A374883 -id:A374883 - cp's OEIS frontend

A144836 a(n) = round(phi^(4^n)) where phi is the golden ratio (A001622).

2, 7, 2207, 23725150497407, 316837008400094222150776738483768236006420971486980607

Offset: 0

Artur Jasinski, Sep 22 2008

Amiram Eldar, Table of n, a(n) for n = 0..6

Cf. A000032, A001566, A001622, A006267, A144837, A144838, A144839.

Cf. A374883 (phi^4).

a := proc(n) option remember; if n = 0 then 2 elif n = 1 then 7 else a(n-1)^4 - 4*a(n-1)^2 + 2 end if; end proc: seq(a(n), n = 0..4); # Peter Bala, Nov 28 2022

Table[Round[GoldenRatio^(4^n)], {n, 0, 5}]
c = (1 + Sqrt[5])/2; Join[{2}, Table[Expand[c^(4^n) + (1 - c)^(4^n)], {n, 1, 5}]] (* Artur Jasinski, Oct 05 2008 *)
Table[Round[2*Cosh[4^n*ArcCosh[Sqrt[5]/2]]], {n, 0, 5}] (* Artur Jasinski, Oct 09 2008 *)
a[n_] := LucasL[4^n]; a[0] = 2; Array[a, 5, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n)=round(((1+sqrt(5))/2)^4^n) \\ Charles R Greathouse IV, Jul 29 2011

a(n) = Lucas(4^n) = A000032(4^n), n>0.
a(n) = phi^(4^n) + (1 - phi)^(4^n) = phi^(4^n) + (-phi)^(-4^n), where phi is golden ratio = (1 + sqrt(5))/2 = 1.6180339887..., n>0. - Artur Jasinski, Oct 05 2008
a(n) = 2*cosh(4^n*arccosh(sqrt(5)/2)), n>0. - Artur Jasinski, Oct 09 2008
a(n+1) = a(n)^4 - 4*a(n-1)^2 + 2 with a(1) = 7. - Peter Bala, Nov 28 2022

Offset corrected by Charles R Greathouse IV, May 15 2013
Offset changed to 0 by Georg Fischer, Sep 02 2022
New name from Peter Bala, Nov 18 2022
Revised by editors, Jul 12 2025

A377291 For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

Original entry on oeis.org

6, 14, 7, 98, 16, 34, 1442, 398, 194, 119, 30, 62, 4354, 1154, 115598, 322, 23, 155234, 48, 98, 10402, 2702, 64514, 727, 482, 3040, 1154, 2114, 70, 142, 21314, 5474, 2498, 1442, 16793602, 674, 48497294, 158402, 47, 48670, 96, 194, 39202, 9998, 1684802, 2599

Offset: 1

Charles L. Hohn, Oct 23 2024

nonn

(a(n)^2-4)/A000037(n) is a square, and as such, a(n) is a member of row x of A298675(x, k), where x is the smallest value >= 3 such that (x^2-4)/A000037(n) is a square. E.g. for n=38: A000037(38)=44, x=20 ((20^2-4)/44 = 3^2), and a(38) = 158402 = A298675(20, 4).

The same row x of A298675(x, k) also results as integer solutions to g+(1/g) where g=(w*sqrt(d) + ceiling(w*sqrt(d)))/2 and d=A000037(n) for integers w >= 1. As such, it follows that g(n) can be expressed as a simple integer arithmetic transformation of sqrt(A000037(n)), e.g. g(1) = 2*sqrt(2)+3 (A156035), g(2) = 4*sqrt(3)+7 (A354129), g(3) = (3*sqrt(5)+7)/2 (A374883), g(4) = 20*sqrt(6)+49, and g(5) = 3*sqrt(7)+8 (A010516+8).

			For n = 5, the first few terms of A374602(5, k) are {4, 5, 11, 28, 62, 79, 175, 446, 988} and the period size is A377290(5) = 4, giving A374602(5, 1+4)/A374602(5, 1) = 62/4 = 15.5, 79/5 = 15.8, 175/11 = 15.909..., 446/28 = 15.928..., 988/62 = 15.935..., ..., to limit 15.937... -> g(5), from which g(5)+(1/g(5)) = 16 -> a(5).

Charles L. Hohn, Table of n, a(n) for n = 1..90

Cf. A374602, A377290, A000037, A298675.

Cf. A156035, A354129, A374883, A010516.

Growth factor g(n) = Lim_{k->oo}(A374602(n, k+A377290(n))/A374602(n, k)).
a(n) = round(g(n)) = ceiling(g(n)) = g(n)+(1/g(n)).
Inverse: g(n) = (sqrt(a(n)^2-4)+a(n))/2.
For d = A000037(n) and x in {1, 2, 4}, when d+x is a square (unless x==4 and d+x is even): a(n) = 4/x*d+2.
For d = A000037(n) and x in {-4, 1, 2, 4}, when n > 3 and d-x is a square (unless x==-4 and d-x is odd): a(n) = (4/abs(x))^2*d^2-16/x*d+2.

A381672 Decimal expansion of the isoperimetric quotient of a regular icosahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

For the definition of isoperimetric quotient of a solid, see A381671.

			0.8287977192520120215005810038129635758617830308723...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273637 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381671 (tetrahedron).
Cf. A000796, A002194, A374883.

First[RealDigits[Pi*GoldenRatio^4/(15*Sqrt[3]), 10, 100]]

Equals Pi*phi^4/(15*sqrt(3)) = A000796*A374883/(15*A002194).

cp's OEIS Frontend

A144836 a(n) = round(phi^(4^n)) where phi is the golden ratio (A001622).

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A377291 For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

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A381672 Decimal expansion of the isoperimetric quotient of a regular icosahedron.

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Mathematica

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