A269962 -id:A269962 - cp's OEIS frontend

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).

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

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

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

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

A339124 a(n) is the number of squares at distance n from the central square of a golden square fractal.

Original entry on oeis.org

1, 4, 12, 28, 60, 132, 300, 692, 1596, 3668, 8412, 19284, 44220, 101428, 232668, 533716, 1224252, 2808180, 6441372, 14775188, 33891324, 77739956, 178319964, 409030356, 938233788, 2152120564, 4936534044, 11323421716, 25973664636, 59578391604

Offset: 0

Rémy Sigrist, Nov 24 2020

For symmetry reasons, a(n) is a multiple of 4 for any n > 0.

Rémy Sigrist, Table of n, a(n) for n = 0..999
Rémy Sigrist, Illustration of initial terms
Wikipedia, Golden square fractal
Index entries for linear recurrences with constant coefficients, signature (3,-2,0,2).

See A337018 for similar sequences.

Cf. A269962 (partial sums).

G.f.: (2*x^4 - 2*x^2 - x - 1)/(2*x^4 - 2*x^2 + 3*x - 1).
a(0) = 1.
a(n) = A269962(n+1) - A269962(n) for any n > 0.
a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-4) for n > 4. - Stefano Spezia, Dec 02 2020

cp's OEIS Frontend

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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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

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A339124 a(n) is the number of squares at distance n from the central square of a golden square fractal.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula