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 1-3 of 3 results.

A269962 Start with a square; at each stage add a square at each expandable vertex so that the ratio of the side of the squares at stage n+1 and at stage n is the golden ratio phi=0.618...; a(n) is the number of squares at n-th stage.

Original entry on oeis.org

1, 5, 17, 45, 105, 237, 537, 1229, 2825, 6493, 14905, 34189, 78409, 179837, 412505, 946221, 2170473, 4978653, 11420025, 26195213, 60086537, 137826493, 316146457, 725176813, 1663410601, 3815531165, 8752065209, 20075486925, 46049151561, 105627543165
Offset: 1

Views

Author

Paolo Franchi, Mar 08 2016

Keywords

Comments

The ratio phi=0.618... is chosen so that from the fourth stage on some squares overlap perfectly. The figure displays some kind of fractal behavior. See illustration.

Links

Crossrefs

Cf. A247618.
Auxiliary sequences: A269963, A269964, A269965.

Programs

  • Mathematica
    RecurrenceTable[{a[n + 1] ==
   4 a[n] - 5 a[n - 1] + 2 a[n - 2] + 2 a[n - 3] - 2 a[n - 4],
  a[1] == 1, a[2] == 5, a[3] == 17, a[4] == 45, a[5] == 105}, a, {n,
  1, 30}]
RecurrenceTable[{a[n + 1] ==
   2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4] + 4,
  a[1] == 1, a[2] == 5, a[3] == 17, a[4] == 45, a[5] == 105}, a, {n,
  1, 30}]
  • PARI
    Vec(x*(1+x)*(1+2*x^2-2*x^3)/((1-x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ Colin Barker, Mar 09 2016

Formula

a(1)=1, for n>=1, a(n) = 2*A269963(n) + 2*A269963(n-1) - 1.
Linear non-homogeneous recurrence relation:
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) + 4.
Linear homogeneous recurrence relation:
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5).
G.f.: x*(1+x)*(1+2*x^2-2*x^3) / ((1-x)*(1-3*x+2*x^2-2*x^4)). - Colin Barker, Mar 09 2016

A269963 Start with a square; at each stage add a square at each expandable vertex so that the ratio of the side of the squares at stage n+1 and at stage n is the golden ratio phi=0.618...; a(n) is the number of squares in a portion of the n-th stage (see below).

Original entry on oeis.org

1, 2, 7, 16, 37, 82, 187, 428, 985, 2262, 5191, 11904, 27301, 62618, 143635, 329476, 755761, 1733566, 3976447, 9121160, 20922109, 47991138, 110082091, 252506316, 579198985, 1328566598, 3047466007, 6990277456, 16034298325, 36779473258, 84364755139
Offset: 1

Views

Author

Paolo Franchi, Mar 09 2016

Keywords

Comments

This is an auxiliary sequence, the main one being A269962.
a(n) gives the number of squares colored red in the illustration.
The ratio phi=0.618... is chosen so that from the fourth stage on some squares overlap perfectly. The figure displays some kind of fractal behavior. See illustration.

Links

Crossrefs

Main sequence: A269962.
Other auxiliary sequences: A269964, A269965.

Programs

  • Mathematica
    RecurrenceTable[{a[n + 1] ==
   2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4],
  a[1] == 1, a[2] == 2, a[3] == 7, a[4] == 16, a[5] == 37}, a, {n, 1,
  30}]
  • PARI
    Vec(x*(1+2*x^2+2*x^3)/((1+x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ Colin Barker, Mar 09 2016

Formula

a(n) = 2*A269964(n) + a(n-1) - 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5).
G.f.: x*(1+2*x^2+2*x^3) / ((1+x)*(1-3*x+2*x^2-2*x^4)). - Colin Barker, Mar 09 2016

A269965 Start with a square; at each stage add a square at each expandable vertex so that the ratio of the side of the squares at stage n+1 and at stage n is the golden ratio phi=0.618...; a(n) is the number of squares in a portion of the n-th stage (see below).

Original entry on oeis.org

1, 3, 10, 26, 63, 145, 332, 760, 1745, 4007, 9198, 21102, 48403, 111021, 254656, 584132, 1339893, 3073459, 7049906, 16171066, 37093175, 85084313, 195166404, 447672720, 1026871705, 2355438303, 5402904310, 12393181766, 28427480091, 65206953349, 149571708488
Offset: 5

Views

Author

Paolo Franchi, Mar 09 2016

Keywords

Comments

This is an auxiliary sequence, the main one being A269962.
a(n) is the number of squares colored red in the illustration.
The ratio phi=0.618... is chosen so that from the fourth stage on some squares overlap perfectly. The figure displays some kind of fractal behavior. See illustration.

Links

Crossrefs

Main sequence: A269962.
Other auxiliary sequences: A269963, A269964.

Programs

  • Mathematica
    RecurrenceTable[{a[n + 1] ==
   2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4] + 5,
  a[5] == 1, a[6] == 3, a[7] == 10, a[8] == 26, a[9] == 63}, a, {n, 5,
   30}]
RecurrenceTable[{a[n + 1] ==
   3 a[n] - a[n - 1] - 3 a[n - 2] + 4 a[n - 3] - 2 a[n - 5],
  a[5] == 1, a[6] == 3, a[7] == 10, a[8] == 26, a[9] == 63,
  a[10] == 145}, a, {n, 5, 30}]
  • PARI
    Vec(x^5*(1+2*x^2+2*x^3)/((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ Colin Barker, Mar 09 2016

Formula

a(1)=a(2)=a(3)=a(4)=0, for n>= 5, a(n) = A269963(n-4)+a(n-1).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) + 5.
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 4*a(n-4) - 2*a(n-6).
G.f.: x^5*(1+2*x^2+2*x^3) / ((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)). - Colin Barker, Mar 09 2016
Showing 1-3 of 3 results.

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