cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Barry E. Williams

Keywords

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

Examples

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

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

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).

Programs

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

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

A372025 Maximum second Zagreb index of maximal 3-degenerate graphs with n vertices.

Original entry on oeis.org

12, 54, 120, 210, 324, 462, 624, 810, 1020, 1254, 1512, 1794, 2100, 2430, 2784, 3162, 3564, 3990, 4440, 4914, 5412, 5934, 6480, 7050, 7644, 8262, 8904, 9570, 10260, 10974, 11712, 12474, 13260, 14070, 14904, 15762, 16644, 17550, 18480, 19434, 20412, 21414, 22440, 23490, 24564, 25662, 26784, 27930
Offset: 3

Views

Author

Allan Bickle, Apr 16 2024

Keywords

Comments

The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.
A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are 3-stars, so the bound also applies to 3-trees.

Examples

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

Links

Crossrefs

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
Cf. A372027 (second Zagreb index of MOPs).

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 1}, {12, 54, 120}, 50] (* Paolo Xausa, Jan 22 2025 *)

Formula

a(n) = 3*(n-1)^2 + 9*(n-3)*(n-1).
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
G.f.: x^3*(6*x^2 - 18*x - 12)/(x - 1)^3. (End)
a(n) = 6*A014107(n-1). Sum_{n>=3} 1/a(n) = (1/2+log(2))/9 = 0.1325719... - R. J. Mathar, Apr 22 2024

A372027 Maximum second Zagreb index of maximal outerplanar graphs with n vertices.

Original entry on oeis.org

12, 33, 61, 96, 135, 181, 233, 291, 355, 425, 501, 583, 671, 765, 865, 971, 1083, 1201, 1325, 1455, 1591, 1733, 1881, 2035, 2195, 2361, 2533, 2711, 2895, 3085, 3281, 3483, 3691, 3905, 4125, 4351, 4583, 4821, 5065, 5315, 5571, 5833, 6101, 6375, 6655, 6941, 7233, 7531
Offset: 3

Views

Author

Allan Bickle, Apr 16 2024

Keywords

Comments

The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.
A maximal outerplanar graph has all vertices on the exterior region, and all other regions triangles. The extremal graphs are fans, except when n=6. Then the extremal graph is the triangular grid with degrees 4,4,4,2,2,2.

Examples

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

Links

Crossrefs

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 1}, {12, 33, 61, 96, 135, 181, 233}, 50] (* Paolo Xausa, Jan 22 2025 *)

Formula

a(n) = 3*n^2 + n - 19 when n is not 3 or 6.
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 9.
G.f.: x^3*(x^6 - 3*x^5 + 3*x^4 + 2*x^2 + 3*x - 12)/(x - 1)^3. (End)
Showing 1-3 of 3 results.

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