A006684 -id:A006684 - cp's OEIS frontend

A133423 Analog of A006684 for the 5x+1 problem (cf. A133419).

1, 5, 17, 23, 41, 53, 65, 71, 77, 95, 221, 317, 365, 383, 3317, 3575, 3605, 6473, 24125, 31901, 39965, 44183, 163733, 317885, 490541, 519113, 558365, 602591, 707735, 753023, 1019615, 1463897, 1597973, 1752575, 4595735, 6197855

Offset: 1

N. J. A. Sloane, Nov 27 2007

nonn

The 5x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, otherwise 5x+1.

N. J. A. Sloane, Table of n, a(n) for n=1..89 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133419, A133424, A133419, ...

A133425 Analog of A006684 for the 7x+1 problem (cf. A133421).

Original entry on oeis.org

1, 7, 11, 19, 31, 43, 163, 283, 403, 1111, 1123, 1243, 1303, 1549, 1963, 4123, 9643, 10003, 11539, 21431, 76963, 97031, 468109, 1351963, 4553323, 4778471, 5163139, 6563551, 7618843, 45214123, 65704243, 161738803, 202903723

Offset: 1

N. J. A. Sloane, Nov 27 2007

nonn

The 7x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, x/5 if x is divisible by 5, otherwise 7x+1.

N. J. A. Sloane, Table of n, a(n) for n=1..53 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, The px+1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A133421, A133426, A133419, ...

A117936 Triangle, rows = inverse binomial transforms of A073133 columns.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 9, 12, 6, 5, 24, 56, 60, 24, 8, 62, 228, 414, 360, 120, 13, 156, 864, 2400, 3480, 2520, 720, 21, 387, 3132, 12606, 27360, 32640, 20160, 5040, 34, 951, 11034, 62220, 190704, 335160, 337680, 181440, 40320, 55, 2323, 38136, 294588, 1229760, 2997120, 4394880, 3820320, 1814400, 362880

Offset: 1

Gary W. Adamson, Apr 03 2006

Left border of the triangle = Fibonacci numbers, right border = factorials. Companion triangle A117937 is generated from Lucas polynomials, using analogous operations.

Note that binomial transforms are defined from offset 1 here. - R. J. Mathar, Aug 16 2019

			First few columns of A073133 are: (1, 1, 1, ...); (1, 2, 3, ...); (2, 5, 10, 17, ...); (3, 12, 33, 72, ...). As sequences, these are f(x), Fibonacci polynomials: (1); (x); (x^2 + 1); (x^3 + 2*x); (x^4 + 3*x^2 + 1); (x^5 + 4*x^3 + 3*x); ... For example, f(x), x = 1,2,3,... using (x^4 + 3*x^2 + 1) generates Column 5 of A073133: (5, 29, 109, 305, ...).
Inverse binomial transforms of the foregoing columns generates the triangle rows:
  1;
  1,  1;
  2,  3,   2;
  3,  9,  12,   6;
  5, 24,  56,  60,  24;
  8, 62, 228, 414, 360, 120;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flatten

Cf. A073133, A117937, A117938.

Cf. A006684 (column 2), A309717 (column 3 halved).

A117936 := proc(n,k)
    add( A073133(i+1,n)*binomial(k-1,i)*(-1)^(i-k-1),i=0..k-1) ;
end proc:
seq(seq(A117936(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Aug 16 2019

(* A = A073133 *) A[, 1] = 1; A[n, k_] := A[n, k] = If[k < 0, 0, n A[n, k - 1] + A[n, k - 2]];
T[n_, k_] := Sum[A[i+1, n] Binomial[k-1, i] (-1)^(i - k - 1), {i, 0, k-1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 01 2020, from Maple *)

@CachedFunction
def A073133(n,k): return 0 if (k<0) else 1 if (k==1) else n*A073133(n,k-1) + A073133(n,k-2)
def A117936(n,k): return sum( (-1)^(j-k+1)*binomial(k-1, j)*A073133(j+1,n) for j in (0..k-1) )
flatten([[A117936(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 23 2021

Inverse binomial transforms of A073133 columns. Such columns are f(x), Fibonacci polynomials.

A091913 Triangle read by rows: a(n,k) = C(n,k)*(2^(n-k) - 1) for k= n, where k=0..max(n-1,0).

Original entry on oeis.org

0, 1, 3, 2, 7, 9, 3, 15, 28, 18, 4, 31, 75, 70, 30, 5, 63, 186, 225, 140, 45, 6, 127, 441, 651, 525, 245, 63, 7, 255, 1016, 1764, 1736, 1050, 392, 84, 8, 511, 2295, 4572, 5292, 3906, 1890, 588, 108, 9, 1023, 5110, 11475, 15240, 13230, 7812, 3150, 840, 135, 10, 2047

Offset: 0

Ross La Haye, Mar 10 2004

Row lengths are 1,1,2,3,4,... = A028310. - M. F. Hasler, Jul 21 2012
Rows: Sum of the n-th row = A001047(n); Sum of the n-th row excluding column 0 = A028243(n+1). Columns: a(n,0) = A000225(n); a(n,1) = A058877(n). Diagonals: a(n,n-2) = A045943(n-1). Also note that the sums of the antidiagonals = A006684.
As an infinite lower triangular matrix * the Bernoulli numbers as a vector (Cf. A027641) = the natural numbers: [1, 2, 3, ...]. The same matrix * the Bernoulli number version starting [1, 1/2, 1/6, ...] = A001787: (1, 4, 12, 32, ...). - Gary W. Adamson, Mar 13 2012

			Triangle begins
   0;
   1;
   3,   2;
   7,   9,   3;
  15,  28,  18,   4;
  31,  75,  70,  30,   5;
  63, 186, 225, 140,  45,   6;
  ...
a(5,3) = 30 because C(5,3) = 10, 2^(5 - 3) - 1 = 3 and 10 * 3 = 30.

Cf. A007318, A038207.

Cf. A001787, A027641, A027642.

For k>=n, a(n, k) = 0; for k < n, a(n, k) = C(n, k) * (2^(n-k) - 1) = Sum [C(n,k) * C(n-k, m), {m=1 to n-k}]. [Formula corrected Aug 22 2006]
The triangle (1; 3,2; 7,9,3; ...) = A007318^2 - A007318, then delete the right border of zeros. - Gary W. Adamson, Nov 16 2007
O.g.f.: 1/( (1 - (1 + x)*t)*(1 - (2 + x)*t) ) = 1 + (3 + 2*x)*t + (7 + 9*x + 3*x^2)*t^2 + .... - Peter Bala, Jul 16 2013

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A309717 Convolve Fibonacci, Pell and bronze Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 1, 6, 28, 114, 432, 1566, 5517, 19068, 65044, 219852, 738316, 2468028, 8222805, 27330858, 90685224, 300521622, 994991716, 3292117698, 10887332473, 35992718136, 118958691528, 393093822744, 1298783453112, 4290755845176, 14174217683209, 46821054068430, 154655837126740

Offset: 0

R. J. Mathar, Aug 16 2019

Index entries for linear recurrences with constant coefficients, signature (6,-8,-6,8,6,1).

Cf. A006684, A006190 (bronze Fibonacci numbers), A117936.

-x^3/( (x^2+2*x-1)*(x^2+3*x-1)*(x^2+x-1) ) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

G.f.: -x^3/( (x^2+2*x-1) * (x^2+3*x-1) * (x^2+x-1) ) = A006190(x) * A000045(x) * A000129(x).
Conjecture: 2*a(n) = A117936(n,3).
2*a(n) = A006190(n) + A000045(n) - 2*A000129(n). - R. J. Mathar, Mar 10 2023, typo corrected by Xiaoyuan Wang and Greg Dresden, May 08 2024

cp's OEIS Frontend

A133423 Analog of A006684 for the 5x+1 problem (cf. A133419).

Views

Author

Keywords

Comments

Links

Crossrefs

A133425 Analog of A006684 for the 7x+1 problem (cf. A133421).

Views

Author

Keywords

Comments

Links

Crossrefs

A117936 Triangle, rows = inverse binomial transforms of A073133 columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A091913 Triangle read by rows: a(n,k) = C(n,k)*(2^(n-k) - 1) for k= n, where k=0..max(n-1,0).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A309717 Convolve Fibonacci, Pell and bronze Fibonacci numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula