A051624 12-gonal (or dodecagonal) numbers: a(n) = n*(5*n-4).

0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, 4089, 4380, 4681, 4992, 5313, 5644, 5985, 6336, 6697, 7068, 7449, 7840, 8241, 8652

Offset

0,3

Comments

Zero followed by partial sums of A017281. - Klaus Brockhaus, Nov 20 2008

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

This is also a star hexagonal number: a(n) = A000384(n) + 6*A000217(n-1). - Luciano Ancora, Mar 30 2015

Starting with offset 1, this is the binomial transform of (1, 11, 10, 0, 0, 0, ...). - Gary W. Adamson, Aug 01 2015

a(n+1) is the sum of the odd numbers from 4n+1 to 6n+1. - Wesley Ivan Hurt, Dec 14 2015

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

The final digit of a(n) equals the final digit of n, A010879(n). - Enrique Pérez Herrero, Nov 13 2022

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

References

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

John Elias, Illustration: compass configuration , Illustration: cross configuration.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.

Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.

J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.

L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.

Wikipedia, Dodecagonal number

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: x*(1+9*x)/(1-x)^3.

a(n) = Sum_{k=0..n-1} 10*k+1. - Klaus Brockhaus, Nov 20 2008

a(n) = 10*n + a(n-1) - 9 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010

a(n) = A131242(10n). - Philippe Deléham, Mar 27 2013

a(10*a(n) + 46*n + 1) = a(10*a(n) + 46*n) + a(10*n+1). - Vladimir Shevelev, Jan 24 2014

E.g.f.: x*(5*x + 1) * exp(x). - G. C. Greubel, Jul 31 2015

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

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

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

Product_{n>=2} (1 - 1/a(n)) = 5/6. - Amiram Eldar, Jan 21 2021

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

Example

The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.

Mathematica

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 *)
Table[n*(5*n - 4), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

Prog

(Magma) [ n eq 1 select 0 else Self(n-1)+10*(n-2)+1: n in [1..43] ]; // Klaus Brockhaus, Nov 20 2008
(PARI) a(n)=(5*n-4)*n \\ Charles R Greathouse IV, Jun 16 2011

Crossrefs

First differences of A007587.

Cf. A093645 ((10, 1) Pascal, column m=2). Partial sums of A017281.

Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).

Cf. A372027 (second Zagreb index of MOPs).

Sequence in context: A372026 A063296 A372027 * A367347 A039338 A118337

Adjacent sequences: A051621 A051622 A051623 * A051625 A051626 A051627

Keyword

nonn,easy,changed

Author

Barry E. Williams

Status

approved