A147993 -id:A147993 - cp's OEIS frontend

A360099 To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, -1, 0, 1, 0, 1, 0, 1, 1, 2, -1, 0, 1, 2, 3, -1, 1, 0, 1, 3, 4, 1, 1, -1, 0, 1, 4, 5, 5, 3, 1, 1, 0, 1, 5, 6, 11, 7, 5, 3, -1, 0, 1, 6, 7, 19, 13, 11, 7, -2, 1, 0, 1, 7, 8, 29, 21, 19, 13, 1, 0, -1, 0, 1, 8, 9, 41, 31, 29, 21, 14, 3, 0, 1, 0, 1, 9, 10, 55, 43, 41, 31, 43, 16, 5, 2, -1

Offset: 0

Alois P. Heinz, Jan 25 2023

The empty bit string is used as binary expansion of 0, so A(0,k) = 0.

			Square array A(n,k) begins:
   0,  0, 0,  0,  0,   0,   0,   0,   0,   0,   0, ...
   1,  1, 1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  -1,  0, 1,  2,  3,   4,   5,   6,   7,   8,   9, ...
   1,  2, 3,  4,  5,   6,   7,   8,   9,  10,  11, ...
  -1, -1, 1,  5, 11,  19,  29,  41,  55,  71,  89, ...
   1,  1, 3,  7, 13,  21,  31,  43,  57,  73,  91, ...
  -1,  1, 5, 11, 19,  29,  41,  55,  71,  89, 109, ...
   1,  3, 7, 13, 21,  31,  43,  57,  73,  91, 111, ...
  -1, -2, 1, 14, 43,  94, 173, 286, 439, 638, 889, ...
   1,  0, 3, 16, 45,  96, 175, 288, 441, 640, 891, ...
  -1,  0, 5, 20, 51, 104, 185, 300, 455, 656, 909, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-6, 10 give: A062157, A145037, A006257, A147991, A147992, A153777, A147993, A359925.
Rows n=0-10 give: A000004, A000012, A023443, A000027(k+1), A165900, A002061, A165900(k+1), A002061(k+1), A083074, A152618, A062158.
Main diagonal gives A360096.
Cf. A030300, A057427.

A:= proc(n, k) option remember; local m;
     `if`(n=0, 0, k*A(iquo(n, 2, 'm'), k)+2*m-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> (l-> add((2*l[i]-1)*k^(i-1), i=1..nops(l)))(Bits[Split](n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

G.f. for column k satisfies g_k(x) = k*(x+1)*g_k(x^2) + x/(1+x).
A(n,k) = k*A(floor(n/2),k)+2*(n mod 2)-1 for n>0, A(0,k)=0.
A(n,k) mod 2 = A057427(n) if k is even.
A(n,k) mod 2 = A030300(n) if k is odd and n>=1.
A(2^(n+1),1) + n = 0.

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

Original entry on oeis.org

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).

A147986 Coefficients of denominator polynomials T(n,x) associated with reciprocation.

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 0, 1, 0, -4, 0, 4, 0, -1, 0, 1, 0, -11, 0, 45, 0, -88, 0, 88, 0, -45, 0, 11, 0, -1, 0, 1, 0, -26, 0, 293, 0, -1896, 0, 7866, 0, -22122, 0, 43488, 0, -60753, 0, 60753, 0, -43488, 0, 22122, 0, -7866, 0, 1896, 0, -293, 0, 26, 0, -1, 0, 1, 0, -57, 0, 1512, 0

Offset: 1

Clark Kimberling, Nov 24 2008

T(n)=S(1)*S(2)*...*S(n-1). The degree of S(n) in x is m=2^(n-1), so that the degree of T(n) is m-1. Write the zeros of T(n) as r(1)

			T(1) = 1
T(2) = x
T(3) = x^3-x
T(4) = x^7-4*x^5+4*x^3-x
so that, as an array, the sequence begins with:
1
1 0
1 0 -1 0
1 0 -4 0 4 0 -1 0

Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147985, 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[t[n], x] // Reverse; Table[row[n], {n, 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).

A147989 Coefficients of factor polynomials U(n,x) associated with reciprocation.

Original entry on oeis.org

1, 1, -1, 1, 1, -3, -1, 1, 1, 1, -7, -4, 13, 4, -7, -1, 1, 1, 1, -15, -11, 83, 45, -220, -88, 303, 88, -220, -45, 83, 11, -15, -1, 1, 1, 1, -31, -26, 413, 293, -3141, -1896, 15261, 7866, -50187, -22122, 115410, 43488, -189036, -60753, 222621, 60753, -189036

Offset: 1

Clark Kimberling, Nov 25 2008

The zeros of U(n,x) and U(n,-x) are the zeros of S(n,x) at A147985.

			U(3) = x^2+x-1;
U(4) = x^4+x^3-3*x^2-x+1;
U(5) = x^8+x^7-7*x^6-4*x^5+13*x^4+4*x^3-7*x^2-x+1;
so that, as an array, the sequence begins with:
1 1 -1
1 1 -3 -1 1
1 1 -7 -4 13 4 -7 -1 1

Robert Israel, Table of n, a(n) for n = 1..9418
Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147985, A147986, A147987, A147988, A147990, A147991, A147992, A147993.

U[3]:= x^2+x-1:
U[4]:= x^4+x^3-3*x^2-x+1:
for n from 5 to 10 do
  U[n]:= normal(U[n-1]*M(U[n-1]) + x*(x^2-1)*mul(U[i]*M(U[i]),i=3..n-2));
od:
seq(seq(coeff(U[m],x,j),j=degree(U[m])..0,-1),m=3..10); # Robert Israel, Jun 30 2015

U[3, x_] = x^2 + x - 1;
U[4, x_] = x^4 + x^3 - 3 x^2 - x + 1;
U[n_, x_] := U[n, x] = U[n-1, x] U[n-1, -x] + x (x^2 - 1) Product[U[k, x] U[k, -x], {k, 3, n-2}];
Table[CoefficientList[U[n, x], x] // Reverse, {n, 3, 7}] // Flatten (* Jean-François Alcover, Mar 25 2019 *)

For n>=5, U(n)=U(n,x)=U(n-1,x)*U(n-1,-x)+x*(x^2-1)*U(3,x)*U(3,-x)*U(4,x)*U(4,-x)*...*U(n-2,x)*U(n-2,-x), where U(3)=x^2+x-1, U(4)=x^4+x^3-3*x^2-x+1.

A147990 Array A147985 (Polynomial coefficients) with zeros deleted.

Original entry on oeis.org

1, 1, -1, 1, -3, 1, 1, -7, 13, -7, 1, 1, -15, 83, -220, 303, -220, 83, -15, 1, 1, -31, 413, -3141, 15261, -50187, 115410, -189036, 222621, -189036, 115410, -50187, 15261, -3141, 413, -31, 1, 1, -63, 1839, -33150, 414861, -3841195, 27378213, -154299168

Offset: 1

Clark Kimberling, Nov 25 2008

			s(1)=x
s(2)=S(2,y)=x-1
s(3)=S(3,y)=x^2-3*x+1
s(4)=S(4,y)=x^4-7*x^3+13*x^2-7*x+1
so that as an array A147990 begins with
1
1 -1
1 -3 1
1 -7 13 -7 1

Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4

Cf. A147985, A147986, A147987, A147988, A147989, A147991, A147992, A147993.

Let s(1)=x and for n>=2, let s(n)=s(n,x)=S(n,y), where y=x^(1/2) and S(n,x)

is as at A147985. Then A147990 gives the coefficients of the polynomials s(n).

A147987 Coefficients of numerator polynomials P(n,x) associated with reciprocation.

Original entry on oeis.org

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, 31, 0, 1, 1, 0, 63

Offset: 1

Clark Kimberling, Nov 24 2008

P(n,1)=A073833(n) for n>=1; P(n,2)=A073833(n+1) for n>=0.
P(n)=P(n-1)^2+P(n-1)*P(n-2)^2-P(n-2)^4 for n>=3.
For n>=3, P(n)=P(n,x)=S(n,i*x), where S(n) is the polynomial at A147985.
Thus all the zeros of P(n,x), for n>=2, are nonreal.

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

Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147985, A147986, A147988, A147989, A147990, A147991, A147992, A147993.

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

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

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

A147988 Coefficients of denominator polynomials Q(n,x) associated with reciprocation.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 4, 0, 1, 0, 1, 0, 11, 0, 45, 0, 88, 0, 88, 0, 45, 0, 11, 0, 1, 0, 1, 0, 26, 0, 293, 0, 1896, 0, 7866, 0, 22122, 0, 43488, 0, 60753, 0, 60753, 0, 43488, 0, 22122, 0, 7866, 0, 1896, 0, 293, 0, 26, 0, 1, 0, 1, 0, 57, 0, 1512, 0, 24858, 0, 284578, 0

Offset: 1

Clark Kimberling, Nov 24 2008

1. Q(n,1)=A073834(n) for n>=1.
2. For n>=3, Q(n)=Q(n,x)=i*T(n,i*x), where T(n) is the polynomial at A147986.
Thus all the zeros of Q(n,x), for n>=2, are nonreal.

			Q(1) = 1
Q(2) = x
Q(3) = x^3+x
Q(4) = x^7+4*x^5+4*x^3+1
so that, as an array, the sequence begins with:
1
1 0
1 0 1 0
1 0 4 0 4 0 1

Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147985, A147986, A147987, A147989, A147990, A147991, A147992, A147993.

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

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

A359925 Numbers with easy multiplication table - the first 9 multiples of these numbers can be derived by either incrementing or decrementing the corresponding digits from the previous multiple.

Original entry on oeis.org

1, 9, 11, 89, 91, 109, 111, 889, 891, 909, 911, 1089, 1091, 1109, 1111, 8889, 8891, 8909, 8911, 9089, 9091, 9109, 9111, 10889, 10891, 10909, 10911, 11089, 11091, 11109, 11111, 88889, 88891, 88909, 88911, 89089, 89091, 89109, 89111, 90889, 90891, 90909, 90911

Offset: 1

Kiran Ananthpur Bacche, Jan 25 2023

This is also the list of numbers having exactly one dot or one antidot in each box in the Decimal Exploding Dots notation.

			a(4) = 89. The first nine multiples of 89 are {089, 178, 267, 356, 445, 534, 623, 712, 801}. The digits in the hundreds place increment by 1, while the digits in the tens and units place decrement by 1. In the Decimal Exploding Dots notation, 89 is represented as DOT-ANTIDOT-ANTIDOT = 100 - 10 - 1 = 89

Global Math Week, Exploding Dots.
Gevorg Hmayakyan, Generalizing a Trig Identity [mentions this sequence]

Cf. A006257, A147991, A147992, A153777, A147993, A256290.

Column k=10 of A360099.

a:= proc(n) option remember;
      `if`(n=0, 0, 10*a(iquo(n, 2, 'm'))+2*m-1)
    end:
seq(a(n), n=1..44);   # Alois P. Heinz, Jan 25 2023

a(n) = 10*a(floor(n/2))+2*(n mod 2)-1 for n>0, a(0)=0. - Alois P. Heinz, Jan 25 2023

a(n) = 2*A256290(n-1) + 1 for n>1. - Hugo Pfoertner, Jan 28 2023

Showing 1-8 of 8 results.

cp's OEIS Frontend

A360099 To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A147986 Coefficients of denominator polynomials T(n,x) associated with reciprocation.

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A147989 Coefficients of factor polynomials U(n,x) associated with reciprocation.

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A147990 Array A147985 (Polynomial coefficients) with zeros deleted.

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A147987 Coefficients of numerator polynomials P(n,x) associated with reciprocation.

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A147988 Coefficients of denominator polynomials Q(n,x) associated with reciprocation.

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A359925 Numbers with easy multiplication table - the first 9 multiples of these numbers can be derived by either incrementing or decrementing the corresponding digits from the previous multiple.

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