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A188551 Numbers located at angle turns in a pentagonal spiral.

%I A188551 #33 Sep 08 2022 08:45:56
%S A188551 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,
%T A188551 71,77,83,84,90,97,104,111,112,119,127,135,143,144,152,161,170,179,
%U A188551 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
%N A188551 Numbers located at angle turns in a pentagonal spiral.
%C A188551 The link illustrates with three figures:
%C A188551 Figure 1 contains the numbers located at angle turns in the pentagonal spiral;
%C A188551 Figure 2 contains the primes in the pentagonal spiral;
%C A188551 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, ...}.
%H A188551 Vincenzo Librandi, <a href="/A188551/b188551.txt">Table of n, a(n) for n = 1..5000</a>
%H A188551 Michel Lagneau, <a href="/A188551/a188551_3.pdf">Illustration of the numbers in the pentagonal spiral</a>
%H A188551 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,2,-2,0,0,0,-1,1).
%F A188551 From _R. J. Mathar_, Apr 12 2011: (Start)
%F A188551 a(n) = a(n-1) + 2*a(n-5) - 2*a(n-6) - a(n-10) + a(n-11).
%F A188551 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)
%p A188551 with(numtheory):
%p A188551 T:=array(1..300): k:=1:
%p A188551 for n from 1 to 50 do:
%p A188551     x1:= 2*n^2 -1:       T[k]:=x1:
%p A188551     x2:= (n+1)*(2*n-1):  T[k+1]:=x2:
%p A188551     x3:=2*n^2+2*n-1:     T[k+2]:=x3:
%p A188551     x4:= 2*n*(n+1):      T[k+3]:=x4:
%p A188551     x5:=n*(2*n+3):       T[k+4]:=x5:
%p A188551     k:=k+5:
%p A188551 od:
%p A188551 for p from 1 to 250 do:
%p A188551     z:= T[p]:
%p A188551     printf(`%d, `, z):
%p A188551 od:
%t A188551 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 *)
%t A188551 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 *)
%o A188551 (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
%K A188551 nonn,easy
%O A188551 1,2
%A A188551 _Michel Lagneau_, Apr 04 2011