A022163 -id:A022163 - cp's OEIS frontend

A249221 Expansion of x(1+5x-2x^3)/(1-6x^2+2*x^4).

1, 5, 6, 28, 34, 158, 192, 892, 1084, 5036, 6120, 28432, 34552, 160520, 195072, 906256, 1101328, 5116496, 6217824, 28886464, 35104288, 163085792, 198190080, 920741824, 1118931904, 5198279360, 6317211264, 29348192512, 35665403776, 165692596352, 201358000128

Offset: 1

Colin Barker, Oct 23 2014

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-2).

Cf. A022163, A022165, A249222.

CoefficientList[Series[(1 + 5 x - 2 x^3)/(1 - 6 x^2 + 2 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 23 2014 *)
LinearRecurrence[{0,6,0,-2},{1,5,6,28},40] (* Harvey P. Dale, Apr 20 2017 *)

Vec((1+5*x-2*x^3)/(1-6*x^2+2*x^4) + O(x^100))

a(n) = 6*a(n-2)-2*a(n-4).

A249222 Expansion of x(1+5x-5x^3)/(1-6x^2+5*x^4).

Original entry on oeis.org

1, 5, 6, 25, 31, 125, 156, 625, 781, 3125, 3906, 15625, 19531, 78125, 97656, 390625, 488281, 1953125, 2441406, 9765625, 12207031, 48828125, 61035156, 244140625, 305175781, 1220703125, 1525878906, 6103515625, 7629394531, 30517578125, 38146972656, 152587890625

Offset: 1

Colin Barker, Oct 23 2014

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-5).

Cf. A022163, A022165, A249221.

CoefficientList[Series[(1 + 5 x - 5 x^3)/(1 - 6 x^2 + 5 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 23 2014 *)

Vec((1+5*x-5*x^3)/(1-6*x^2+5*x^4) + O(x^100))

a(n)=round((9-(-1)^n)*5^(n\2)/8) \\ Tani Akinari, Oct 26 2014

a(n) = 6*a(n-2)-5*a(n-4).

A022162 First column of spectral array W(sqrt(3/2)).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 11, 13, 14, 15, 17, 18, 20, 22, 23, 24, 26, 28, 29, 30, 31, 34, 35, 36, 37, 40, 41, 42, 44, 45, 47, 48, 50, 51, 53, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 71, 73, 74, 75, 77, 78, 80, 82, 83, 84, 86, 88, 89, 90, 91, 94, 95, 96, 97

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000
A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.

Cf. A022163.

[Floor(Sqrt(3/2)*Floor(n*Sqrt(3/2))): n in [1..50]]; // G. C. Greubel, May 27 2018

Table[Floor[Sqrt[3/2]*Floor[Sqrt[3/2]*n]], {n, 1, 50}] (* G. C. Greubel, May 27 2018 *)

a(n) = floor(sqrt(3/2)*floor(sqrt(3/2)*n)); \\ Michel Marcus, Mar 05 2014

More terms from Michel Marcus, Mar 05 2014

A249309 First row of spectral array W(Pi/2).

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 20, 35, 54, 96, 150, 264, 414, 726, 1140, 1997, 3136, 5495, 8631, 15121, 23752, 41612, 65363, 114513, 179876, 315132, 495008, 867223, 1362230, 2386544, 3748774, 6567622, 10316396

Offset: 1

Colin Barker, Oct 25 2014

nonn

A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.

Cf. A007068, A022159, A022161, A022163, A022165.

\\ The first row of the generalized Wythoff array W(h),
\\   where h is an irrational number between 1 and 2.
row1(h, m) = {
  my(
    a=vector(m, n, floor(n*h)),
    b=setminus(vector(m, n, n), a),
    w=[a[1]^2, b[a[1]]],
    j=3
  );
  while(1,
    if(j%2==1,
      if(w[j-1]<=#a, w=concat(w, a[w[j-1]]), return(w))
    ,
      if(w[j-2]<=#b, w=concat(w, b[w[j-2]]), return(w))
    );
    j++
  );
  w
}
allocatemem(10^9)
row1(Pi/2, 10^7)

A249697 First row of spectral array W(Pi-2).

Original entry on oeis.org

1, 8, 9, 64, 73, 516, 589, 4160, 4749, 33540, 38289, 270416, 308704, 2180232, 2488936, 17578149

Offset: 1

Colin Barker, Nov 04 2014

A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.

Cf. A007068, A022159, A022161, A022163, A022165, A249309.

\\ Row i of the generalized Wythoff array W(h),
\\ where h is an irrational number between 1 and 2,
\\ and m is the number of terms in the vectors a and b.
row(h, i, m) = {
  my(
    a=vector(m, n, floor(n*h)),
    b=vector(m, n, floor(n*h/(h-1))),
    w=[a[a[i]], b[a[i]]],
    j=3
  );
  while(1,
    if(j%2==1,
      if(w[j-1]<=#a, w=concat(w, a[w[j-1]]), return(w))
    ,
      if(w[j-2]<=#b, w=concat(w, b[w[j-2]]), return(w))
    );
    j++
  )
}
allocatemem(10^9)
row(Pi-2, 1, 10^7)

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A249221 Expansion of x*(1+5*x-2*x^3)/(1-6*x^2+2*x^4).

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A249222 Expansion of x*(1+5*x-5*x^3)/(1-6*x^2+5*x^4).

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A022162 First column of spectral array W(sqrt(3/2)).

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A249309 First row of spectral array W(Pi/2).

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A249697 First row of spectral array W(Pi-2).

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A249221 Expansion of x(1+5x-2x^3)/(1-6x^2+2*x^4).

A249222 Expansion of x(1+5x-5x^3)/(1-6x^2+5*x^4).