cp's OEIS Frontend

Reid, Alan W., and Colin MacLachlan. "The Arithmetic of hyperbolic 3-manifolds." Graduate Texts in Mathematics 219, Springer (2003).

Row reversal of A261363 (which is the main entry).

These sums can be formed by taking A047999 as a lower triangular matrix times an all-1's lower triangular matrix.

Alternative formula for e^(n*Pi/2) is i^(-n*i), where i = sqrt(-1). Substitute 2i for n in each identity, resulting in e^(Pi*i) = -1; Euler's formula.

A121905 is the bisection of the sequence, ceiling(e^(n*Pi)).

The format of A233932 has a Gray code property of one term change in the next row. Using the production matrix shown below, we can obtain an array with row sums of any target sequence.

As an unsigned sequence a(n) this is identical with the one of A155586(n+1), for n >= 0, but the triangle is not a simple signed version of A155586. See the formula.

This lower triangular Riordan matrix T of Toeplitz type is the inverse of the Riordan matrix (c(x), x) = |A106270|, also of Toeplitz type.

This is the partial sum triangle of triangle A143929.

The main diagonal is the INVERT transform of the first column (offset 1 in both sequences).

The length of row n is A037225(n), for n >= 0.

The doubling sequence modulo N = 2*n+1, for n >= 0, has entries DS(N, s(N,i), j) = s(N,i)*2^j (mod N), with j >= 0, and certain positive odd integer seeds s(N, i), for i = 1, 2, ..., S(N) = A037226((N-1)/2), where gcd(s(N, i), N) = 1 (restricted seeds modulo N). These doubling sequences are periodic with period length P(N) = A002326((N-1)/2) (order of 2 modulo N). Only the periods (cycles) {DS(N, s(N, i), j)}_{j=0..P(N)-1}, for i = 1, 2, ..., S(N), are listed.

N = 1 (n=0) is special: one takes here the restricted residue system modulo N not as [0] but as [1]. The order of 2 modulo 1 is 1, because 2^1 == 1 (mod 1) (== 0 (mod 1)).

In order to obtain the complete system of doubling sequences one starts with seed s(N, 1) = 1, and if all numbers from the smallest positive reduced residue system modulo N (called RRS(N), given in row N of A038566) are obtained, i.e., if P(N) = #RRS(N) = phi(N) = A000010(N), then the system is complete. Otherwise the smallest missing number from RRS(N) is taken as new seed s(N, 2), etc. until the system is complete. This means that the number of seeds needed is S(N) = phi(N)/P(N) = A037226((N-1)/2)).

The irregular subtriangle where only seed s(N, 1) = 1 has been used is given in A201908. But there 0 (not 1) for N = 1 has been used.

From Gary W. Adamson and Wolfdieter Lang, Dec 15 2020: (Start)

The cycles in row n, for N = 2*n + 1, of period length P(N) = A002326((N-1)/2) give the periods of the iterated doubling function D(x) = frac(2*x) with seeds x = s(N, i)/N, for i = 1, 2, ..., S(N) = A037226((N-1)/2), after multiplication with N. This is the doubling function used in the Devaney reference, pp. 24-25, 27, 125. 132, 171,289.

Each cycle in row n can also be used to find from the base 2 version of its first entry (the seed s = s(N, i)) divided by N the other entries by repeated application of a cyclic left shift by one step (called sigma operation) to the period of the base 2 expression of s/N. E.g., n = 7, N = 15, P(N) = 4, s = 1: (1/15){10->2} = .repeat(0001), then (.repeat(0010)){2->10} = 2/10, (.repeat(0100)){2->10} = 4/10 and (.repeat(1000)){2->10} = 8/15. Similarly for s = 7: from (7/15)_{10->2} = .repeat(0111) one obtains by repeated sigma operations 14/15, 13/15 and 11/15. The proof uses the elementary formulas for the conversion from base 10 to base 2, and the reverse one, from base 2 to base 10. See also a comment on the period length P(N) given in A002326. (End)

The length of row n is phi(2*n+1)/2 with phi = A000010, for n > = 1.

There are c(2*n+1) = A135303(n) cycles in row n, and the length of each cycle is k(2*n+1) = A003558(n). c(2*n+1)*k(2*n+1) = phi(n)/2 (in Hilton and Pedersen [HP], the coach theorem, p. 262).

In the construction of the cycles a modified modular equivalence relation, called mod* b, for b = 2*n + 1, with n >= 1, proposed in the Brändli and Beyne [BB] paper, is used. It is defined as mod*(a, b) = mod(a, b) (with values in I(b) = {0, 1, ..., b-1}) for integer numbers a, if mod(a, b) <= (b-1)/2 and mod*(a, b) = mod(-a, b) if mod(a, b) > (b-1)/2. The equivalence relation k ~ m if mod*(k, b) = mod*(m,b) is multiplicative (not additive).

The MDS(b) cycles are obtained from the doubling sequence {a(b,i)*2^j}A003558(n)%20for%20each%20odd%20a(b,%20i)%20with%20gcd(a(b,%20i),%20b)%20=%201%20and%20a(b,%20i)%20%3C=%20(b-1)/2.%20The%20first%20cycle%20is%20Cy*(b,1)%20obtained%20for%20a(b,%201)%20=%201.%20If%20not%20all%20odd%20numbers%20relatively%20prime%20to%20b%20and%20%3C=%20(b-1)/2%20are%20present,%20the%20next%20cycle%20Cy*(b,2)%20uses%20the%20smallest%20missing%20reduced%20odd%20number,%20etc.%20The%20number%20of%20such%20inputs%20a(b,%20i),%20hence%20cycles,%20is%20c*(b),%20coinciding%20with%20c(2*n+1)%20=%20A135303(n),%20and%20all%20odd%20and%20even%20positive%20numbers%20with%20gcd(m,%20b)%20=%201%20and%20m%20%3C=%20(b-1)/2%20appear%20just%20once%20in%20the%20complete%20cycle%20system%20for%20b,%20called%20Cy*(b)%20=%20%7BCy*(b,i)%7D">{j=1..P(b)} evaluated mod*b: Cy*(b,i)_j = mod*(a(b,i)*2^j, b), with certain inputs (seeds) a(b, i). P(b), the length of the period independent, coincides with k(2*n+1) = A003558(n) for each odd a(b, i) with gcd(a(b, i), b) = 1 and a(b, i) <= (b-1)/2. The first cycle is Cy*(b,1) obtained for a(b, 1) = 1. If not all odd numbers relatively prime to b and <= (b-1)/2 are present, the next cycle Cy*(b,2) uses the smallest missing reduced odd number, etc. The number of such inputs a(b, i), hence cycles, is c*(b), coinciding with c(2*n+1) = A135303(n), and all odd and even positive numbers with gcd(m, b) = 1 and m <= (b-1)/2 appear just once in the complete cycle system for b, called Cy*(b) = {Cy*(b,i)}{i=1..c(b)}.

Such modified doubling sequences have been considered in the Kappraff-Adamson paper using iterations of x^2 - 2, and also in comments and examples by Adamson like the Aug 25 2019 comment in A065941, where it is named "r-t table" (for roots trajectory).

The recurrence relation for the cycle Cy*(b,i) with elements {c_j(b,i)} is

c(b, i)j = mod*(2*c(b, i){j-1}, b), for j = 2, 3, ..., P(b), and input c(b, i)1, for i = 1, 2, ..., c(b). This input is here not a(b, i) but 2*a(b, i) if a(b, i) <= floor((b-1)/4) and b - 2*a(b, i) otherwise. Observe that c(b, i){P(b)} = a(b, i).

The complete cycle system Cy*(b) is equivalent to the complete coach system of Hilton and Pedersen [HP], with the coach number c(b) = c*(b) and k(b) = P(b). The odd numbers of each cycle Cy*(b,i), read backwards (anticyclically), give the first row of the coach; and the number of steps to go from one odd number to the next odd number, ending with the starting number, gives the second row of the coach with sum k(b). See [HP] pp. 102-103 for the modified coach system and the extended list, eq. (7. 12), in the proof of their quasi-order theorem.

The complete cycle system of Schick (unsigned) and Brändli-Beyne SBB(b) is also equivalent to Cy*(b). Each cycle Cy*(b,i) gives the SBB(b,i) cycle (q(b, i)j) with elements obtained from q(b, i)_j = b - 2*c(b,i){P(b)-1+j}, for j = 0, 1, 2, ..., P(b)-1. Cyclicity is used: c(b,i){P(b)+k} = c(b,i){k}.

For more details and proofs see the arxiv paper by W. Lang.

From Gary W. Adamson, Mar 11 2020: (Start)

The upper left entry of M^n gives the Catalan numbers A000108. Extracting 2 X 2, 3 X 3, and 4 X 4 submatrices from M; then generating sequences from the upper left entries of M^n, we obtain the following sequences:

1, 1, 2, 5, 13, ... = A001519 and the convergent is 2.61803... = 2 + 2*cos(2*Pi/5) = (2*cos(Pi/5))^2.

1, 1, 2, 5, 14, 42, 131, ... = A080937 and the convergent is 3.24697... = 2 + 2*cos(2*Pi/7) = (2*cos(Pi/7))^2.

1, 1, 2, 5, 14, 42, 132, 429, 1429, ... = A080938 and the convergent is 3.53208... = 2 + 2*cos(2*Pi/9) = (2*cos(Pi/9))^2. (End)

The characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) = S(N, 2-x) - S(N-1, 2-x), with Chebyshev's S polynomial (see A049310) evaluated at 2-x. Proof by determinant expansion, to obtain the recurrence Phi(N, x) - (x-2)*Phi(N-1, x) - Phi(N-2, x), for N >= 2, and Phi(0, x) = 1 and Phi(1, x) = 1 - x, that is Phi(-1, x) = 1. The trace is tr(M_N) = 1 + 2^(N-1) = A000051(N-1), and Det(M_N) = 1. - Wolfdieter Lang, Mar 13 2020

The explicit form of the characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) := Det(M_N - x*1_N) = Sum_{k=0..N} binomial(N+k, 2*k)*(-x)^k = Sum_{k=0..N} A085478(N, k)*(-x)^k, for N >= 0, with Phi(0, x) := 1. Proof from the recurrence given in the preceding comment. - Wolfdieter Lang, Mar 25 2020

For the proofs of the 2 X 2, 3 X 3 and 4 X 4 conjectures, see the comments in the respective A-numbers A001519, A080937 and A080938. - Wolfdieter Lang, Mar 30 2020

Replace the main diagonal 1,2,2,2,... of the matrix M with 1,0,0,0,...; 1,1,1,1,...; 1,3,3,3,...; 1,2,1,2,...; 1,2,3,4,...; 1,0,1,0...; and 1,1,0,0,1,1,0,0,.... Take powers of M and extract the upper left terms, resulting in respectively: A001405, A001006, A033321, A176677, A006789, A090344, and A007902. - Gary W. Adamson, Apr 12 2022

The statement that the upper left entry of M^n is a Catalan number is equivalent to Exercise 41 of R. Stanley, "Catalan Numbers." - Richard Stanley, Feb 28 2023

If the upper left 1 in matrix M is replaced with 3, taking powers of the resulting matrix and extracting the upper left terms apparently results in sequence A001700. - Gary W. Adamson, Apr 03 2023

Row sums = the binomial transform of the Catalan numbers, A007317.

Right border = the Motzkin numbers, A001006.