Paolo Franchi - cp's OEIS frontend

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).

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

Paolo Franchi, Mar 09 2016

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.

Colin Barker, Table of n, a(n) for n = 5..1000
Paolo Franchi, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,4,0,-2).

Main sequence: A269962.

Other auxiliary sequences: A269963, A269964.

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}]

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

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

A269964 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, 1, 3, 5, 11, 23, 53, 121, 279, 639, 1465, 3357, 7699, 17659, 40509, 92921, 213143, 488903, 1121441, 2572357, 5900475, 13534515, 31045477, 71212113, 163346335, 374683807, 859449705, 1971405725, 4522010435, 10372587467, 23792640941, 54575559337

Offset: 1

Paolo Franchi, Mar 09 2016

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.

Colin Barker, Table of n, a(n) for n = 1..1000
Paolo Franchi, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,4,0,-2).

Main sequence: A269962.

Other auxiliary sequences: A269963, A269965.

RecurrenceTable[{a[n + 1] ==
   2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4] - 2,
  a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 5, a[5] == 11}, a, {n, 1,
  30}]
RecurrenceTable[{a[n + 1] ==
   3 a[n] - a[n - 1] - 3 a[n - 2] + 4 a[n - 3] - 2 a[n - 5],
  a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 5, a[5] == 11,
  a[6] == 23}, a, {n, 1, 30}]

Vec(x*(1-2*x+x^2-2*x^4)/((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ Colin Barker, Mar 09 2016

a(n) = 2*a(n-2) + 2*a(n-3) + 2*A269965(n) + 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) - 2.
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 4*a(n-4) - 2*a(n-6).
G.f.: x*(1-2*x+x^2-2*x^4) / ((1-x)*(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

Paolo Franchi, Mar 09 2016

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.

Colin Barker, Table of n, a(n) for n = 1..1000
Paolo Franchi, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,2,2).

Main sequence: A269962.

Other auxiliary sequences: A269964, A269965.

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}]

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

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

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

Paolo Franchi, Mar 08 2016

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.

Colin Barker, Table of n, a(n) for n = 1..1000
Paolo Franchi, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,2,-2).

Cf. A247618.

Auxiliary sequences: A269963, A269964, A269965.

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}]

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

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

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User: Paolo Franchi

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).

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A269964 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).

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Author

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Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

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).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula