OFFSET
1,2
COMMENTS
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.
LINKS
FORMULA
G.f.: x^2*(-9+x)/(x-1)^3 . - R. J. Mathar, Sep 15 2021
Sum_{n>=2} 1/a(n) = 24/25 -3*log(2)/5 -Pi/10. - R. J. Mathar, May 30 2022
EXAMPLE
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:
99 64--65--66--67--68--69--70--71--72
| | |
98 63 36--37--38--39--40--41--42 73
| | | | |
97 62 35 16--17--18--19--20 43 74
| | | | | | |
96 61 34 15 4---5---6 21 44 75
| | | | | | | | |
95 60 33 14 3 *0* 7 22 45 76
| | | | | | | | | |
94 59 32 13 2---1 8 23 46 77
| | | | | | | |
93 58 31 12--11--10--*9* 24 47 78
| | | | | |
92 57 30--29--28--27-*26*-25 48 79
| | | |
91 56--55--54--53--52-*51*-50--49 80
| |
90--89--88--87--86--85-*84*-83--82--81
PROG
(C) int a(int n) { return 4*n*n-3*n-1; }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zachary Dove, Apr 19 2021
STATUS
approved