A010532 -id:A010532 - cp's OEIS frontend

A133273 Indices of centered decagonal numbers which are also decagonal numbers.

1, 10, 171, 3060, 54901, 985150, 17677791, 317215080, 5692193641, 102142270450, 1832868674451, 32889493869660, 590178020979421, 10590314883759910, 190035489886698951, 3410048503076821200, 61190837565496082641, 1098025027675852666330, 19703259660599851911291

Offset: 1

Richard Choulet, Oct 16 2007

Numbers k such that 80*k^2 - 80*k + 25 is a square.

Also the indices of centered square numbers which are also centered pentagonal numbers. - Colin Barker, Jan 01 2015

Colin Barker, Table of n, a(n) for n = 1..798
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Cf. A001107, A001844, A005891, A010532, A062786, A128922.

LinearRecurrence[{19,-19,1},{1,10,171},20] (* Harvey P. Dale, Oct 09 2020 *)

Vec(x*(-1+9*x)/((-1+x)*(1-18*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 01 2015

a(n+2) = 18*a(n+1) - a(n) - 8.
a(n+1) = 9*a(n) - 4 + sqrt(80*a(n)^2 - 80*a(n) + 25).
G.f.: x*(-1+9*x)/(-1+x)/(1 - 18*x + x^2). - R. J. Mathar, Nov 14 2007
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). - Colin Barker, Jan 01 2015
Product_{n>=2} (1 - 1/a(n)) = 2/sqrt(5) (= A010532 / 10). - Amiram Eldar, Dec 02 2024

More terms from Paolo P. Lava, Nov 25 2008

A378388 Decimal expansion of the surface area of a tetrakis hexahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 27 2024

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

			11.925695879998878380848926233233473255683297917928...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Tetrakis Hexahedron.

Cf. A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1), A378389 (dihedral angle).
Cf. A377341 (surface area of a truncated octahedron with unit edge).
Cf. A002163, A208899.

First[RealDigits[16*Sqrt[5]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TetrakisHexahedron", "SurfaceArea"], 10, 100]]

Equals (16/3)*sqrt(5) = (16/3)*A002163 = 16*A208899.

A041142 Numerators of continued fraction convergents to sqrt(80).

Original entry on oeis.org

8, 9, 152, 161, 2728, 2889, 48952, 51841, 878408, 930249, 15762392, 16692641, 282844648, 299537289, 5075441272, 5374978561, 91075098248, 96450076809, 1634276327192, 1730726404001, 29325898791208, 31056625195209, 526231901914552, 557288527109761

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..199
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Cf. A010532, A041143.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (8+9*x+8*x^2-x^3)/(1-18*x^2+x^4) )); // G. C. Greubel, Apr 16 2019

CoefficientList[Series[(8+9*x+8*x^2-x^3)/(1-18*x^2+x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 29 2013 *)

Vec((8+9*x+8*x^2-x^3)/(1-18*x^2+x^4) + O(x^30)) \\ Colin Barker, Mar 27 2016

((8+9*x+8*x^2-x^3)/(1-18*x^2+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 16 2019

G.f.: (8+9*x+8*x^2-x^3)/(1-18*x^2+x^4).
a(n) = 18*a(n-2) - a(n-4).
a(n) = (-3*(-2-sqrt(5))^(n+1) + 5*(2-sqrt(5))^(n+1) - 3*(-2+sqrt(5))^(n+1) + 5*(2+sqrt(5))^(n+1))/4. - Colin Barker, Mar 27 2016
a(n) = (5 - 3*(-1)^(n+1))*Lucas(3*(n+1))/4. - Ehren Metcalfe, Apr 15 2019

More terms from Colin Barker, Nov 05 2013

First term 1 removed in b-file, formulas and programs by Georg Fischer, Jul 01 2019

A041143 Denominators of continued fraction convergents to sqrt(80).

Original entry on oeis.org

1, 1, 17, 18, 305, 323, 5473, 5796, 98209, 104005, 1762289, 1866294, 31622993, 33489287, 567451585, 600940872, 10182505537, 10783446409, 182717648081, 193501094490, 3278735159921, 3472236254411, 58834515230497, 62306751484908, 1055742538989025

Offset: 0

N. J. A. Sloane

This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 16 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [First term 0 removed by _Georg Fischer_, Jul 01 2019]
Eric W. Weisstein, MathWorld: Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Cf. A041142, A020837, A040071, A010532, A001076, A002530.

List([0..30], n-> (5 +3*(-1)^n)*Fibonacci(3*(n+1))/16 ); # G. C. Greubel, Jul 02 2019

I:=[1,1,17,18]; [n le 4 select I[n] else 18*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

with(numtheory): cf := cfrac(sqrt(80),25): seq(nthdenom(cf,n), n=0..24); # Peter Luschny, Jul 06 2019

Denominator/@Convergents[Sqrt[80], 30] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 18 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,18,0]^n*[1;1;17;18])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

a(n) = (5 + 3*(-1)^n)*fibonacci(3*(n+1))/16 \\ Georg Fischer, Jul 01 2019

from sympy import continued_fraction_convergents as convergents, continued_fraction_iterator as cf, sqrt, denom
denominators = (denom(c) for c in convergents(cf(sqrt(80))))
print([next(denominators) for  in range(30)]) # _Ehren Metcalfe, Jul 03 2019

[(5 +3*(-1)^n)*fibonacci(3*(n+1))/16 for n in (0..30)] # G. C. Greubel, Jul 02 2019

G.f.: (1 + x - x^2) / (1 - 18*x^2 + x^4).
a(n) = 18*a(n-2) - a(n-4).
From Peter Bala, May 28 2014: (Start)
Let alpha = 2 + sqrt(5) and beta = 2 - sqrt(5) be the roots of the equation x^2 - 4*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n even, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n odd.
a(n) = A001076(n+1) for n even; a(n) = 1/4*A001076(n+1) for n odd.
a(n) = Product_{k = 1..floor(n/2)} ( 16 + 4*cos^2(k*Pi/(n+1)) ).
Recurrence equations: a(0) = 1, a(1) = 1 and for n >= 1, a(2*n) = 16*a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = a(2*n) + a(2*n - 1). (End)
a(n) = (5 + 3*(-1)^n)*Fibonacci(3*(n+1))/16. - Ehren Metcalfe, Apr 15 2019

First term 0 removed from b-file, formulas and programs by Georg Fischer, Jul 01 2019

A249600 Decimal expansion of 1/phi + 1/phi^3 + 1/phi^5, where phi is the Golden Ratio.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 02 2014

1/phi + 1/phi^3 + 1/phi^5 + 1/phi^7 + 1/phi^9 + ... = 1.

David Snook, Email, Mar 31 2009

Cf. A094214, A249601.

With[{g=GoldenRatio},RealDigits[1/g+1/g^3+1/g^5,10,120][[1]]] (* Harvey P. Dale, Jul 16 2017 *)

Equals A010532 minus 8. - R. J. Mathar, Nov 09 2014

A378389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a tetrakis hexahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 27 2024

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

			2.498091544796508851659834154562180246155658808...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Tetrakis hexahedron.

Cf. A378388 (surface area), A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1).

Cf. A156546 and A195698 (dihedral angles of a truncated octahedron), A195729.

First[RealDigits[ArcCos[-4/5], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TetrakisHexahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-4/5).

Equals 2*A195729. - Amiram Eldar, Nov 27 2024

A338574 Decimal expansion of the Prime Zeta Function at 3/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Nov 03 2020

			0.849562683621566446350893065... = A020765 +A020784 +A010532/10. + ...

Cf. A085548.

Mathematica
```
N[PrimeZetaP[3/2],70]
```

A377606 Decimal expansion of -30arcsin((5 - 4sqrt(5))/15).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 03 2024

Dehn invariant of an icosidodecahedron with unit edge length and (negated) of a (small) rhombicosidodecahedron with unit edge length.

			7.9824014167848074172162128505631888010390657928...

Eric Weisstein's World of Mathematics, Dehn Invariant.
Eric Weisstein's World of Mathematics, Icosidodecahedron.
Eric Weisstein's World of Mathematics, Small Rhombicosidodecahedron.
Index entries for transcendental numbers

Cf. A002163, A010532.

First[RealDigits[-30*ArcSin[(5 - Sqrt[80])/15], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["Icosidodecahedron", "DehnInvariant"], 10, 100]]

30*asin((4*sqrt(5)-5)/15) \\ Charles R Greathouse IV, Nov 21 2024

Equals -30*arcsin((5 - 4*A002163)/15) = -30*arcsin((5 - A010532)/15).

A387189 Decimal expansion of the smallest dihedral angle, in radians, in a pentagonal bipyramid (Johnson solid J_13).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 21 2025

This is the dihedral angle between triangular faces at the edge where the two pyramidal parts of the solid meet.

Also the dihedral angle between triangular faces in a pentagonal orthobicupola (Johnson solid J_30).

			1.3047162795687363719907812632876451487306158399...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Pentagonal bipyramid.
Wikipedia, Pentagonal orthobicupola.

Cf. A236367 (J_13 smallest dihedral angle).
Cf. other J_30 dihedral angles: A105199, A377995, A377996.
Cf. A179641 (J_13 volume), A120011 (J_13 surface area, divided by 10).
Cf. A384624 (J_30 volume), A384625 (J_30 surface area).
Cf. A010532, A386852.

First[RealDigits[ArcCos[(Sqrt[80] - 5)/15], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J13", "DihedralAngles"]], 10, 100]]

Equals arccos((4*sqrt(5) - 5)/15) = arccos((A010532 - 5)/15).

Equals 2*A386852.

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A133273 Indices of centered decagonal numbers which are also decagonal numbers.

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A378388 Decimal expansion of the surface area of a tetrakis hexahedron with unit shorter edge length.

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A041142 Numerators of continued fraction convergents to sqrt(80).

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A041143 Denominators of continued fraction convergents to sqrt(80).

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A249600 Decimal expansion of 1/phi + 1/phi^3 + 1/phi^5, where phi is the Golden Ratio.

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A378389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a tetrakis hexahedron.

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A338574 Decimal expansion of the Prime Zeta Function at 3/2.

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A377606 Decimal expansion of -30arcsin((5 - 4sqrt(5))/15).