George Plousos - cp's OEIS frontend

User: George Plousos

George Plousos has authored 2 sequences.

A376177 Triangle, read by rows, where T(n,k) = T(n-1,k-1) + 2*T(n,k-1) when k > 0, else T(n,0) = T(n-1,n-1) when n > 0, with T(0,0) = 1.

1, 1, 3, 3, 7, 17, 17, 37, 81, 179, 179, 375, 787, 1655, 3489, 3489, 7157, 14689, 30165, 61985, 127459, 127459, 258407, 523971, 1062631, 2155427, 4372839, 8873137, 8873137, 17873733, 36005873, 72535717, 146134065, 294423557, 593219953, 1195313043, 1195313043, 2399499223, 4816872179, 9669750231, 19412036179, 38970206423, 78234836403, 157062892759, 315321098561, 315321098561

Offset: 0

George Plousos and Paul D. Hanna, Sep 22 2024

This triangle was found by George Plousos while exploring a variation of Aitken's array (A011971).

			G.f.: A(x,y) = 1 + (3*y + 1)*x + (17*y^2 + 7*y + 3)*x^2 + (179*y^3 + 81*y^2 + 37*y + 17)*x^3 + (3489*y^4 + 1655*y^3 + 787*y^2 + 375*y + 179)*x^4 + (127459*y^5 + 61985*y^4 + 30165*y^3 + 14689*y^2 + 7157*y + 3489)*x^5 + (8873137*y^6 + 4372839*y^5 + 2155427*y^4 + 1062631*y^3 + 523971*y^2 + 258407*y + 127459)*x^6 + ...
which is defined by A(x,y) = (B(x) - 2*(B(x*y) - 1)/x) / (1 - (2+x)*y),
where B(x) = 1 + x*B( 2*x/(1-x) )/(1-x) is the g.f. B(x) for A126443,
B(x) = 1 + x + 3*x^2 + 17*x^3 + 179*x^4 + 3489*x^5 + 127459*x^6 + 8873137*x^7 + 1195313043*x^8 + 315321098561*x^9 + ... + A126443(n)*x^n + ...
This triangle begins
  1,
  1, 3,
  3, 7, 17,
  17, 37, 81, 179,
  179, 375, 787, 1655, 3489,
  3489, 7157, 14689, 30165, 61985, 127459,
  127459, 258407, 523971, 1062631, 2155427, 4372839, 8873137,
  8873137, 17873733, 36005873, 72535717, 146134065, 294423557, 593219953, 1195313043,
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..1275

Cf. A011971, A126443.

{A126443(n) = if(n==0, 1, sum(k=0, n-1, binomial(n-1, k) * 2^k * A126443(k)))}
{T(n,k) = sum(j=0,k, binomial(k,j) * 2^j * A126443(n-k+j) )}
for(n=0,10, for(k=0,n, print1(T(n,k),", ")); print(""))

If k > 0, T(n,k) = T(n-1,k-1) + 2*T(n,k-1), else if n > 0, T(n,0) = T(n-1,n-1), with T(0,0) = 1.
T(n,k) = Sum_{j=0..k} binomial(k,j) * 2^j * A126443(n-k+j), where A126443(m) = Sum_{k=0..m-1} binomial(m-1, k) * 2^k * A126443(k) for m > 0 with A126443(0) = 1.
G.f. A(x,y) = (B(x) - 2*(B(x*y) - 1)/x) / (1 - (2+x)*y), where B(x) = 1 + x*B( 2*x/(1-x) )/(1-x) is the g.f. B(x) for A126443 given therein by Ilya Gutkovskiy.

A334904 a(n) is the least integer b such that the fractions (b^0)/p, (b^1)/p, ..., (b^(r-1))/p where p is the n-th prime, produce the A006556(n) distinct cycles.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 3, 2, 3, 2, 1, 2, 1, 1, 2, 7, 2, 3, 2, 3, 1, 2, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 7, 1, 2, 3, 2, 1, 2, 1, 1, 7, 7, 3, 1, 1, 1, 6, 2, 3, 2, 2, 2, 11, 1, 2, 2, 1, 2, 2, 2, 7, 1, 2, 1, 1, 1, 2, 3, 7, 1, 2, 7, 1, 3, 2, 3, 3, 1, 2, 2, 13

Offset: 1

George Plousos, May 15 2020

With the exception of the prime numbers 2 and 5, the values of r mentioned above form the sequence A006556.

Detection of all different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p=n-th prime. If for the n-th prime p the number of different cycles of digits is equal to r, then there will be the smallest integer b in the interval 0 < b < p with the following property: The fractions (b^0)/p, (b^1)/p, ..., (b^(r-1))/p will produce r different cycles of digits. In this case the term a(n) of the sequence becomes equal to b.

			For n=13, prime(13)=41, there are A006556(13)=8 cycles.
With b=3, we get (normally, these fractions should be in the form (b^k mod p)/p):
  frac(3^0 / 41) = 0.02439 (1)
  frac(3^1 / 41) = 0.07317 (2)
  frac(3^2 / 41) = 0.21951 (3)
  frac(3^3 / 41) = 0.65853 (4)
  frac(3^4 / 41) = 0.97560 (5)
  frac(3^5 / 41) = 0.92682 (6)
  frac(3^6 / 41) = 0.78048 (7)
  frac(3^7 / 41) = 0.34146 (8=r)
So a(13) = 3.

Cf. A006556.

\\ default(realprecision, 1000)
nbc(p) = (p-1)/znorder(Mod(10, p));
len(p) = znorder(Mod(10, p));
pad(x, sz) = {while(#digits(x) != sz, x*=10); x;}
cmpc(x,y) = {if (x==y, return (0)); my(dx=digits(x), dy=digits(y), v=dx); for (k=1, #dx, v=vector(#v, k, if (k==#v, v[1], v[k+1])); if (v == dy, return (0));); return (1);}
decimals(x, sz) = pad(floor(1.0*10^sz*x), sz);
a(n) = {my(p=prime(n)); if ((p==2), return (1)); if ((p==5), return (2)); my(sz=len(p), nb=nbc(p), m=1); while (#vecsort(vector(f(p), k, decimals((m^(k-1) % p)/p, sz)),cmpc,8) != nb, m++); m;} \\ Michel Marcus, May 29 2020

cp's OEIS Frontend

User: George Plousos

A376177 Triangle, read by rows, where T(n,k) = T(n-1,k-1) + 2*T(n,k-1) when k > 0, else T(n,0) = T(n-1,n-1) when n > 0, with T(0,0) = 1.

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