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

Author

Michel Lagneau, Apr 04 2011

Keywords

• nonn
• easy

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).

Programs

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

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 *)

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)