login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A269964 Start with a square; at each stage add a square at each expandable vertex so that the ratio between 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). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

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).

FORMULA

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

MATHEMATICA

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}]

PROG

(PARI) 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

CROSSREFS

Main sequence: A269962.

Other auxiliary sequences: A269963, A269965.

Sequence in context: A113281 A037446 A113151 * A305412 A094810 A139376

Adjacent sequences:  A269961 A269962 A269963 * A269965 A269966 A269967

KEYWORD

nonn,easy

AUTHOR

Paolo Franchi, Mar 09 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 2 11:14 EST 2021. Contains 349440 sequences. (Running on oeis4.)