

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
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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

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


FORMULA

a(n) = a(n1) + 2*a(n5)  2*a(n6)  a(n10) + a(n11).
G.f.: x*(1+x)*(1+x^2)*(x^2x+1)*(x^3x1) / ((x^4+x^3+x^2+x+1)^2*(x1)^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*n1): T[k+1]:=x2:
x3:=2*n^2+2*n1: 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(n1)+2*Self(n5)2*Self(n6)Self(n10)+Self(n11): n in [1..90]]; // Vincenzo Librandi, Aug 18 2018


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



