The On-Line Encyclopedia of Integer Sequences (OEIS)

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Revision History for A067518

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A067518

Number of spanning trees in n X n X 2 grid.
(history; published version)

#29 by Vaclav Kotesovec at Sun Mar 17 11:52:41 EDT 2024

STATUS

editing

approved

#28 by Vaclav Kotesovec at Sun Mar 17 11:52:15 EDT 2024

FORMULA

a(n) ~ c * d^n * 2^(n^2) * exp(4*n^2*(G/Pi + m/Pi^2)) / sqrt(n), where m = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dy dx = A340422 = 2.5516988064039243609935616786056293143254369265492957275912213393835172..., d = (2*sqrt(2)-3)*(2+sqrt(3))*(sqrt(15)-4) = 0.08133113706589390743806107..., c = 5^(1/4) * Gamma(1/4) / (sqrt(3) * (2*Pi)^(3/4)) = 0.788729432659299631982768... and G is Catalan's constant A006752. Equivalently, m = Pi * Integral_{x=0..Pi/2} (log(1 + sqrt(1 + 2/(3 - 2*cos(x)^2))) + log((1 + 2*sin(x)^2)/4)/2) dx. - Vaclav Kotesovec, Jan 06 2021, updated Mar 16 17 2024

STATUS

approved

editing

#27 by Vaclav Kotesovec at Sat Mar 16 12:54:20 EDT 2024

STATUS

editing

approved

#26 by Vaclav Kotesovec at Sat Mar 16 12:53:36 EDT 2024

FORMULA

a(n) ~ c * d^n * 2^(n^2) * exp(4*n^2*(G/Pi + m/Pi^2)) / sqrt(n), where m = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dy dx = A340422 = 2.5516988064039243609935616786056293143254369265492957275912213393835172..., d = (2*sqrt(2)-3)*(2+sqrt(3))*(sqrt(15)-4) = 0.08133113706589390743806107..., c = 0.788729432659299631982768... and G is Catalan's constant A006752. Equivalently, m = Pi * Integral_{x=0..Pi/2} (log(1 + sqrt(1 + 2/(3 - 2*cos(x)^2))) + log((1 + 2*sin(x)^2)/4)/2) dx. - Vaclav Kotesovec, Jan 06 2021, updated Mar 16 2024

STATUS

approved

editing

#25 by Michael De Vlieger at Tue Feb 28 23:46:48 EST 2023

STATUS

proposed

approved

#24 by Jon E. Schoenfield at Tue Feb 28 23:22:23 EST 2023

STATUS

editing

proposed

#23 by Jon E. Schoenfield at Tue Feb 28 23:22:22 EST 2023

FORMULA

a(n) ~ c * d^n * 2^(n^2) * exp(4*n^2*(G/Pi + m/Pi^2)) / sqrt(n), where m = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dy dx = A340422 = 2.5516988064039243609935616786056293143254369265492957275912213393835172..., d = 0.08133113706589390743806107..., c = 0.788729432659299631982768... and G is the Catalan's constant A006752. Equivalently, m = Pi * Integral_{x=0..Pi/2} (log(1 + sqrt(1 + 2/(3 - 2*cos(x)^2))) + log((1 + 2*sin(x)^2)/4)/2) dx. - Vaclav Kotesovec, Jan 06 2021

STATUS

approved

editing

#22 by Vaclav Kotesovec at Thu Jan 07 07:15:37 EST 2021

STATUS

editing

approved

#21 by Vaclav Kotesovec at Thu Jan 07 07:15:30 EST 2021

FORMULA

a(n) ~ c * d^n * 2^(n^2) * exp(4*n^2*(G/Pi + m/Pi^2)) / sqrt(n), where m = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dx dy dx = A340422 = 2.5516988064039243609935616786056293143254369265492957275912213393835172..., d = 0.08133113706589390743806107..., c = 0.788729432659299631982768... and G is the Catalan's constant A006752. Equivalently, m = Pi * Integral_{x=0..Pi/2} (log(1 + sqrt(1 + 2/(3 - 2*cos(x)^2))) + log((1 + 2*sin(x)^2)/4)/2) dx. - Vaclav Kotesovec, Jan 06 2021

#20 by Vaclav Kotesovec at Thu Jan 07 06:45:24 EST 2021

FORMULA

a(n) ~ c * d^n * 2^(n^2) * exp(4*n^2*(G/Pi + m/Pi^2)) / sqrt(n), where m = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dx dy = A340422 = 2.5516988064039243609935616786056293143254369265492957275912213393835172..., d = 0.08133113706589390743806107..., c = 0.788729432659299631982768... and G is the Catalan's constant A006752. Equivalently, m = Pi * Integral_{x=0..Pi/2} (log(1 + sqrt(1 + 2/(3 - 2*cos(x)^2))) + log((1 + 2*sin(x)^2)/4)/2) dx. - Vaclav Kotesovec, Jan 06 2021

STATUS

approved

editing