A079273 Octo numbers (a polygonal sequence): a(n) = 5*n^2 - 6*n + 2 = (n-1)^2 + (2*n-1)^2.

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

Offset: 1

Author

Matthew Vandermast, Feb 06 2003

Keywords

• nonn
• easy
• nice

Comments

a(n+1) = a(n) + 10*n - 1, and n + a(n) is always congruent to 2 mod 10 (notice pattern of final digits). a(n) = the n-th hex number (3*n^2 - 3*n + 1) added to the (2n-2)-nd triangular number (2*n^2 - 3*n + 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) = 5*n^2 + 4*n + 1 is also the number of ways of realizing the amount 10n using only coins with values 1, 2 and 5. - 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

For n > 1, a(n) is the Wiener index of the caterpillar of diameter 3 where each internal vertex has attached n - 2 pendent vertices. - Christian Barrientos, Mar 31 2023

Examples

			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
		

Links

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

Crossrefs

Cf. A000217 (triangular numbers), A000567 (octagonal numbers), A003215 (hex numbers).

Row n=3 of A183134. - Alois P. Heinz, Aug 31 2011

Cf. A016873.

Programs

Magma

[n*(5*n-6) +2: n in [1..50]]; // G. C. Greubel, Apr 19 2023

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 *)

PARI

a(n)=5*n^2-6*n+2 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[n*(5*n-6) +2 for n in range(1,51)] # G. C. Greubel, Apr 19 2023

Formula

a(n) = 10*n + a(n-1) - 11 for n > 1, a(1)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(1) = 1, a(2) = 10, a(3) = 29. - Harvey P. Dale, May 03 2011

G.f.: x*(1 + 7*x + 2*x^2)/(1 - x)^3. - Alois P. Heinz, Sep 01 2011

E.g.f.: -2 + (2 - x + 5*x^2)*exp(x). - G. C. Greubel, Apr 19 2023

5*a(n) = A016873(n-1)^2 + 1. - Charlie Marion, May 10 2024