A104457 -id:A104457 - cp's OEIS frontend

A107387 Expansion of x(1-2x-x^2)/( (1-x)(1+x)(1-3*x+x^2)).

0, 1, 1, 2, 3, 7, 16, 41, 105, 274, 715, 1871, 4896, 12817, 33553, 87842, 229971, 602071, 1576240, 4126649, 10803705, 28284466, 74049691, 193864607, 507544128, 1328767777, 3478759201, 9107509826, 23843770275, 62423800999, 163427632720, 427859097161

Offset: 0

Roger L. Bagula, May 24 2005, corrected Sep 04 2008

The real roots of the denominator function are x = +-1, A104457 and A132338.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

I:=[ 0,1,1,2]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-3) +Self(n-4): n in [1..40]] // Vincenzo Librandi, Aug 22 2012

LinearRecurrence[{3,0,-3,1},{0,1,1,2},50] (* Vincenzo Librandi, Aug 22 2012 *)
CoefficientList[Series[x (1-2x-x^2)/((1-x)(1+x)(1-3x+x^2)),{x,0,40}],x] (* Harvey P. Dale, May 31 2020 *)

a(n) = 1 + A001654(n-2).

Replaced with a regular sequence by R. J. Mathar, Aug 31 2011

A168259 Eigensequence of triangle A168258.

Original entry on oeis.org

1, 2, 6, 14, 38, 96, 254, 656, 1724, 4492, 11776, 30774, 80608, 210892, 552226, 1445374, 3784308, 9906482, 25936206, 67899344, 177764618, 465387226, 1218404344, 3189806746, 8351034954, 21863248282, 57238759726, 149852900454, 392320072078, 1027106974446, 2689001192594, 7039895709776

Offset: 1

Gary W. Adamson, Nov 21 2009

Eigensequence of triangle A168258, derived from the following operation: Shift down triangle A168258, so that rows begin [1; 1; 1,1; 2,2,1; ...] = triangle M. Then take lim_{n->oo} M^n, resulting in a left-shifted vector. Delete the first 1, getting (1, 2, 6, 14, 38, 96, ...) = this sequence.

a(n)/a(n-1) apparently tends to phi^2=A104457. a(19)/a(18) = 2.618104...

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Cf. A104457, A168258, A168260.

T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688
mat(nn) = my(ma=matrix(nn+1, nn, n, k, T(n-1, k-1)), mb=matrix(nn, nn, n, k, n>=k)); ma*mb; \\ A168258
shiftm(m, nn) = my(shm=matrix(nn+1, nn+1)); shm[1,1]=1; for (n=1, nn, for(k=1, nn, shm[n+1,k] = m[n,k];);); shm;
lista(nn) = my(m=mat(nn), shm=shiftm(m, nn), shmnn=shm^nn); vector(nn, k, shmnn[k+1, 1]); \\ Michel Marcus, Nov 19 2022

Edited and more terms from Michel Marcus, Nov 19 2022

A168317 Eigensequence of triangle A168316.

Original entry on oeis.org

1, 1, 3, 6, 16, 39, 103, 263, 690, 1791, 4693, 12247, 32073, 83869, 219598, 574658, 1504540, 3938272, 10310703

Offset: 1

Gary W. Adamson, Nov 22 2009

Conjectured convergent of a(n)/a(n-1) = phi^2 = 2.6180339... E.g.: a(19)/a(18) = 10310703/3938272 = 2.6180779....

Cf. A104457, A168316, A168318.

Equals lim_{n->oo} M^2, the left shifted vector considered as a sequence where M = triangle A168316 as an infinite lower triangular matrix, then shifted down one row, and inserting a "1" at top.

A180014 Decimal expansion of Pi/(2*phi^2).

Original entry on oeis.org

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

Offset: 0

Frank M Jackson, Aug 06 2010

This is the first of the three angles (in radians) of a unique triangle that is right angled and where the angles are in a Geometric Progression - pi/(2*phi^2), pi/(2*phi), pi/2. The angles (in degrees) are approx 34.377, 55.623, 90.

			0.5999908074321633305578888766584034632812497527645287607337781876828268345598596...

Index entries for transcendental numbers

RealDigits[N[Pi/(2(GoldenRatio)^2),100]][[1]]

Pi/4*(3-sqrt(5)) \\ Charles R Greathouse IV, Jul 29 2011

pi/(2*phi^2) = A019669 / A104457 = (3 - sqrt(5)) * Pi/4.

Partially edited by R. J. Mathar, Aug 07 2010

Mathematica program edited by Harvey P. Dale, Jul 10 2012

A183137 a(n) = [1/s] + [2/s] + ... + [n/s], where s = (golden ratio)^2 = (3+sqrt(5))/2 and [] = floor.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 10, 13, 16, 20, 24, 28, 33, 38, 44, 50, 56, 63, 70, 78, 86, 94, 103, 112, 121, 131, 141, 152, 163, 174, 186, 198, 210, 223, 236, 250, 264, 278, 293, 308, 324, 340, 356, 373, 390, 407, 425, 443, 462, 481, 500, 520, 540, 561, 582, 603, 625

Offset: 1

Clark Kimberling, Dec 26 2010

nonn

A183136(n) + a(n) = A000217(n+1) (the triangular numbers).

			a(7) = 7 = 0+0+1+1+1+2+2.

Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See p. 4.

Cf. A001622, A104457, A183136, A000217, A005206, A060143, A060144.

Accumulate[With[{c=GoldenRatio^2},Floor[Range[60]/c]]]  (* Harvey P. Dale, Apr 20 2011 *)

a(n+1) = a(n) + n - A005206(n). - John Furey, Jun 03 2015

A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 04 2021

The moments of inertia of the five Platonic solids were apparently first calculated by the Canadian physicist John Satterly (1879-1963) in 1957.
The moment of inertia of a solid regular dodecahedron with a uniform mass density distribution, mass M, and edge length L is I = c*M*L^2, where c is this constant.
The corresponding values of c for the other Platonic solids are:
Tetrahedron: 1/20 (= A020761/10).
Octahedron: 1/10 (= A000007).
Cube: 1/6 (= A020793).
Icosahedron: (3 + sqrt(5))/20 (= A104457/10).

			0.60735550374163932719985924360173257727394705341616...

P. K. Aravind, Gravitational collapse and moment of inertia of regular polyhedral configurations, American Journal of Physics, Vol. 59, No. 7 (1991), pp. 647-652.
Frédéric Perrier, Moments of inertia of Archimedean solids, 2015.
John Satterly, Moments of Inertia about Selected Axes of Regular Polygons, Right Pyramids on Regular Polygonal Bases, and of the Platonic and Some Archimedian Polyhedra, American Journal of Physics, Vol. 25, No. 7 (1957), pp. 489-490.
John Satterly, The Moments of Inertia of Some Polyhedra, The Mathematical Gazette, Vol. 42, No. 339 (1958), pp. 11-13.
Wikipedia, List of moments of inertia.
Wikipedia, Moment of inertia.
Wikipedia, Regular dodecahedron.
Index entries for sequences related to moment of inertia.

Cf. A000007, A001622, A020761, A020793, A104457.

Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798.

RealDigits[(95 + 39*Sqrt[5])/300, 10, 100][[1]]

Equals (95 + 39*sqrt(5))/300.

Equals (28 + 39*phi)/150, where phi is the golden ratio (A001622).

A385446 Decimal expansion of -7 + 10*phi, with the golden section phi = A001622.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jul 01 2025

This constant d gives the imaginary part of -2*11*Z = c + d*i, where Z is the fixed point of a complex function w (of the loxodromic type) mapping vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumcribed by the unit circle with center at the origin, and D_1 = i (the complex unit), D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-3*Pi*i/5) = -((2 - phi) + s*i)/2  and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D2, D3) of the first triangle).
The real part c = (-1 + 3*phi)*s is given in A385445.
For details see A385445, and eqs.(5a,b) of the linked paper there.

			9.18033988749894848204586834365638117720309179805762862...

Cf. A001622, A182007, A385445.

RealDigits[10*GoldenRatio - 7, 10, 120][[1]] (* Amiram Eldar, Jul 02 2025 *)

Equals -7 + 10*phi, an integer in the quadratic number field Q(sqrt(5)).

Equals 10*A176055-12 = 10*A104457-17 = 10*A001622-7 . - R. J. Mathar, Jul 06 2025

A171969 Smallest prime greater than phi^n.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 31, 47, 79, 127, 211, 331, 523, 853, 1367, 2207, 3581, 5779, 9371, 15131, 24481, 39607, 64081, 103687, 167771, 271451, 439217, 710663, 1149857, 1860503, 3010363, 4870861, 7881221, 12752053, 20633279

Offset: 1

Michel Lagneau, Nov 19 2010

nonn

			The first prime > phi^6 = 17.94427... is 19, so a(6) = 19.

Charles R Greathouse IV, Table of n, a(n) for n = 1..4784
M. Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fibonacci Quarterly, 4 (1966), 157-162.

Cf. A001622, A104457.

p[n_] := Module[{r, i}, r = 2; i = 1; While[r <= n, i = i + 1; r = Prime[i]];  r]; Table[p[GoldenRatio^n], {n, 1, 35}]
NextPrime[GoldenRatio^Range[40]] (* Harvey P. Dale, Dec 06 2013 *)

a(n)=nextprime(((1+sqrt(5))/2)^n) \\ Charles R Greathouse IV, Jul 29 2011

A192044 Decimal approximation of x such that f(x)=r+1, where f is the Fibonacci function described in Comments and r=(golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 21 2011

f(x)=(r^x-r^(-x*cos[pi*x]))/sqrt(5), where r=(golden ratio)=(1+sqrt(5))/2. This function, a variant of the Binet formula, gives Fibonacci numbers for integer values of x; e.g., f(3)=2, f(4)=3, f(5)=5.

			3.70822831961181544622795697604762903141444780151470467124724

Cf. A192038, A104457.

r = GoldenRatio; s = 1/Sqrt[5];
f[x_] := s (r^x - r^-x Cos[Pi x]);
x /. FindRoot[Fibonacci[x] == r+1, {x, 5}, WorkingPrecision -> 100]
RealDigits[%, 10]
(Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
Fibonacci[x], {x, -7, 7}]
(* Peter J. C. Moses, Jun 21 2011 *)

A257671 Numbers of the form floor(r^i + s^j), where r = (1 + sqrt(5))/2, s = r^2, i >= 0, j >= 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 17, 18, 19, 20, 22, 24, 29, 30, 31, 35, 46, 47, 48, 49, 51, 53, 58, 64, 76, 77, 78, 82, 93, 122, 123, 124, 125, 127, 129, 134, 140, 152, 169, 199, 200, 201, 205, 216, 245, 321, 322, 323, 324, 326, 328, 333, 339, 351, 368

Offset: 1

Clark Kimberling, May 03 2015

Cf. A001622, A104457.

Cf. A257672.

mx = 400; r = GoldenRatio; s = r/(r - 1); Union@ Flatten@ Table[ Floor[r^i + s^j], {i, Log[r, mx]}, {j, 0, Log[s, mx - r^i]}] (* iterators modified by Robert G. Wilson v, May 07 2015 *)

cp's OEIS Frontend

A107387 Expansion of x*(1-2*x-x^2)/( (1-x)*(1+x)*(1-3*x+x^2)).

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A168259 Eigensequence of triangle A168258.

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A168317 Eigensequence of triangle A168316.

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A180014 Decimal expansion of Pi/(2*phi^2).

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A183137 a(n) = [1/s] + [2/s] + ... + [n/s], where s = (golden ratio)^2 = (3+sqrt(5))/2 and [] = floor.

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A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

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A385446 Decimal expansion of -7 + 10*phi, with the golden section phi = A001622.

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A171969 Smallest prime greater than phi^n.

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A107387 Expansion of x(1-2x-x^2)/( (1-x)(1+x)(1-3*x+x^2)).