%I #24 May 30 2022 13:11:54

%S 0,9,26,51,84,125,174,231,296,369,450,539,636,741,854,975,1104,1241,

%T 1386,1539,1700,1869,2046,2231,2424,2625,2834,3051,3276,3509,3750,

%U 3999,4256,4521,4794,5075,5364,5661,5966,6279,6600,6929,7266,7611,7964,8325

%N a(n) = (n-1)*(4*n+1).

%C A polynomial curve. However, write 0, 1, 2, ... in a square spiral, with 0 at the origin and 1 immediately below it; sequence gives numbers parallel to the negative y-axis (see Example section). This sequence only encounters composite numbers as it expands to infinity.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A164754(n+1) + 1 = A001107(n+1), for n >= 2.

%F G.f.: x^2*(-9+x)/(x-1)^3 . - _R. J. Mathar_, Sep 15 2021

%F Sum_{n>=2} 1/a(n) = 24/25 -3*log(2)/5 -Pi/10. - _R. J. Mathar_, May 30 2022

%e On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence, not including "0", will lie parallel to the negative y-axis, located within the fourth quadrant, as seen in the example below:

%e 99 64--65--66--67--68--69--70--71--72

%e | | |

%e 98 63 36--37--38--39--40--41--42 73

%e | | | | |

%e 97 62 35 16--17--18--19--20 43 74

%e | | | | | | |

%e 96 61 34 15 4---5---6 21 44 75

%e | | | | | | | | |

%e 95 60 33 14 3 *0* 7 22 45 76

%e | | | | | | | | | |

%e 94 59 32 13 2---1 8 23 46 77

%e | | | | | | | |

%e 93 58 31 12--11--10--*9* 24 47 78

%e | | | | | |

%e 92 57 30--29--28--27-*26*-25 48 79

%e | | | |

%e 91 56--55--54--53--52-*51*-50--49 80

%e | |

%e 90--89--88--87--86--85-*84*-83--82--81

%p A343560 := n -> 4*n^2 - 3*n - 1;

%p seq(A343560(n), n = 1 .. 50);

%o (C) int a(int n) { return 4*n*n-3*n-1; }

%Y Cf. A164754, A001107, A180863.

%K nonn,easy

%O 1,2

%A _Zachary Dove_, Apr 19 2021