A305491 -id:A305491 - cp's OEIS frontend

A060755 a(n) = log_2(A060723(n)).

0, 0, 0, 1, 0, 2, 2, 3, 0, 4, 4, 5, 3, 6, 6, 7, 3, 8, 8, 9, 7, 10, 10, 11, 8, 12, 12, 13, 11, 14, 14, 15, 10, 16, 16, 17, 15, 18, 18, 19, 16, 20, 20, 21, 19, 22, 22, 23, 19, 24, 24, 25, 23, 26, 26, 27, 24, 28, 28, 29, 27, 30, 30, 31, 25, 32, 32, 33, 31, 34, 34, 35, 32

Offset: 0

Vladeta Jovovic, Apr 24 2001

nonn

Cf. A060723, A305491.

Table[IntegerExponent[Denominator[Simplify[((1/2 (Sqrt[3] + 1))^x - (1/2 (Sqrt[3] - 1))^x Cos[Pi x])/Sqrt[3]]], 2], {x, 0, 72}]  (* Peter Luschny, Jun 02 2018 *)

Data corrected by Peter Luschny, Jun 02 2018

A060723 a(n) is the denominator of r(n) where r(n) is the sequence of rational numbers defined by the recursion: r(0) = 0, r(1) = 1 and for n>1 r(n) = r(n-1) + r(n-2)/2. From this definition it is clear that a(n) is always a power of 2 (see A060755).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 4, 8, 1, 16, 16, 32, 8, 64, 64, 128, 8, 256, 256, 512, 128, 1024, 1024, 2048, 256, 4096, 4096, 8192, 2048, 16384, 16384, 32768, 1024, 65536, 65536, 131072, 32768, 262144, 262144, 524288, 65536, 1048576, 1048576, 2097152

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

It can be proved that r(n) is an integer (i.e. a(n) = 1) if and only if n is one of 0, 1, 2, 4, 8.

			The sequence r(n) begins 0, 1, 1, 3/2, 2, 11/4, 15/4, 41/8, 7, 153/16, 209/16, 571/32, 363/16, 2023/64, 2749/64, 7521/128, 5135/64, ...

Cf. A060755, A305491 (numerators).

Denominator[RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+a[n-2]/2},a,{n,50}]] (* Harvey P. Dale, Mar 07 2016 *)
Table[Denominator[Simplify[((1/2(1 + Sqrt[3]))^x - (1/2(Sqrt[3] - 1))^x Cos[Pi x])/ Sqrt[3]]], {x, 0, 43}]  (* Peter Luschny, Jun 02 2018 *)

r(n) = (((1/2)*(sqrt(3) + 1))^n - ((1/2)*(sqrt(3) - 1))^n * cos(Pi*n))/sqrt(3). - Peter Luschny, Jun 02 2018

More terms from Vladeta Jovovic, Apr 24 2001

A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).

Original entry on oeis.org

0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, 915814, 3504600, 13419898, 51371996, 196683278, 752970528, 2882724002, 11036241700, 42251551414, 161756794728, 619274449354, 2370846461804, 9076614069086, 34749153370800

Offset: 0

Peter Luschny, Jun 02 2018

			Array ((1+y)^n - (1-y)^n)/y with y = sqrt(k).
[k\n]
[1]   1, 2, 4,  8, 16, 32,   64,  128,    256,   512,   1024, ...
[2]   0, 2, 4, 10, 24, 58,  140,  338,    816,  1970,   4756, ...
[3]   0, 2, 4, 12, 32, 88,  240,  656,   1792,  4896,  13376, ...
[4]   0, 2, 4, 14, 40, 122, 364,  1094,  3280,  9842,  29524, ...
[5]   0, 2, 4, 16, 48, 160, 512,  1664,  5376, 17408,  56320, ...
[6]   0, 2, 4, 18, 56, 202, 684,  2378,  8176, 28242,  97364, ...
[7]   0, 2, 4, 20, 64, 248, 880,  3248, 11776, 43040, 156736, ...
[8]   0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, ...
[9]   0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, ...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7).

Let f(n, y) = ((1 + y)^n - (1 - y)^n)/y.
f(n,      1 ) = A000079(n);
f(n, sqrt(2)) = A163271(n+1);
f(n, sqrt(3)) = A028860(n+2);
f(n,      2 ) = A152011(n) for n>0;
f(n, sqrt(5)) = A103435(n);
f(n, sqrt(6)) = A083694(n);
f(n, sqrt(7)) = A274520(n);
f(n, sqrt(8)) = a(n);
f(n,      3 ) = A192382(n+1);
Cf. A305491.
Equals 2 * A015519.

egf :=  (n,x) -> 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n):
ser := series(egf(8,x), x, 26):
seq(n!*coeff(ser,x, n), n=0..24);

Table[Simplify[((1 + Sqrt[8])^n - (1 - Sqrt[8])^n)/ Sqrt[8]], {n, 0, 24}]

concat(0, Vec(2*x / (1 - 2*x - 7*x^2) + O(x^40))) \\ Colin Barker, Jun 05 2018

E.g.f.: 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n) for n = 8.
From Colin Barker, Jun 02 2018: (Start)
G.f.: 2*x / (1 - 2*x - 7*x^2).
a(n) = 2*a(n-1) + 7*a(n-2) for n>1.
(End)

cp's OEIS Frontend

A060755 a(n) = log_2(A060723(n)).

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A060723 a(n) is the denominator of r(n) where r(n) is the sequence of rational numbers defined by the recursion: r(0) = 0, r(1) = 1 and for n>1 r(n) = r(n-1) + r(n-2)/2. From this definition it is clear that a(n) is always a power of 2 (see A060755).

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A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).

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