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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.

%I A079273 #71 Feb 16 2025 08:32:48
%S A079273 1,10,29,58,97,146,205,274,353,442,541,650,769,898,1037,1186,1345,
%T A079273 1514,1693,1882,2081,2290,2509,2738,2977,3226,3485,3754,4033,4322,
%U A079273 4621,4930,5249,5578,5917,6266,6625,6994,7373,7762,8161,8570,8989,9418,9857,10306
%N A079273 Octo numbers (a polygonal sequence): a(n) = 5*n^2 - 6*n + 2 = (n-1)^2 + (2*n-1)^2.
%C A079273 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.
%C A079273 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
%C A079273 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
%C A079273 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
%H A079273 Alois P. Heinz, <a href="/A079273/b079273.txt">Table of n, a(n) for n = 1..1000</a>
%H A079273 Leo Tavares, <a href="/A079273/a079273.png">Illustration: Compacted Hexagons</a>
%H A079273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexNumber.html">Hex Number</a>
%H A079273 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A079273 a(n) = 10*n + a(n-1) - 11 for n > 1, a(1)=1. - _Vincenzo Librandi_, Aug 08 2010
%F A079273 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
%F A079273 G.f.: x*(1 + 7*x + 2*x^2)/(1 - x)^3. - _Alois P. Heinz_, Sep 01 2011
%F A079273 E.g.f.: -2 + (2 - x + 5*x^2)*exp(x). - _G. C. Greubel_, Apr 19 2023
%F A079273 5*a(n) = A016873(n-1)^2 + 1. - _Charlie Marion_, May 10 2024
%e A079273 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.
%e A079273 a(4) = 58:
%e A079273      O O O O
%e A079273     O O O O O
%e A079273    O O O O O O
%e A079273   O O O O O O O
%e A079273   O O O O O O O
%e A079273   O O O O O O O
%e A079273   O O O O O O O
%e A079273    O O O O O O
%e A079273     O O O O O
%e A079273      O O O O
%t A079273 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 *)
%o A079273 (PARI) a(n)=5*n^2-6*n+2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A079273 (Magma) [n*(5*n-6) +2: n in [1..50]]; // _G. C. Greubel_, Apr 19 2023
%o A079273 (SageMath) [n*(5*n-6) +2 for n in range(1,51)] # _G. C. Greubel_, Apr 19 2023
%Y A079273 Cf. A000217 (triangular numbers), A000567 (octagonal numbers), A003215 (hex numbers).
%Y A079273 Row n=3 of A183134. - _Alois P. Heinz_, Aug 31 2011
%Y A079273 Cf. A016873.
%K A079273 nonn,easy,nice
%O A079273 1,2
%A A079273 _Matthew Vandermast_, Feb 06 2003