A247719 -id:A247719 - cp's OEIS frontend

A095214 a(n) = floor((Pi/sqrt(2))^n).

1, 2, 4, 10, 24, 54, 120, 266, 593, 1317, 2926, 6501, 14441, 32081, 71266, 158315, 351688, 781255, 1735512, 3855339, 8564410, 19025337, 42263673, 93886277, 208562871, 463310210, 1029216515, 2286344247, 5078979923, 11282656623

Offset: 0

Jason Earls, Jun 22 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A247719 (Pi/sqrt(2)).

R:= RealField(20); [Floor((Pi(R)/Sqrt(2))^n): n in [0..30]]; // G. C. Greubel, Sep 27 2018

Floor[(Pi/Sqrt[2])^Range[0,30]] (* Harvey P. Dale, May 12 2014 *)

vector(30, n, n--; floor((Pi/sqrt(2))^n)) \\ G. C. Greubel, Sep 27 2018

A256924 Decimal expansion of log(Pi/sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 13 2015

Previous name, "Decimal expansion of Sum_{k>=1} (zeta(2k)/k)*(2/3)^(2k)", was wrong. See A377008 for the correct sequence with this name. - Amiram Eldar, Oct 12 2024

			0.798156295569427519434811290623970427609544745735183944453283...

Cf. A247719, A377008.

RealDigits[Log[Pi/Sqrt[2]], 10, 105] // First

PARI
```
log(Pi/sqrt(2))
```

Equals log(A247719).

Name corrected by Amiram Eldar, Oct 12 2024

A380141 Decimal expansion of the real part of (-1)^sqrt(i), negated, where i is the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jan 23 2025

			0.0656897647351535320902668799676610103365089153475...

Math Beast, Oxford university entrance exam question, YouTube video, 2025.

A380142 is the imaginary part.

Cf. A247719.

evalf[140](-Re((-1)^sqrt(I)));  # Alois P. Heinz, Jan 23 2025

Prepend[RealDigits[Re[(-1)^Sqrt[I]], 10, 87][[1]], 0] (* Shenghui Yang, Jan 23 2025 *)

PARI
```
real((-1)^sqrt(I))
```

Equals -cos(Pi/sqrt(2))/exp(Pi/sqrt(2)).

A318614 Scaled g.f. S(u) = Sum_{n>0} a(n)16(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).

Original entry on oeis.org

1, 6, 76, 1260, 24276, 515592, 11721072, 280020312, 6945369860, 177358000248, 4635276570288, 123449340098448, 3339525750984528, 91535631253610400, 2537277723600799680, 71015600640006437040, 2004523477053308685540, 57003431104378084982040

Offset: 1

Bradley Klee, Aug 30 2018

nonn

Area interior to the central loop of u = 2*H = x^2 + y^2 - (1/2)*(x^4 + y^4) equals to Pi*S(u), when u in [0,1/2].

			Singular Value: S(1/2) = 1/sqrt(2).
N=4, h=1/sqrt(2) Quantization: S(u) = (n+1/2)*h/N.
  n  |                  u
==================================================
  0  |  0.08544689553344134756293807606337...
  1  |  0.23840989875904155311088418238272...
  2  |  0.36638282702449450473835851051425...
  3  |  0.46595506694324457665483887176081...

E. Heller, The Semiclassical Way to Dynamics and Spectroscopy, Princeton University Press, 2018, page 204.

Eric Weisstein's World of Mathematics, Plane Division by Ellipses.
Eric Weisstein's World of Mathematics, Circle Ellipse Intersection.

Cf. A318417, A010503, A247719, A000888.

a:=[1,6];; for n in [3..20] do a[n]:=(1/(n*(n-1)^2))*(12*(n-1)*(2*n-3)^2*a[n-1]-(128*(n-2)*(2*n-5)*(2*n-3)*a[n-2])); od; a; # Muniru A Asiru, Sep 24 2018

RecurrenceTable[{(n-1)^2*n*a[n] - 12*(n-1)*(2*n-3)^2*a[n-1] + 128*(n-2)*(2*n-5)*(2*n-3)*a[n-2] == 0, a[1] == 1, a[2] == 6}, a, {n, 1, 1000}]

(n-1)^2*n*a(n) - 12*(n-1)*(2*n-3)^2*a(n-1) + 128*(n-2)*(2*n-5)*(2*n-3)*a(n-2) == 0.

a(n) = A000108(n-1)*A098410(n-1).

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A095214 a(n) = floor((Pi/sqrt(2))^n).

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A256924 Decimal expansion of log(Pi/sqrt(2)).

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A380141 Decimal expansion of the real part of (-1)^sqrt(i), negated, where i is the imaginary unit.

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A318614 Scaled g.f. S(u) = Sum_{n>0} a(n)*16*(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).

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A318614 Scaled g.f. S(u) = Sum_{n>0} a(n)16(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).