A188551 Numbers located at angle turns in a pentagonal spiral.

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

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 *)
LinearRecurrence[{1, 0, 0, 0, 2, -2, 0, 0, 0, -1, 1}, {1, 2, 3, 4, 5, 7, 9, 11, 12, 14, 17}, 80] (* Harvey P. Dale, Jun 17 2021 *)

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: A331230 A258456 A230843 * A045782 A184112 A064005

Adjacent sequences: A188548 A188549 A188550 * A188552 A188553 A188554

Keyword

nonn,easy

Author

Michel Lagneau, Apr 04 2011

Status

approved