A127814 -id:A127814 - cp's OEIS frontend

A147985 Coefficients of numerator polynomials S(n,x) associated with reciprocation.

1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 1, 0, -7, 0, 13, 0, -7, 0, 1, 1, 0, -15, 0, 83, 0, -220, 0, 303, 0, -220, 0, 83, 0, -15, 0, 1, 1, 0, -31, 0, 413, 0, -3141, 0, 15261, 0, -50187, 0, 115410, 0, -189036, 0, 222621, 0, -189036, 0, 115410, 0, -50187, 0, 15261, 0, -3141, 0, 413, 0

Offset: 1

Clark Kimberling, Nov 24 2008

S(n)=U(n-1)V(n-1) where U(n-1)=S(n-1)+S(1)*S(2)*...*S(n-2) and V(n-1)=S(n-1)-S(1)*S(2)*...*S(n-2), for n>=2. If U(n) and V(n) are written as polynomials U(n,x) and V(n,x), then V(n,x)=U(n,-x). See A147989 for coefficients of U(n).
S(n)=S(n-1)^2+S(n-1)*S(n-2)^2-S(n-2)^4 for n>2. (The Gorskov-Wirsting polynomials also have this recurrence; see H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, 84, AMS, pp. 183-190.)
For n>0, the 2^(n-1) zeros of S(n) are real. If r is a zero of S(n), then -r and 1/r are zeros of S(n).
If r is a zero of S(n), then the numbers z satisfying r=z-1/z and r=z+1/z are zeros of S(n+1).
If n>2, then S(n,1)=1 and S(n,2)=A127814(n).
S(n,2^(1/2))=-1 for n>2 and S(n,2^(-1/2))=-2^(1-n) for n>1.

			S(1)=x
S(2)=x^2-1=(x-1)(x+1)
S(3)=x^4-3*x^2+1=(x^2+x-1)(x^2-x-1)
S(4)=x^8-7*x^6+13*x^4-7*x^2+1=(x^4+x^3-3*x^2-x+1)(x^4-x^3-3*x^2+x+1),
so that, as an array, sequence begins with
1 0
1 0 -1
1 0 -3 0 1
1 0 -7 0 13 0 -7 0 1

Peter J. C. Moses, Rows n = 1..13 of irregular triangle, flattened
Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147986, A147987, A147988, A147989, A147990, A147991, A147992, A147993.

s[1] = x; t[1] = 1; s[n_] := s[n] = s[n-1]^2 - t[n-1]^2; t[n_] := t[n] = s[n-1]*t[n-1]; row[n_] := CoefficientList[s[n], x] // Reverse; Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Apr 22 2013 *)

The basic idea is to iterate the reciprocation-difference mapping x/y -> x/y-y/x.

Let x be an indeterminate, S(1)=x, T(1)=1 and for n>1, define S(n)=S(n-1)^2-T(n-1)^2 and T(n)=S(n-1)*T(n-1), so that S(n)/T(n)=S(n-1)/T(n-1)-T(n-1)/S(n-1).

A127815 a(n) = denominator of b(n), where b(1) = 2, b(n) = b(n-1) - 1/b(n-1).

Original entry on oeis.org

1, 2, 6, 30, 330, 257070, 128005692870, 23279147893155496537470, 388475314992168993748220639081347493631827670, 102339769648127358726761918460732576814168548432921287355299929744910591862606847215978930

Offset: 1

Leroy Quet, Jan 30 2007

			A127814/A127815 = 2, 3/2, 5/6, -11/30, 779/330, 497941/257070, 181860254581/128005692870, ...

Cf. A127814.

f[l_List] := Append[l, l[[ -1]] - 1/l[[ -1]]];Denominator[Nest[f, {2}, 10]] (* Ray Chandler, Feb 07 2007 *)

For n >= 2, a(n) = a(n-1)*A127814(n-1).

Extended by Ray Chandler, Feb 07 2007

A242995 a(n) = a(n-1)^2 + a(n-1)*a(n-2)^2 - a(n-2)^4 with a(1) = 2, a(2) = 3.

Original entry on oeis.org

2, 3, 5, -11, -779, 497941, 181860254581, 16687694789137362648661, -263439569256003706800705587722279993788907979

Offset: 1

Michael Somos, Aug 17 2014

sign

G. C. Greubel, Table of n, a(n) for n = 1..13
M. Chamberland and M. Martelli, Unbounded Orbits and Binary Digits, Grinnell College.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A127814, A242996.

I:=[2,3]; [n le 2 select I[n] else Self(n-1)^2 + Self(n-1)*Self(n-2)^2 - Self(n-2)^4: n in [1..10]]; // G. C. Greubel, Aug 05 2018

RecurrenceTable[{a[n] == a[n-1]^2 + a[n-1]*a[n-2]^2 - a[n-2]^4, a[1] == 2, a[2] == 3}, a, {n, 1, 10}] (* G. C. Greubel, Aug 05 2018 *)

{a(n) = if( n<1, 0, if( n<3, n+1, my(x = a(n-2)^2, y = a(n-1)); y^2 + x*y - x^2))};

0 = a(n)^2*(a(n+1) - a(n)^2) - (a(n+2) - a(n+1)^2) for all n>0.
abs(a(n)) = abs(A127814(n)) for all n>0.
a(n) = A242996(n+1) / A242996(n) for all n>0.

A125676 a(n) = floor(abs(b(n))), where b(1) = 2, b(n) = b(n-1) - 1/b(n-1).

Original entry on oeis.org

2, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 1, 1, 0, 3, 3, 3, 2, 2, 1, 1, 0, 1, 0, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 22, 22, 22, 22, 22, 22, 22, 22

Offset: 1

Leroy Quet, Jan 30 2007

nonn

Cf. A127814, A127815.

f[l_List] := Append[l, l[[ -1]] - 1/l[[ -1]]];Floor /@ Abs /@ Nest[f, {2}, 30] (* Ray Chandler, Feb 08 2007 *)

lista(nn) = my(b=2); print1(2); for(n=2, nn, print1(", ", floor(abs(b-=1/b)))); \\ Jinyuan Wang, Aug 10 2021

a(9)-a(31) from Ray Chandler, Feb 08 2007

More terms from Jinyuan Wang, Aug 10 2021

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A147985 Coefficients of numerator polynomials S(n,x) associated with reciprocation.

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A127815 a(n) = denominator of b(n), where b(1) = 2, b(n) = b(n-1) - 1/b(n-1).

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A242995 a(n) = a(n-1)^2 + a(n-1)*a(n-2)^2 - a(n-2)^4 with a(1) = 2, a(2) = 3.

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A125676 a(n) = floor(abs(b(n))), where b(1) = 2, b(n) = b(n-1) - 1/b(n-1).

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