A140138 -id:A140138 - cp's OEIS frontend

A140080 Fix e = 3; a(n) = minimal Hamming distance between the binary representation of n and the binary representation of any multiple ke (0 <= k <= n/e) which is a child of n.

0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 03 2008

nonn

A number m is a child of n if the binary representation of n has a 1 in every position where the binary representation of m has a 1.

In other words, this tells us how closely (in Hamming weight) we can approximate n "from below" by a multiple of e.

			If n = 14 = 1110_2, take k=2, ke = 6 = 110_2, which is Hamming distance 1 from n. This is the best we can do, so a(14) = 1.

Nadia Heninger and N. J. A. Sloane, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Fortran program for this and related sequences

For e=2 and 4 through 9 see A000035 and A140081 through A140086.

Cf. A140137, A140138, A140200-A140206.

Fortran
```
See Sloane link.
```

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.

Original entry on oeis.org

0, 1, 0, 3, 4, 1, 0, 9, 24, 22, 8, 1, 0, 27, 108, 171, 136, 57, 12, 1, 0, 81, 432, 972, 1200, 886, 400, 108, 16, 1, 0, 243, 1620, 4725, 7920, 8430, 5944, 2810, 880, 175, 20, 1, 0, 729, 5832, 20898, 44280, 61695, 59472, 40636, 19824, 6855, 1640, 258, 24, 1

Offset: 0

Franck Maminirina Ramaharo, Feb 26 2018

T(n,k) is the number of state diagrams having k components of n connected summed trefoil knots.

Row sums gives A001018.

			The triangle T(n, k) begins:
n\k 0     1      2      3       4       5       6      7        8       9
0:  0     1
1:  0     3      4      1
2:  0     9     24     22       8       1
3:  0    27    108    171     136      57      12       1
4:  0    81    432    972    1200     886     400     108      16       1

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.

Michael De Vlieger, Table of n, a(n) for n = 0..10301 (rows 0 <= n <= 100, flattened.)
Ryo Hanaki, Pseudo diagrams of knots, links and spatial graphs, Osaka Journal of Mathematics, Vol. 47 (2010), 863-883.
Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Carolina Medina, Jorge Ramírez-Alfonsín and Gelasio Salazar, On the number of unknot diagrams, arXiv:1710.06470 [math.CO], 2017.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row 2: row 5 of A158454.
Row 3: row 2 of A220665.
Row 4: row 5 of A219234.

row[n_] := CoefficientList[x*(x^2 + 4*x + 3)^n, x]; Array[row, 7, 0] // Flatten (* Jean-François Alcover, Mar 16 2018 *)

g(x, y) := taylor(x/(1 - y*(x^2 + 4*x + 3)), y, 0, 10)$
a : makelist(ratcoef(g(x, y), y, n), n, 0, 10)$
T : []$
for i:1 thru 11 do
  T : append(T, makelist(ratcoef(a[i], x, n), n, 0, 2*i - 1))$
T;

T(n, k) = polcoeff(x*(x^2 + 4*x + 3)^n, k);
tabf(nn) = for (n=0, nn, for (k=0, 2*n+1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 03 2018

T(n,k) = coefficients of x*(x^2 + 4*x + 3)^n.
T(n,k) = T(n-1,k-2) + 4*T(n-1,k-1) + 3*T(n-1,k), with T(n,0) = 0, T(n,1) = 3^n and  T(n,2) =  4*n*3^(n-1).
T(n,n+k+1) = A152842(2*n,n+k) and T(n,n-k) = A152842(2*n,n+k+1), for n >= k >= 0.
T(n,1) =  A000244(n).
T(n,2) =  A120908(n).
T(n,n+1) = A069835(n).
T(n,2*n-1) = A139272(n).
T(n,2*n) = A008586(n).
T(n,2*n-2) = A140138(4*n) = A185872(2n,2) for n >= 1.
G.f.: x/(1 - y*(x^2 + 4*x + 3)).

Typo in row 6 corrected by Jean-François Alcover, Mar 16 2018

A120956 G.f. A(x) satisfies x / Series_Reversion(x*A(x)) = (A(x) + 1+x)/2.

Original entry on oeis.org

1, 1, 2, 8, 50, 412, 4120, 47840, 628130, 9164600, 146786980, 2557718352, 48147082520, 973612557504, 21050077835440, 484637221115520, 11839623684281890, 305949448095405252, 8339153054042801704

Offset: 0

Paul D. Hanna, Jul 19 2006

nonn

The g.f. for A120955 = x / Series_Reversion(x*A(x)) = (A(x) + 1+x)/2.
a(n) = 2 (mod 4) when n = 2^k for k > 0. - Paul D. Hanna, Sep 21 2019
a(n) = 4 (mod 8) when n = A140138(k) for k > 0. - Paul D. Hanna, Sep 21 2019

			A(x) = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 412*x^5 + 4120*x^6 +...
The g.f. of A120955 is:
x/series_reversion(x*A(x)) = 1 + x + x^2 + 4*x^3 + 25*x^4 + 206*x^5 +...
Compare terms to see that A120955(n) = a(n)/2 for n>=2.
A(x*A(x)) = 1 + x + 3*x^2 + 14*x^3 + 92*x^4 + 774*x^5 +...
A(x)*(2-x) = 2 + x + 3*x^2 + 14*x^3 + 92*x^4 + 774*x^5 +...
Contribution from _Paul D. Hanna_, Sep 04 2010: (Start)
Let G(x) = x*A(x), then
A(x) = 1 + G(x)/2 + G(G(x))/2^2 + G(G(G(x)))/2^3 + G(G(G(G(x))))/2^4 + G(G(G(G(G(x)))))/2^5 +...
The table of coefficients in the iterations of G(x) = x*A(x) begin:
[1, 1, 2, 8, 50, 412, 4120, 47840, 628130, ...];
[1, 2, 6, 27, 170, 1380, 13580, 155568, 2020526, ...];
[1, 3, 12, 63, 422, 3482, 34208, 389007, 5010678, ...];
[1, 4, 20, 122, 892, 7690, 76900, 878032, 11284106, ...];
[1, 5, 30, 210, 1690, 15490, 160464, 1864844, 24130948, ...];
[1, 6, 42, 333, 2950, 29002, 315184, 3775392, 49699640, ...];
[1, 7, 56, 497, 4830, 51100, 587104, 7318983, 98962072, ...];
[1, 8, 72, 708, 7512, 85532, 1043032, 13621120, 190640924, ...];
[1, 9, 90, 972, 11202, 137040, 1776264, 24394608, 355390206, ...]; ...
in which the following sum along column k equals a(k+1):
a(2) = 2 = 1/2 + 2/4 + 3/8+ 4/16 + 5/32 + 6/64 +...
a(3) = 8 = 2/2 + 6/4 + 12/8 + 20/16 + 30/32 + 42/64 + ...
a(4) = 50 = 8/2 + 27/4 + 63/8 + 122/16 + 210/32 + 333/64 +...
a(5) = 412 = 50/2 + 170/4 + 422/8 + 892/16 + 1690/32 + 2950/64 +... (End)

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A120955.

{a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,t); A[ #A]=subst(Vec(serreverse(x/Ser(A)))[ #A],t,0)); Vec(serreverse(x/Ser(A)))[n+1]}
for(n=0,30, print1(a(n),", "))

/* Prints N terms using x/Series_Reversion(x*A(x)) = (A(x) + 1+x)/2 */
N = 30; {A=[1,1]; for(i=1,N, A = concat(A, -2*Vec(x/serreverse(x*Ser(concat(A,0))))[#A+1]); print1(i,",") );A} \\ Paul D. Hanna, Sep 21 2019

a(n) = 2*A120955(n) for n>=2.
G.f. A(x) satisfies:
(1) A( 2x/(A(x) + 1+x) ) = (A(x) + 1+x)/2.
(2) A(x) = F(x*A(x)) and F(x) = A(x/F(x)) where F(x) = g.f. of A120955.
(3) A(x) = (1 + A(x*A(x))) / (2-x).
(4) A(x) = 1 + Sum_{n>=0} G_n(x)/2^(n+1) where G(x)=x*A(x) and G_{n+1}(x) = G_n(x*A(x)) denotes iteration with G_0(x)=x. [From Paul D. Hanna, Sep 04 2010]

cp's OEIS Frontend

A140080 Fix e = 3; a(n) = minimal Hamming distance between the binary representation of n and the binary representation of any multiple ke (0 <= k <= n/e) which is a child of n.

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A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.

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A120956 G.f. A(x) satisfies x / Series_Reversion(x*A(x)) = (A(x) + 1+x)/2.

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A140137 Numbers n such that A140080(n) = 1.

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A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.