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%I #118 Apr 16 2024 13:53:16
%S 0,1,12,33,64,105,156,217,288,369,460,561,672,793,924,1065,1216,1377,
%T 1548,1729,1920,2121,2332,2553,2784,3025,3276,3537,3808,4089,4380,
%U 4681,4992,5313,5644,5985,6336,6697,7068,7449,7840,8241,8652
%N 12-gonal (or dodecagonal) numbers: a(n) = n*(5*n-4).
%C Zero followed by partial sums of A017281. - _Klaus Brockhaus_, Nov 20 2008
%C Sequence found by reading the line from 0, in the direction 0, 12, ... and the parallel line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized 12-gonal numbers A195162. - _Omar E. Pol_, Jul 18 2012
%C This is also a star hexagonal number: a(n) = A000384(n) + 6*A000217(n-1). - _Luciano Ancora_, Mar 30 2015
%C Starting with offset 1, this is the binomial transform of (1, 11, 10, 0, 0, 0, ...). - _Gary W. Adamson_, Aug 01 2015
%C a(n+1) is the sum of the odd numbers from 4n+1 to 6n+1. - _Wesley Ivan Hurt_, Dec 14 2015
%C For n >= 2, a(n) is the number of intersection points of all unit circles centered on the inner lattice points of an (n+1) X (n+1) square grid. - _Wesley Ivan Hurt_, Dec 08 2020
%C The final digit of a(n) equals the final digit of n, A010879(n). - _Enrique Pérez Herrero_, Nov 13 2022
%C a(n-1) is the maximum second Zagreb index of maximal 2-degenerate graphs with n vertices. (The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.) - _Allan Bickle_, Apr 16 2024
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
%H T. D. Noe, <a href="/A051624/b051624.txt">Table of n, a(n) for n = 0..1000</a>
%H John Elias, <a href="/A051624/a051624.png">Illustration: compass configuration</a> , <a href="/A051624/a051624_1.png">Illustration: cross configuration</a>.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H Allan Bickle, <a href="https://ajc.maths.uq.edu.au/pdf/89/ajc_v89_p167.pdf">Zagreb Indices of Maximal k-degenerate Graphs</a>, Australas. J. Combin. 89 1 (2024) 167-178.
%H J. Estes and B. Wei, <a href="https://doi.org/10.1007/s10878-012-9515-6">Sharp bounds of the Zagreb indices of k-trees</a>, J Comb Optim 27 (2014), 271-291.
%H L. Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n39">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dodecagonal_number">Dodecagonal number</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(1+9*x)/(1-x)^3.
%F a(n) = Sum_{k=0..n-1} 10*k+1. - _Klaus Brockhaus_, Nov 20 2008
%F a(n) = 10*n + a(n-1) - 9 (with a(0)=0). - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = A131242(10n). - _Philippe Deléham_, Mar 27 2013
%F a(10*a(n) + 46*n + 1) = a(10*a(n) + 46*n) + a(10*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F E.g.f.: x*(5*x + 1) * exp(x). - _G. C. Greubel_, Jul 31 2015
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=12. - _G. C. Greubel_, Jul 31 2015
%F Sum_{n>=1} 1/a(n) = sqrt(1 + 2/sqrt(5))*Pi/8 + 5*log(5)/16 + sqrt(5)*log((1 + sqrt(5))/2)/8 = 1.177956057922663858735173968... . - _Vaclav Kotesovec_, Apr 27 2016
%F a(n) + 4*(n-1)^2 = (3*n-2)^2. Let P(k,n) be the n-th k-gonal number. Then, in general, P(4k,n) + (k-1)^2*(n-1)^2 = (k*n-k+1)^2. - _Charlie Marion_, Feb 04 2020
%F Product_{n>=2} (1 - 1/a(n)) = 5/6. - _Amiram Eldar_, Jan 21 2021
%F a(n) = (3*n-2)^2 - (2*n-2)^2. In general, if we let P(k,n) = the n-th k-gonal number, then P(4k,n) = (k*n-(k-1))^2 - ((k-1)*n-(k-1))^2. - _Charlie Marion_, Nov 11 2021
%e The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.
%t RecurrenceTable[{a[0]==0, a[1]==1, a[2]==12, a[n]== 3*a[n-1] - 3*a[n-2] + a[n-3]}, a, {n, 30}] (* _G. C. Greubel_, Jul 31 2015 *)
%t Table[n*(5*n - 4), {n, 0, 100}] (* _Robert Price_, Oct 11 2018 *)
%o (Magma) [ n eq 1 select 0 else Self(n-1)+10*(n-2)+1: n in [1..43] ]; // _Klaus Brockhaus_, Nov 20 2008
%o (PARI) a(n)=(5*n-4)*n \\ _Charles R Greathouse IV_, Jun 16 2011
%Y First differences of A007587.
%Y Cf. A093645 ((10, 1) Pascal, column m=2). Partial sums of A017281.
%Y Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
%Y Cf. A372027 (second Zagreb index of MOPs).
%K nonn,easy,changed
%O 0,3
%A _Barry E. Williams_
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