Mohamed Sabba - cp's OEIS frontend

User: Mohamed Sabba

Mohamed Sabba has authored 5 sequences.

A362534 Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems.

1, 1, 6, 6, 15, 15, 140, 140, 315, 315, 1386, 1386, 3003, 3003, 51480, 51480, 109395, 109395, 92378, 92378, 969969, 969969, 2704156, 2704156, 16900975, 16900975, 70204050, 70204050, 145422675, 145422675, 4808643120, 4808643120, 9917826435, 9917826435, 40838108850, 40838108850

Offset: 1

Mohamed Sabba, Apr 24 2023

In spin physics and NMR, these numbers appear as the numerators of the ratio of different classes of upper bounds on the transfer of z-magnetization in AXn spin systems from a group of spin-1/2 nuclei Xn to a single spin-1/2 nucleus A.
The symmetry-constrained upper bounds are given by the function f(n):
(1) for even n, f(n) = (2^(1-n))*n*binomial(n-1, n/2)
(2) for odd n, f(n) = (2^(1-n))*n*binomial(n-1, (n-1)/2)
The adiabatic bounds are given by the function g(n):
(3) for even n, g(n) = 2*(1-(2^(-n))*binomial(n, n/2))
(4) for odd n, g(n) = 2*(1-(2^(-n))*binomial(n, (n-1)/2))
Where we have the relation:
(5) g(n) = 2*(1 - f(n+1)/(n+1))
The sequence a(n) is defined as the numerator of f(n)/g(n):
(6) a(n) = numerator(f(n)/g(n))
(7) for even n, f(n)/g(n) = (n/2)/(2^(n)*binomial(n, n/2)^(-1) - 1)
(8) for odd n, f(n)/g(n) = ((n+1)/2)/(2^(n)*binomial(n, (n+1)/2)^(-1) - 1)
The first few values of the upper symmetry-constrained bounds f(n) are {1, 1, 3/2, 3/2, 15/8, 15/8, 35/16, 35/16, 315/128, 315/128, ...} which appears to be related to A086116 and A001803.
The first few values of the upper adiabatic bounds g(n) are {1, 1, 5/4, 5/4, 11/8, 11/8, 93/64, 93/64, 193/128, 193/128, ...} which appears to be related to A141244 and A120778.
The first few values of f(n)/g(n) are {1, 1, 6/5, 6/5, 15/11, 15/11, 140/93, 140/93, 315/193, 315/193, ...}
Conjecture: the numerator of g(n) is the denominator of f(n)/g(n).

G. C. Chingas, A. N. Garroway, W. B. Moniz, and R. D. Bertrand, Adiabatic J cross-polarization in liquids for signal enhancement in NMR, Journal of Chemical Physics, 102:8 (1980), 2526-2528 (page 1, equation 2 gives an expression for the adiabatic bounds).
Malcolm H. Levitt, Symmetry constraints on spin dynamics: Application to hyperpolarized NMR, Journal of Chemical Physics, 102:8 (2016).
Ole W. Sørensen, A universal bound on spin dynamics, Journal of Magnetic Resonance, 262 (1990) (appendix gives proof of the expression for symmetry constrained bounds).

Cf. A001803, A086116, A120778, A141244 (denominators but shifted).

Table[Numerator[Ceiling[n/2]  (2^n Binomial[n, Ceiling[n/2]]^-1 - 1 )^-1], {n, 1, 20}]

a(n) = numerator(ceil(n/2)/(2^(n)*binomial(n,ceil(n/2))^(-1) - 1)); \\ Michel Marcus, Apr 25 2023

a(n) = numerator(ceiling(n/2)/(2^(n)*binomial(n,ceiling(n/2))^(-1) - 1)).

A262957 Numerators of the n-th iteration of the alternating continued fraction formed from the positive integers, starting with (1 - ...).

Original entry on oeis.org

2, 3, 19, 64, 538, 2833, 29169, 210308, 2572158, 23595915, 334778571, 3732092084, 60305234822, 791741083537, 14359827157009, 217037153818264, 4366918714540522, 74685204276602819, 1651116684587556019, 31524723785455714840, 759659139498065625218, 16017463672140861567617

Offset: 1

Mohamed Sabba, Nov 19 2015

As n->inf, a(n)/A263295(n) converges to 0.57663338973... (A346590); this number has a surprisingly elegant standard continued fraction representation of [0; 1, 1, 2, 1, 3, 4, 1, 5, 6, 1, 7, 8, ...].
From Robert Israel, Dec 22 2015: (Start)
a(n) is the numerator of b(n)/c(n) where
b(1) = 2, b(2) = 3, c(1) = 3, c(2) = 5,
b(n+1) = (((-1)^n*(n-1)+n*(n+2))*b(n) - (1+(-1)^n*(n+1))*b(n-1))/(n-(-1)^n),
c(n+1) = (((-1)^n*(n-1)+n*(n+2))*c(n) - (1+(-1)^n*(n+1))*c(n-1))/(n-(-1)^n).
Conjecture: b(n) and c(n) are coprime for all n, so that a(n) = b(n).
I have verified this for n <= 10000. (End)

			(1-1/(2+1)) = 2/3, so a(1) = 2;
(1-1/(2+1/(3-1))) = 3/5, so a(2) = 3;
(1-1/(2+1/(3-1/(4+1)))) = 19/33, so a(3) = 19;
(1-1/(2+1/(3-1/(4+1/(5-1))))) = 64/111, so a(4) = 64.

Robert Israel, Table of n, a(n) for n = 1..448
Peter Bala, A note on A262957 and A263295

Same principle as A244279 and A244280 - except here we begin with subtraction rather than addition.

Cf. A263295 (denominators), A346590.

P[1]:= 2: P[2]:= 3:
Q[1]:= 3; Q[2]:= 5;
for i from 2 to 100 do
  P[i+1]:= ((-1)^i*(i-1) + i^2 + 2*i)/(i-(-1)^i)*P[i] + (1 + (i+1)*(-1)^i)/((-1)^i-i)*P[i-1];
  Q[i+1]:= ((-1)^i*(i-1) + i^2 + 2*i)/(i-(-1)^i)*Q[i] + (1 + (i+1)*(-1)^i)/((-1)^i-i)*Q[i-1];
od:
seq(numer(P[i]/Q[i]),i=1..100); # Robert Israel, Dec 22 2015

a(n) = if(n%2==0, s=-1, s=1); t=1; while(n>-1, t=n+1+s/t; n--; s=-s); denominator(t=1/t)
vector(30, n, a(n)) \\ Mohamed Sabba, Dec 22 2015

More terms from Mohamed Sabba, Dec 22 2015

A263295 Denominators of the n-th iteration of the alternating continued fraction formed from the positive integers, starting with (1 - ...).

Original entry on oeis.org

3, 5, 33, 111, 933, 4913, 50585, 364717, 4460647, 40920133, 580574377, 6472209467, 104581586665, 1373040648769, 24902871413201, 376386726269561, 7573128424949291, 129519388933667493, 2863373356440803473, 54670305859684290279, 1317404009250178503245

Offset: 1

Mohamed Sabba, Nov 20 2015

As n->inf, A262957(n)/a(n) converges to 0.57663338973018...; this number has a surprisingly elegant standard continued fraction representation: [0; 1, 1, 2, 1, 3, 4, 1, 5, 6, 1, 7, 8, ...].

			(1-1/(2+1)) = 2/3, so a(1) = 3;
(1-1/(2+1/(3-1))) = 3/5, so a(2) = 5;
(1-1/(2+1/(3-1/(4+1)))) = 19/33, so a(3) = 33;
(1-1/(2+1/(3-1/(4+1/(5-1))))) = 64/111, so a(4) = 111.

Jinyuan Wang, Table of n, a(n) for n = 1..448

Same principle as A244279 and A244280 - except here we begin with subtraction rather than addition.

Cf. A262957 (numerators).

a(n) = if(n%2==0, s=-1, s=1); t=1; while(n>0, t=n+1+s/t; n--; s=-s); denominator(t=1/t)
vector(30, n, a(n)) \\ corrected by Mohammed Sabba, Dec 22 2015

More terms from Mohamed Sabba, Dec 22 2015

A244280 Denominators of the n-th iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).

Original entry on oeis.org

2, 2, 11, 27, 202, 870, 8129, 50681, 570638, 4673558, 61724211, 627102091, 9514420518, 115483788186, 1980202320561, 27962630844865, 534877446987082, 8615820301234778, 181912525664114699, 3292162161484924619, 76056192127792619858, 1527880958525256735838

Offset: 1

Mohamed Sabba, Jun 24 2014

As n-->inf, a(n) converges to 0.628736607098954801603428...

This is the result of taking the denominator of a continued fraction with alternating signs a(n) = 1/(1+1/(2-1/(3+1/(4-...1/(n +/- 1))))), where addition follows an odd number and subtraction follows an even number.

			a(1) = 1/(1+1) = 1/2;
a(2) = 1/(1+1/(2-1)) = 1/2;
a(3) = 1/(1+1/(2-1/(3+1))) = 7/11;
a(4) = 1/(1+1/(2-1/(3+1/(4-1)))) = 17/27.

Robert Israel, Table of n, a(n) for n = 1..449

Cf. A244279 (Numerators).

seq(denom(numtheory:-cfrac([0, [1,1], seq([(-1)^j,j],j=2..n),[(-1)^(n+1),1]])), n = 1..40); # Robert Israel, Jan 17 2016

a(n) = if(n%2==0,s=-1,s=1); t=1; while(n>0, t=n+s/t; n--; s=-s); denominator(t=1/t)
vector(30, n, a(n)) \\ Colin Barker, Jul 20 2014

More terms from Colin Barker, Jul 20 2014

A244279 Numerators of the n-th iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).

Original entry on oeis.org

1, 1, 7, 17, 127, 547, 5111, 31865, 358781, 2938437, 38808271, 394282041, 5982064475, 72608885159, 1245025688399, 17581129642961, 336297031232409, 5417081623572649, 114375064174857015, 2069902867431592833, 47819312187294567447, 960634689914268797707

Offset: 1

Mohamed Sabba, Jun 24 2014

As n-->inf, a(n)/A244280(n) converges to 0.628736607098954801603428... ; this number has a surprisingly elegant standard continued fraction representation of [0; 1, 1, 1, 2, 3, 1, 4, 5, 1, 6, 7, 1, 8, 9...].

			a(1) = 1/(1+1) = 1/2;
a(2) = 1/(1+1/(2-1)) = 1/2;
a(3) = 1/(1+1/(2-1/(3+1))) = 7/11;
a(4) = 1/(1+1/(2-1/(3+1/(4-1)))) = 17/27.

Robert Israel, Table of n, a(n) for n = 1..449

Cf. A244280 (Denominators).

seq(numer(numtheory:-cfrac([0, [1,1], seq([(-1)^j,j],j=2..n),[(-1)^(n+1),1]])), n = 1..40); # Robert Israel, Jan 17 2016

a(n) = if(n%2==0,s=-1,s=1); t=1; while(n>0, t=n+s/t; n--; s=-s); numerator(t=1/t)
vector(30, n, a(n)) \\ Colin Barker, Jul 20 2014

This is the result of taking the numerator of a continued fraction with alternating signs a(n) = 1/(1+1/(2-1/(3+1/(4-...1/(n +/- 1))))), where addition follows an odd number and subtraction follows an even number.

More terms from Colin Barker, Jul 20 2014

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User: Mohamed Sabba

A362534 Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems.

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A262957 Numerators of the n-th iteration of the alternating continued fraction formed from the positive integers, starting with (1 - ...).

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A263295 Denominators of the n-th iteration of the alternating continued fraction formed from the positive integers, starting with (1 - ...).

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A244280 Denominators of the n-th iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).

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A244279 Numerators of the n-th iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).

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