A332133 -id:A332133 - cp's OEIS frontend

A385257 Decimal expansion of the surface area of a gyroelongated triangular bicupola with unit edge.

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

Offset: 2

Paolo Xausa, Jun 24 2025

The gyroelongated triangular bicupola is Johnson solid J_44.

			14.660254037844386467637231707529361834714026269...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated triangular bicupola.

Cf. A385256 (volume).
Cf. A002194, A010527.
Essentially the same of A332133, A375193 and A010527.

First[RealDigits[6 + 5*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J44", "SurfaceArea"], 10, 100]]

Equals 6 + 5*sqrt(3) = 6 + 5*A002194 = 6 + 10*A010527.

Equals the largest root of x^2 - 12*x - 39.

A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 30 2024

			0.133974596215561353236276829247063816528597373...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Sagitta
Index entries for algebraic numbers, degree 2.

Essentially the same as A334843.
Cf. A010527 (apothem), A104956 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375068 (pentagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A010527, A019913, A152422, A332133.

First[RealDigits[Tan[Pi/12]/2, 10, 100]]

tan(Pi/12)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(4*x^2-8*x+1)[1] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/12)/2 = A019913/2.
Equals 1 - sqrt(3)/2 = 1 - A010527.
Equals A152422^2 = (1 - A332133)^2. - Hugo Pfoertner, Jul 30 2024
Equals A334843-1/2. - R. J. Mathar, Aug 02 2024

A375193 Decimal expansion of the apothem (inradius) of a regular 12-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 04 2024

Apart from the first digit the same as A010527.

			1.8660254037844386467637231707529361834714026269...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 2.

Cf. A188887 (circumradius), A375194 (sagitta), A178809 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon).
Cf. A019884, A019913, A019973, A332133, A375069.

First[RealDigits[Cot[Pi/12]/2, 10, 100]]

sqrt(3)/2 + 1 \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/12)/2 = (2 + sqrt(3))/2 = A019973/2.
Equals 1/(2*tan(Pi/12)) = 1/(2*A019913).
Equals A188887*cos(Pi/12) = A188887*A019884.
Equals A188887 - A375194.
Equals A332133^2 = 2 - A375069. - Hugo Pfoertner, Aug 04 2024

A064324 a(n) = a(n-1) + floor(a(n-2)/2) with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 22, 30, 41, 56, 76, 104, 142, 194, 265, 362, 494, 675, 922, 1259, 1720, 2349, 3209, 4383, 5987, 8178, 11171, 15260, 20845, 28475, 38897, 53134, 72582, 99149, 135440, 185014, 252734, 345241, 471608, 644228, 880032

Offset: 0

Henry Bottomley, Sep 11 2001

nonn

a(n)/a(n-1) approaches (1+sqrt(3))/2 = 1.3660254... = A332133 for large n.

			a(5) = a(4)+floor(a(3)/2) = 4+floor(3/2) = 5.

Harry J. Smith, Table of n, a(n) for n = 0..400

Cf. A064323, A332133, A182229, A182230.

[n le 2 select n else Self(n-1)+Floor(Self(n-2)/2): n in [1..45]]; // Bruno Berselli, Apr 20 2012

RecurrenceTable[{a[n] == a[n-1] + Floor[a[n-2]/2], a[0] == 1, a[1] == 2}, a, {n, 0, 50}] (* G. C. Greubel, May 04 2019 *)
nxt[{a_,b_}]:={b,Floor[a/2]+b}; NestList[nxt,{1,2},50][[;;,1]] (* Harvey P. Dale, Jul 28 2023 *)

{ for (n=0, 400, if (n>1, a=a1 + a2\2; a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b064324.txt", n, " ", a) ) }; \\ Harry J. Smith, Sep 11 2009

a(n) = A064323(n) + 1.

A064323 a(n) = a(n-1)+ceiling(a(n-2)/2) with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 75, 103, 141, 193, 264, 361, 493, 674, 921, 1258, 1719, 2348, 3208, 4382, 5986, 8177, 11170, 15259, 20844, 28474, 38896, 53133, 72581, 99148, 135439, 185013, 252733, 345240, 471607, 644227, 880031, 1202145

Offset: 0

Henry Bottomley, Sep 11 2001

nonn

a(n)/a(n-1) approaches (1+sqrt(3))/2 = 1.3660254... = A332133 for large n.

			a(5) = a(4)+ceiling(a(3)/2) = 3+ceiling(2/2) = 4.

Harry J. Smith, Table of n, a(n) for n = 0..400

Cf. A064324, A332133.

[n le 2 select n-1 else Self(n-1)+Ceiling(Self(n-2)/2): n in [1..45]]; // Bruno Berselli, Apr 20 2012

a:= proc(n) option remember;
      `if`(n<2, n, a(n-1)+ceil(a(n-2)/2))
    end:
seq(a(n), n=0..48);  # Alois P. Heinz, Jan 26 2023

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+Ceiling[a[n-2]/2]},a,{n,50}] (* Harvey P. Dale, Nov 06 2013 *)

for (n=0, 400, if (n>1, a=a1 + ceil(a2/2); a2=a1; a1=a, if (n, a=a1=1, a=a2=0)); write("b064323.txt", n, " ", a) )  \\ Harry J. Smith, Sep 11 2009

first(n)=if(n<2, return([0,1][1..n+1])); my(v=vector(n+1)); v[2]=1; for(k=3,n+1, v[k]=v[k-1]+(v[k-2]+1)\2); v \\ Charles R Greathouse IV, Jan 26 2023

a(n) = A064324(n)-1.

cp's OEIS Frontend

A385257 Decimal expansion of the surface area of a gyroelongated triangular bicupola with unit edge.

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A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

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A375193 Decimal expansion of the apothem (inradius) of a regular 12-gon with unit side length.

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A064324 a(n) = a(n-1) + floor(a(n-2)/2) with a(0)=1, a(1)=2.

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A064323 a(n) = a(n-1)+ceiling(a(n-2)/2) with a(0)=0, a(1)=1.

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