A292179 -id:A292179 - cp's OEIS frontend

A292178 Decimal expansion of: Sum_{n>=1} -1 / (n * (1/2 - 2^n)^n).

6, 2, 6, 6, 3, 6, 1, 3, 8, 7, 8, 9, 4, 3, 6, 3, 3, 9, 7, 1, 9, 2, 2, 4, 1, 1, 7, 2, 8, 0, 9, 6, 2, 6, 5, 9, 2, 4, 4, 0, 8, 3, 3, 3, 8, 4, 3, 4, 3, 3, 6, 9, 0, 0, 2, 6, 3, 1, 3, 2, 9, 0, 6, 2, 4, 9, 2, 3, 0, 1, 1, 1, 6, 8, 1, 4, 8, 8, 7, 4, 8, 3, 9, 5, 1, 4, 3, 9, 6, 9, 5, 4, 5, 8, 9, 7, 7, 2, 3, 8, 0, 9, 0, 9, 9, 7, 7, 7, 3, 6, 8, 4, 8, 2, 9, 5, 1, 0, 8, 4, 7, 1, 7, 2, 5, 0, 4, 4, 9, 4, 3, 7, 7, 4, 3, 5, 3, 4, 8, 8, 3, 9, 5, 5, 5, 7, 3, 6, 7, 4

Offset: 0

Paul D. Hanna, Oct 05 2017

This constant plus A292179 equals log(2), due to the identity (at x = 1/2):
Sum_{n=-oo..+oo, n<>0} (x - x^n)^n / n = -log(1-x).
More generally, it appears that Sum_{n = -oo..+oo, n <> 0} (x - a^n)^n / n = -log(1 - x) for |x| < 1 and |a| < 1. - Peter Bala, Nov 03 2024

			Constant t = 0.62663613878943633971922411728096265924408333843433690026313290...
where t = 2/(1*3) - 4/(2*7^2) + 8/(3*15^3) - 16/(4*31^4) + 32/(5*63^5) - 64/(6*127^6) + 128/(7*255^7) - 256/(8*511^8) + 512/(9*1023^9) - 1024/(10*2047^10) + 2048/(11*4095^11) - 4096/(12*8191^12) + 8192/(13*16383^13) - 16384/(14*32767^14) + 32768/(15*65535^15) +...
Also,
log(2) - t = 0/(1*2) + 1^2/(2*2^4) + 3^3/(3*2^9) + 7^4/(4*2^16) + 15^5/(5*2^25) + 31^6/(6*2^36) + 63^7/(7*2^49) + 127^8/(8*2^64) + 255^9/(9*2^81) + 511^10/(10*2^100) + 1023^11/(11*2^121) + 2047^12/(12*2^144) + 4095^13/(13*2^169) + 8191^14/(14*2^196) + 16383^15/(15*2^225) +... (constant A292179)

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A292179.

Constant: Sum_{n>=1} -(-1)^n * 2^n / (n * (2^(n+1) - 1)^n).

Constant: log(2) - Sum_{n>=1} (2^(n-1) - 1)^n / (n * 2^(n^2)).

A293381 Decimal expansion of Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).

Original entry on oeis.org

3, 0, 0, 0, 4, 9, 6, 8, 9, 8, 5, 9, 8, 6, 4, 7, 3, 2, 8, 7, 1, 8, 7, 7, 5, 0, 1, 5, 8, 5, 0, 5, 5, 7, 2, 3, 0, 5, 4, 1, 5, 8, 5, 5, 9, 3, 5, 3, 4, 9, 9, 5, 4, 3, 7, 9, 8, 6, 8, 9, 7, 0, 1, 4, 6, 0, 9, 1, 4, 7, 5, 4, 4, 3, 3, 9, 8, 7, 1, 3, 8, 1, 0, 6, 9, 9, 6, 9, 7, 1, 2, 3, 4, 1, 9, 4, 4, 5, 0, 5, 4, 4, 0, 4, 4, 9, 9, 3, 4, 7, 5, 5, 7, 6, 9, 0, 0, 6, 7, 4

Offset: 0

Paul D. Hanna, Oct 12 2017

This constant plus A293382 equals log(2), due to the identity:

Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 1/2, y = 1/3.

			Constant t = 0.3000496898598647328718775015850557230541585593534995437986897...
such that
t = (3 - 2)/(1*2*3) + (3^2 - 2)^2/(2*2^2*3^4) + (3^3 - 2)^3/(3*2^3*3^9) + (3^4 - 2)^4/(4*2^4*3^16) + (3^5 - 2)^5/(5*2^5*3^25) + (3^6 - 2)^6/(6*2^6*3^36) + (3^7 - 2)^7/(7*2^7*3^49) +...+ (3^n - 2)^n / (n * 2^n * 3^(n^2)) +...
More explicitly,
t = 1/(1*2*3) + 7^2/(2*4*3^4) + 25^3/(3*8*3^9) + 79^4/(4*16*3^16) + 241^5/(5*32*3^25) + 727^6/(6*64*3^36) + 2185^7/(7*128*3^49) + 6559^8/(8*256*3^64) + 19681^9/(9*512*3^81) + 59047^10/(10*1024*3^100) + 177145^11/(11*2048*3^121) + 531439^12/(12*4096*3^144) +...
Also,
log(2) - t = 2/(2*3-1) - 2^2/(2*(2*3^2-1)^2) + 2^3/(3*(2*3^3-1)^3) - 2^4/(4*(2*3^4-1)^4) + 2^5/(5*(2*3^5-1)^5) - 2^6/(6*(2*3^6-1)^6) + 2^7/(7*(2*3^7-1)^7) - 2^8/(8*(2*3^8-1)^8) +...+ -(-1)^n * 2^n / (n * (2*3^n - 1)^n) +...

Cf. A002162, A293382, A292178, A292179, A293383, A293384.

{t = suminf(n=1, 1.*(3^n - 2)^n / (n * 2^n * 3^(n^2)) )}
for(n=1,120, print1(floor(10^n*t)%10,", "))

Constant: Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).

Constant: log(2) - Sum_{n>=1} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).

A293382 Decimal expansion of Sum_{n>=1} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).

Original entry on oeis.org

3, 9, 3, 0, 9, 7, 4, 9, 0, 7, 0, 0, 0, 8, 0, 5, 7, 6, 5, 4, 5, 3, 5, 4, 6, 1, 9, 8, 7, 3, 1, 2, 0, 8, 4, 5, 0, 2, 1, 3, 4, 1, 5, 7, 5, 0, 0, 6, 7, 5, 5, 7, 1, 0, 3, 2, 1, 9, 9, 0, 3, 0, 8, 0, 3, 2, 4, 7, 8, 8, 6, 7, 5, 3, 5, 7, 0, 7, 5, 7, 7, 4, 9, 8, 8, 6, 6, 3, 5, 5, 7, 6, 2, 2, 2, 4, 2, 3, 6, 9, 9, 7, 9, 5, 6, 4, 8, 7, 5, 4, 4, 9, 9, 3, 7, 8, 5, 0, 5, 9

Offset: 0

Paul D. Hanna, Oct 13 2017

This constant plus A293381 equals log(2), due to the identity:

Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 1/2, y = 1/3.

			Constant t = 0.3930974907000805765453546198731208450213415750067557103219903...
such that
t = 2/(2*3-1) - 2^2/(2*(2*3^2-1)^2) + 2^3/(3*(2*3^3-1)^3) - 2^4/(4*(2*3^4-1)^4) + 2^5/(5*(2*3^5-1)^5) - 2^6/(6*(2*3^6-1)^6) + 2^7/(7*(2*3^7-1)^7) - 2^8/(8*(2*3^8-1)^8) +...+ -(-1)^n * 2^n / (n * (2*3^n - 1)^n) +...
More explicitly,
t = 2/5 - 4/(2*17^2) + 8/(3*53^3) - 16/(4*161^4) + 32/(5*485^5) - 64/(6*1457^6) + 128/(7*4373^7) - 256/(8*13121^8) + 512/(9*39365^9) - 1024/(10*118097^10) +...
Also,
log(2) - t = (3 - 2)/(1*2*3) + (3^2 - 2)^2/(2*2^2*3^4) + (3^3 - 2)^3/(3*2^3*3^9) + (3^4 - 2)^4/(4*2^4*3^16) + (3^5 - 2)^5/(5*2^5*3^25) + (3^6 - 2)^6/(6*2^6*3^36) + (3^7 - 2)^7/(7*2^7*3^49) +...+ (3^n - 2)^n / (n * 2^n * 3^(n^2)) +...

Cf. A002162, A293381, A292178, A292179, A293383, A293384.

{t = suminf(n=1, -1.*(-1)^n * 2^n / (n * (2*3^n - 1)^n) )}
for(n=1,120, print1(floor(10^n*t)%10,", "))

Constant: Sum_{n>=1} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).

Constant: log(2) - Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).

A293383 Decimal expansion of Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

Original entry on oeis.org

3, 9, 2, 7, 7, 1, 5, 7, 5, 5, 5, 5, 0, 6, 7, 5, 1, 1, 8, 5, 9, 1, 1, 1, 8, 7, 7, 2, 6, 1, 2, 2, 8, 0, 9, 1, 3, 4, 2, 7, 2, 3, 4, 4, 9, 0, 4, 2, 2, 6, 3, 4, 8, 6, 2, 0, 2, 3, 8, 8, 3, 4, 3, 8, 7, 3, 1, 7, 5, 1, 9, 7, 9, 9, 7, 0, 9, 7, 5, 9, 1, 8, 4, 9, 7, 0, 7, 2, 1, 8, 1, 6, 3, 4, 7, 6, 2, 4, 5, 5, 1, 3, 2, 1, 8, 9, 6, 7, 0, 1, 3, 5, 2, 4, 8, 6, 2, 6, 6, 3

Offset: 0

Paul D. Hanna, Oct 13 2017

This constant plus A293384 equals log(3), due to the identity:

Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 2/3, y = 1/2.

			Constant t = 0.3927715755550675118591118772612280913427234490422634862023883438....
such that
t = (2^2 - 3)/(1*3*2) + (2^3 - 3)^2/(2*3^2*2^4) + (2^4 - 3)^3/(3*3^3*2^9) + (2^5 - 3)^4/(4*3^4*2^16) + (2^6 - 3)^5/(5*3^5*2^25) + (2^7 - 3)^6/(6*3^6*2^36) + (2^8 - 3)^7/(7*3^7*2^49) + (2^9 - 3)^8/(8*3^8*2^64) + (2^10 - 3)^9/(9*3^9*2^81) +...+ (2^(n+1) - 3)^n/(n * 3^n * 2^(n^2)) +...
More explicitly,
t = 1/(1*3*2) + 5^2/(2*9*2^4) + 13^3/(3*27*2^9) + 29^4/(4*81*2^16) + 61^5/(5*243*2^25) + 125^6/(6*729*2^36) + 253^7/(7*2187*2^49) + 509^8/(8*6561*2^64) + 1021^9/(9*19683*2^81) + 2045^10/(10*59049*2^100) + 4093^11/(11*177147*2^121) + 8189^12/(12*531441*2^144) +...
Also,
log(3) - t = 3/(1*2*(3-1)) - 3^2/(2*4*(3*2-1)^2) + 3^3/(3*8*(3*2^2-1)^3) - 3^4/(4*16*(3*2^3-1)^4) + 3^5/(5*32*(3*2^4-1)^5) - 3^6/(6*64*(3*2^5-1)^6) + 3^7/(7*128*(3*2^6-1)^7) +...+ -(-1)^n*3^n/(n*2^n*(3*2^(n-1) - 1)^n) +...

Cf. A293384, A293381, A293382, A292178, A292179, A002391 (log 3).

{t = suminf(n=1, 1.*(2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)) )}
for(n=1,120, print1(floor(10^n*t)%10,", "))

Constant: Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

Constant: log(3) - Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

A293384 Decimal expansion of Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

Original entry on oeis.org

7, 0, 5, 8, 4, 0, 7, 1, 3, 1, 1, 3, 0, 4, 2, 1, 7, 9, 5, 3, 6, 1, 3, 3, 3, 5, 9, 6, 6, 1, 2, 9, 7, 6, 1, 3, 3, 0, 4, 7, 6, 7, 1, 0, 8, 7, 8, 0, 4, 8, 5, 9, 6, 5, 5, 3, 2, 3, 0, 5, 9, 8, 9, 7, 6, 4, 3, 1, 9, 0, 9, 5, 2, 2, 1, 5, 1, 1, 3, 7, 5, 0, 2, 3, 9, 0, 8, 5, 3, 6, 6, 5, 0, 2, 5, 5, 8, 4, 3, 2, 7, 4, 7, 5, 1, 0, 6, 2, 0, 5, 2, 4, 3, 3, 0, 0, 3, 0, 7, 8

Offset: 0

Paul D. Hanna, Oct 13 2017

This constant plus A293383 equals log(3), due to the identity:

Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 2/3, y = 1/2.

			Constant t = 0.7058407131130421795361333596612976133047671087804859655323059...
such that
t = 3/(1*2*(3-1)) - 3^2/(2*4*(3*2-1)^2) + 3^3/(3*8*(3*2^2-1)^3) - 3^4/(4*16*(3*2^3-1)^4) + 3^5/(5*32*(3*2^4-1)^5) - 3^6/(6*64*(3*2^5-1)^6) + 3^7/(7*128*(3*2^6-1)^7) - 3^8/(8*256*(3*2^7-1)^8) +...+ -(-1)^n*3^n/(n*2^n*(3*2^(n-1) - 1)^n) +...
More explicitly,
t = 3/(1*2*2) - 9/(2*4*5^2) + 27/(3*8*11^3) - 81/(4*16*23^4) + 243/(5*32*47^5) - 729/(6*64*95^6) + 2187/(7*128*191^7) - 6561/(8*256*383^8) + 19683/(9*512*767^9) - 59049/(10*1024*1535^10) + 177147/(11*2048*3071^11) - 531441/(12*4096*6143^12) +...
Also,
log(3) - t = (2^2 - 3)/(1*3*2) + (2^3 - 3)^2/(2*3^2*2^4) + (2^4 - 3)^3/(3*3^3*2^9) + (2^5 - 3)^4/(4*3^4*2^16) + (2^6 - 3)^5/(5*3^5*2^25) + (2^7 - 3)^6/(6*3^6*2^36) + (2^8 - 3)^7/(7*3^7*2^49) + (2^9 - 3)^8/(8*3^8*2^64) + (2^10 - 3)^9/(9*3^9*2^81) +...+ (2^(n+1) - 3)^n/(n * 3^n * 2^(n^2)) +...

Cf. A293383, A293381, A293382, A292178, A292179, A002391 (log 3).

{t = suminf(n=1, -1.*(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n) )}
for(n=1,120, print1(floor(10^n*t)%10,", "))

Constant: Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

Constant: log(3) - Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

cp's OEIS Frontend

A292178 Decimal expansion of: Sum_{n>=1} -1 / (n * (1/2 - 2^n)^n).

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A293381 Decimal expansion of Sum_{n>=1} (3^n - 2)^n / (n * 2^n * 3^(n^2)).

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A293382 Decimal expansion of Sum_{n>=1} -(-1)^n * 2^n / (n * (2*3^n - 1)^n).

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A293383 Decimal expansion of Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

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A293384 Decimal expansion of Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

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