

A079273


Octo numbers (a polygonal sequence): a(n) = 5*n^2  6*n + 2 = (n1)^2 + (2*n1)^2.


4



1, 10, 29, 58, 97, 146, 205, 274, 353, 442, 541, 650, 769, 898, 1037, 1186, 1345, 1514, 1693, 1882, 2081, 2290, 2509, 2738, 2977, 3226, 3485, 3754, 4033, 4322, 4621, 4930, 5249, 5578, 5917, 6266, 6625, 6994, 7373, 7762, 8161, 8570, 8989, 9418, 9857, 10306
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

a(n+1) = a(n)+10n1 and n+a(n) is always congruent to 2 mod 10 (notice pattern of final digits). a(n)= the nth hex number (3n^23n+1) added to the (2n2)nd triangular number (2n^23n+1). The formula for the nth octo number can be written as (2n1)^2 + (n1)^2; compare to formula for nth octagonal number, n(3n2)= (2n1)^2  (n1)^2.
a(n+1) = 5n^2+4n+1 is also the number of ways of realizing the amount 10n using only coins with values 1, 2 and 5. [From Francois Brunault (brunault(AT)gmail.com), Nov 24 2009]
a(n) is the number of length 6 nary words beginning with the first character of the alphabet, that can be built by repeatedly inserting doublets into the initially empty word.  Alois P. Heinz, Sep 01 2011


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hex Number
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 10*n + a(n1)  11 for n>1, a(1)=1.  Vincenzo Librandi, Aug 08 2010
a(1) = 1, a(2) = 10, a(3) = 29; for n>3, a(n) = 3*a(n1)  3*a(n2) + a(n3).  Harvey P. Dale, May 03 2011
G.f.: (2*x^2 + 7*x + 1)*x / (x  1)^3.  Alois P. Heinz, Sep 01 2011


EXAMPLE

a(4) = 58 because 58 dots can be arranged into a simple octagonal pattern with 4 dots on each side, its rows from top to bottom containing 4,5,6,7,7,7,7,6,5 and 4 dots respectively. The pattern is similar to the pattern for hex numbers (see link), with the exception that while the nth hex figure has only 1 row of length 2n1 dots (the maximum length) in the center, the nth octo figure has n such rows.
a(4) = 58:
.. O O O O
. O O O O O
.O O O O O O
O O O O O O O
O O O O O O O
O O O O O O O
O O O O O O O
.O O O O O O
. O O O O O
.. O O O O


MATHEMATICA

Table[5n^26n+2, {n, 50}] (* or *) LinearRecurrence[{3, 3, 1}, {1, 10, 29}, 150] (* Harvey P. Dale, Apr 06 2011 & May 03 2011 *)


PROG

(PARI) a(n)=5*n^26*n+2 \\ Charles R Greathouse IV, Oct 07 2015


CROSSREFS

Cf. A000217 (triangular numbers), A000567 (octagonal numbers), A003215 (hex numbers).
Row n=3 of A183134.  Alois P. Heinz, Aug 31 2011
Sequence in context: A143190 A009771 A068197 * A271991 A048469 A031129
Adjacent sequences: A079270 A079271 A079272 * A079274 A079275 A079276


KEYWORD

nonn,easy,nice


AUTHOR

Matthew Vandermast, Feb 06 2003


STATUS

approved



