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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
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
KEYWORD
nonn,easy
AUTHOR
Michel Lagneau, Apr 04 2011
STATUS
approved

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Last modified May 18 12:18 EDT 2024. Contains 372630 sequences. (Running on oeis4.)