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A188551 Numbers located at angle turns in a pentagonal spiral. 2
1, 2, 3, 4, 5, 7, 9, 11, 12, 14, 17, 20, 23, 24, 27, 31, 35, 39, 40, 44, 49, 54, 59, 60, 65, 71, 77, 83, 84, 90, 97, 104, 111, 112, 119, 127, 135, 143, 144, 152, 161, 170, 179, 180, 189, 199, 209, 219, 220, 230, 241, 252, 263, 264, 275, 287, 299, 311, 312, 324, 337, 350, 363, 364, 377, 391, 405, 419, 420, 434, 449, 464, 479, 480 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The link illustrates with three figures:

Figure 1 contains the numbers located at angle turns in the pentagonal spiral;

Figure 2 contains the primes in the pentagonal spiral;

Figure 3 shows a variety of sequences that are associated with the numbers on the lines and diagonals in the pentagonal spiral. For example, the sequence A033537 given by the formula n(2n+5) generates {0, 7, 18, 33, 52, 75, ...} and the corresponding line in the spiral passes through {7, 18, 33, 52, 75, ...}.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Michel Lagneau, Illustration of the numbers in the pentagonal spiral

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,2,-2,0,0,0,-1,1).

FORMULA

From R. J. Mathar, Apr 12 2011: (Start)

a(n) = a(n-1) + 2*a(n-5) - 2*a(n-6) - a(n-10) + a(n-11).

G.f.: x*(1+x)*(1+x^2)*(x^2-x+1)*(x^3-x-1) / ((x^4+x^3+x^2+x+1)^2*(x-1)^3 ). (End)

MAPLE

with(numtheory):

T:=array(1..300): k:=1:

for n from 1 to 50 do:

    x1:= 2*n^2 -1:       T[k]:=x1:

    x2:= (n+1)*(2*n-1):  T[k+1]:=x2:

    x3:=2*n^2+2*n-1:     T[k+2]:=x3:

    x4:= 2*n*(n+1):      T[k+3]:=x4:

    x5:=n*(2*n+3):       T[k+4]:=x5:

    k:=k+5:

od:

for p from 1 to 250 do:

    z:= T[p]:

    printf(`%d, `, z):

od:

MATHEMATICA

CoefficientList[Series[(1 + x) (1 + x^2) (x^2 - x + 1) (x^3 - x - 1) / ((x^4 + x^3 + x^2 + x + 1)^2 (x - 1)^3), {x, 0, 80}], x] (* Vincenzo Librandi, Aug 18 2018 *)

PROG

(MAGMA) I:=[1, 2, 3, 4, 5, 7, 9, 11, 12, 14, 17]; [n le 11 select I[n] else Self(n-1)+2*Self(n-5)-2*Self(n-6)-Self(n-10)+Self(n-11): n in [1..90]]; // Vincenzo Librandi, Aug 18 2018

CROSSREFS

Sequence in context: A319237 A258456 A230843 * A045782 A184112 A064005

Adjacent sequences:  A188548 A188549 A188550 * A188552 A188553 A188554

KEYWORD

nonn,easy

AUTHOR

Michel Lagneau, Apr 04 2011

STATUS

approved

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Last modified September 15 18:22 EDT 2019. Contains 327082 sequences. (Running on oeis4.)