|
| |
|
|
A126587
|
|
a(n) is the number of integer lattice points inside the right triangle with legs 3n and 4n (and hypotenuse 5n).
|
|
7
|
|
|
|
3, 17, 43, 81, 131, 193, 267, 353, 451, 561, 683, 817, 963, 1121, 1291, 1473, 1667, 1873, 2091, 2321, 2563, 2817, 3083, 3361, 3651, 3953, 4267, 4593, 4931, 5281, 5643, 6017, 6403, 6801, 7211, 7633, 8067, 8513, 8971, 9441, 9923, 10417, 10923, 11441
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Row sums of triangle A193832. - Omar E. Pol, Aug 22 2011
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Zak Seidov Inside points
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
|
|
|
FORMULA
|
a(n) = A186424(2*n-1).
By Pick's theorem, a(n) = 6*n^2 - 4*n + 1. - Nick Hobson (nickh(AT)qbyte.org), Mar 13 2007
O.g.f.: x*(3+8*x+x^2)/(1-x)^3 = -1 - 12/(-1+x)^3 - 11/(-1+x) - 22/(-1+x)^2. - R. J. Mathar, Dec 10 2007
|
|
|
EXAMPLE
|
At n=1, three lattice points (1,1), (1,2) and (2,1) are inside the triangle with vertices at the points (0,0), (3n,0) and (0,4n); hence a(1)=3.
|
|
|
MATHEMATICA
|
nip[a_, b_]:=Sum[Floor[b-b*i/a-10^-6], {i, a-1}] Table[nip[3k, 4k], {k, 100}]
Table[6*n^2-4*n+1, {n, 1, 50}] (* G. C. Greubel, Mar 06 2018 *)
|
|
|
PROG
|
(MAGMA) [6*n^2 - 4*n + 1: n in [1..50] ]; // Vincenzo Librandi, May 23 2011
(PARI) a(n)=6*n^2-4*n+1 \\ Charles R Greathouse IV, Jun 17 2017
|
|
|
CROSSREFS
|
Sequence in context: A226492 A092347 A215429 * A108126 A106256 A091624
Adjacent sequences: A126584 A126585 A126586 * A126588 A126589 A126590
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Zak Seidov, Jan 05 2007
|
|
|
STATUS
|
approved
|
| |
|
|
