] is a(n) = [3, 10, 42, 216, ...]. - Michael Somos, Mar 04 2004

Stirling transform of a(n) = [2, 10, 42, 216, ...] is A052875(n+1) = [2, 12, 74, ...]. - Michael Somos, Mar 04 2004

A related sequence also arises in evaluating indefinite integrals of sec(x)^(2k+1), k=0, 1, 2, ... Letting u = sec(x) and d = sqrt(u^2-1), one obtains a(0) = log(u+d) 2*k*a(k) = (2*k-1)*u^(2*k-1)*d + a(k-1). Viewing these as polynomials in u gives 2^k*k!*a(k) = a(0) + d*Sum(i=0..k-1){ (2*i+1)*i!*2^i*u^(2*i+1) }, which is easily proved by induction. Apart from the power of 2, which could be incorporated into the definition of u (or by looking at erf(ix/2)/ i (i=sqrt(-1)), the sum's coefficients form our series and are the reciprocals of the power series terms for -sqrt(-Pi/4)*erf(ix/2)). This yields a direct but somewhat mysterious relationship between the power series of erf(x) and integrals involving sec(x). - William A. Huber (whuber(AT)quantdec.com), Mar 14 2002

When written in factoradic ("factorial base"), this sequence from a(1) onwards gives the smallest number containing two adjacent digits, increasing when read from left to right, whose difference is n-1. - Christian Perfect, May 03 2016

a(n-1)^2 is the number of permutations p of [1..2n] such that Sum_{i=1..2n} abs(p(i)-i) = 2n^2-2. - Fang Lixing, Dec 07 2018

A standard series for the calculation of coordinates on a clothoid (also called cornuspiral):

x = s*(a(0) - (tau^2/a(2)) + (tau^4/a(4)) - (tau^6/a(6)) + ...)

y = s*((tau/a(1)) + (tau^3/a(3)) - (tau^5/a(5)) + ...).

s is the arclength from the clothoids origin to the desired point p(x,y). The tangent at the clothoids origin intersects with the tangent at the point p(x,y) with an angle of tau. - Thomas Scheuerle, Oct 13 2021

a(n) = P_n(1) where P_n(x) is the Pidduck polynomials. - Michael Somos, May 27 2023