cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A249310 Expansion of x*(1+7*x-6*x^3)/(1-8*x^2+6*x^4).

Original entry on oeis.org

1, 7, 8, 50, 58, 358, 416, 2564, 2980, 18364, 21344, 131528, 152872, 942040, 1094912, 6747152, 7842064, 48324976, 56167040, 346116896, 402283936, 2478985312, 2881269248, 17755181120, 20636450368, 127167537088, 147803987456, 910809209984, 1058613197440
Offset: 1

Views

Author

Colin Barker, Oct 25 2014

Keywords

Comments

It seems that this is also the first row of the spectral array W(sqrt(10)-2).
It also seems that, for all k>0, the first row of W(sqrt(k^2+1)-k+1) has a generating function of the form x*(1+(2*k+1)*x-2*k*x^3)/(1-(2*k+2)*x^2+2*k*x^4).

Links

Crossrefs

Cf. A007068 (k=1), A022165 (k=2), A249311 (k=4), A249312 (k=5), A249313 (k=6).

Programs

  • Mathematica
    CoefficientList[Series[(1 + 7 x - 6 x^3)/(1 - 8 x^2 + 6 x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 25 2014 *)
LinearRecurrence[{0,8,0,-6},{1,7,8,50},30] (* Harvey P. Dale, Sep 22 2019 *)
  • PARI
    Vec((1+7*x-6*x^3)/(1-8*x^2+6*x^4) + O(x^100))

A249311 Expansion of x*(1+9*x-8*x^3)/(1-10*x^2+8*x^4).

Original entry on oeis.org

1, 9, 10, 82, 92, 748, 840, 6824, 7664, 62256, 69920, 567968, 637888, 5181632, 5819520, 47272576, 53092096, 431272704, 484364800, 3934546432, 4418911232, 35895282688, 40314193920, 327476455424, 367790649344, 2987602292736, 3355392942080, 27256211283968
Offset: 1

Views

Author

Colin Barker, Oct 25 2014

Keywords

Comments

It seems that this is also the first row of the spectral array W(sqrt(17)-3).
It also seems that, for all k>0, the first row of W(sqrt(k^2+1)-k+1) has a generating function of the form x*(1+(2*k+1)*x-2*k*x^3)/(1-(2*k+2)*x^2+2*k*x^4).

Links

Crossrefs

Cf. A007068 (k=1), A022165 (k=2), A249310 (k=3), A249312 (k=5), A249313 (k=6).

Programs

  • PARI
    Vec(x*(1+9*x-8*x^3)/(1-10*x^2+8*x^4) + O(x^100))

A249312 Expansion of x*(1+11*x-10*x^3)/(1-12*x^2+10*x^4).

Original entry on oeis.org

1, 11, 12, 122, 134, 1354, 1488, 15028, 16516, 166796, 183312, 1851272, 2034584, 20547304, 22581888, 228054928, 250636816, 2531186096, 2781822912, 28093683872, 30875506784, 311812345504, 342687852288, 3460811307328, 3803499159616, 38411612232896
Offset: 1

Views

Author

Colin Barker, Oct 25 2014

Keywords

Comments

It seems that this is also the first row of the spectral array W(sqrt(26)-4).
It also seems that, for all k>0, the first row of W(sqrt(k^2+1)-k+1) has a generating function of the form x*(1+(2*k+1)*x-2*k*x^3)/(1-(2*k+2)*x^2+2*k*x^4).

Links

Crossrefs

Cf. A007068 (k=1), A022165 (k=2), A249310 (k=3), A249311 (k=4), A249313 (k=6).

Programs

  • Mathematica
    LinearRecurrence[{0,12,0,-10},{1,11,12,122},40] (* Harvey P. Dale, Feb 02 2015 *)
  • PARI
    Vec(x*(1+11*x-10*x^3)/(1-12*x^2+10*x^4) + O(x^100))

Formula

a(1)=1, a(2)=11, a(3)=12, a(4)=122, a(n)=12*a(n-2)-10*a(n-4). - Harvey P. Dale, Feb 02 2015

A249313 Expansion of x*(1+13*x-12*x^3)/(1-14*x^2+12*x^4).

Original entry on oeis.org

1, 13, 14, 170, 184, 2224, 2408, 29096, 31504, 380656, 412160, 4980032, 5392192, 65152576, 70544768, 852375680, 922920448, 11151428608, 12074349056, 145891492352, 157965841408, 1908663749632, 2066629591040, 24970594586624, 27037224177664, 326684359217152
Offset: 1

Views

Author

Colin Barker, Oct 25 2014

Keywords

Comments

It seems that this is also the first row of the spectral array W(sqrt(37)-5).
It also seems that, for all k>0, the first row of W(sqrt(k^2+1)-k+1) has a generating function of the form x*(1+(2*k+1)*x-2*k*x^3)/(1-(2*k+2)*x^2+2*k*x^4).

Links

Crossrefs

Cf. A007068 (k=1), A022165 (k=2), A249310 (k=3), A249311 (k=4), A249312 (k=5).

Programs

  • Mathematica
    CoefficientList[Series[x (1+13x-12x^3)/(1-14x^2+12x^4),{x,0,30}],x] (* or *) LinearRecurrence[{0,14,0,-12},{1,13,14,170},30] (* Harvey P. Dale, Oct 19 2018 *)
  • PARI
    Vec(x*(1+13*x-12*x^3)/(1-14*x^2+12*x^4) + O(x^100))

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

Original entry on oeis.org

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

Views

Author

Colin Barker, Oct 23 2014

Keywords

Links

Crossrefs

Cf. A022163, A022165, A249222.

Programs

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

Formula

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

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

Views

Author

Colin Barker, Oct 23 2014

Keywords

Links

Crossrefs

Cf. A022163, A022165, A249221.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 5 x - 5 x^3)/(1 - 6 x^2 + 5 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 23 2014 *)
  • PARI
    Vec((1+5*x-5*x^3)/(1-6*x^2+5*x^4) + O(x^100))
  • PARI
    a(n)=round((9-(-1)^n)*5^(n\2)/8) \\ Tani Akinari, Oct 26 2014

Formula

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

A022164 First column of spectral array W(sqrt(5)-1).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 33, 34, 35, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 53, 54, 55, 56, 59, 60, 61, 63, 65, 66, 67, 69, 71, 72, 74, 75, 77, 79, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 95, 97, 98, 100
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A022165.

Programs

  • Magma
    [Floor((Sqrt(5)-1)*Floor(n*(Sqrt(5)-1))): n in [1..50]]; // G. C. Greubel, May 27 2018
  • Mathematica
    Table[Floor[(Sqrt[5] - 1)*Floor[(Sqrt[5] - 1)*n]], {n, 1, 50}] (* G. C. Greubel, May 27 2018 *)
  • PARI
    a(n) = floor((sqrt(5)-1)*floor((sqrt(5)-1)*n)); \\ Michel Marcus, Mar 05 2014

Extensions

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

Views

Author

Colin Barker, Oct 25 2014

Keywords

Links

Crossrefs

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

Programs

  • PARI
    \\ 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

Views

Author

Colin Barker, Nov 04 2014

Keywords

Links

Crossrefs

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

Programs

  • PARI
    \\ 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)
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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