Peter McNair - cp's OEIS frontend

User: Peter McNair

Peter McNair has authored 2 sequences.

A365066 Decimal expansion of the constant 1/0! - 1/1! + 1/2! + 1/3! - 1/4! + 1/5! + 1/6! - 1/7! + ...

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

Offset: 0

Peter McNair, Aug 19 2023

			0.63455111826122554275761424130960772236307995025163265587548911687697314...

Cf. A143820.

Digits:=105: evalf(sum(1/(3*n)!-1/(3*n+1)!+1/(3*n+2)!, n=0..infinity)); # Michal Paulovic, Aug 20 2023

RealDigits[E/3 - (4*Sin[Sqrt[3]/2-Pi/6])/(3*Sqrt[E]), 10, 105][[1]]

suminf(n=0,1/(3*n)!-1/(3*n+1)!+1/(3*n+2)!) \\ Michal Paulovic, Aug 20 2023

Equals e - 2*A143820.
Equals Sum_{n>=0} (-1)^(2^((n-1) mod 3) mod 2) / n! = e/3 - 4*sin(sqrt(3)/2 - Pi/6) / (3*sqrt(e)).
Equals Sum_{n>=0} 1/(3*n)! - 1/(3*n+1)! + 1/(3*n+2)!. - Michal Paulovic, Aug 19 2023

A352526 a(n) = Product_{k=0..n} Nimsum (2*k + 2), with Nimsum (2 + 2) = 0 replaced by 1.

Original entry on oeis.org

2, 2, 12, 48, 480, 3840, 53760, 645120, 11612160, 185794560, 4087480320, 81749606400, 2125489766400, 51011754393600, 1530352631808000, 42849873690624000, 1456895705481216000, 46620662575398912000, 1771585177865158656000, 63777066403145711616000, 2678636788932119887872000

Offset: 0

Peter McNair, Mar 19 2022

nonn

Nimsum 2*k + 2 = A004443(2*k).
Sum_{n>0} 1/a(n) = 1/sqrt(e) = A092605.
Sum_{n>0} 1/a(2*n-1) = sinh(1/2) = A334367.
Sum_{n>0} 1/a(2*n) = cosh(1/2) - 2*sinh(1/2).
a(n)/2^n = abs(A265376(n+1)) = Product_{k=0..n} Nimsum k + 1, with Nimsum 1 + 1 = 0 replaced by 1, n > 0.

Cf. A004443, A092605, A265376, A334367.

a[n_] := Product[If[k == 1, 1, BitXor[2*k, 2]], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Mar 19 2022 *)

PARI
```
a(n) = 2*prod(k=2,n,bitxor(2*k, 2))
```

a(n) = 2*Product_{k=2..n} A004443(2*k).
a(n) = 2^(n-1)*(n+1)!/floor((n+1)/2), n > 0.
a(n) = 2^(n-1)*(1+(-1)^n)*((n-1)!+n!)-((-1)^n-1)*(2*n)!!/2, n > 0.
a(n) = 2*a(n-1)*(n+(-1)^n), n > 1, with a(1) = 2.

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User: Peter McNair

A365066 Decimal expansion of the constant 1/0! - 1/1! + 1/2! + 1/3! - 1/4! + 1/5! + 1/6! - 1/7! + ...

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