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A079273 Octo numbers (a polygonal sequence): a(n) = 5*n^2 - 6*n + 2 = (n-1)^2 + (2*n-1)^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)+10n-1 and n+a(n) is always congruent to 2 mod 10 (notice pattern of final digits). a(n)= the n-th hex number (3n^2-3n+1) added to the (2n-2)nd triangular number (2n^2-3n+1). The formula for the n-th octo number can be written as (2n-1)^2 + (n-1)^2; compare to formula for n-th octagonal number, n(3n-2)= (2n-1)^2 - (n-1)^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 n-ary 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(n-1) - 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(n-1) - 3*a(n-2) + a(n-3). - 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 n-th hex figure has only 1 row of length 2n-1 dots (the maximum length) in the center, the n-th 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^2-6n+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^2-6*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

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)