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 1-2 of 2 results.

A153792 12 times pentagonal numbers: a(n) = 6*n*(3*n-1).

Original entry on oeis.org

0, 12, 60, 144, 264, 420, 612, 840, 1104, 1404, 1740, 2112, 2520, 2964, 3444, 3960, 4512, 5100, 5724, 6384, 7080, 7812, 8580, 9384, 10224, 11100, 12012, 12960, 13944, 14964, 16020, 17112, 18240, 19404, 20604, 21840, 23112, 24420
Offset: 0

Views

Author

Omar E. Pol, Jan 01 2009

Keywords

Comments

For n>=1, a(n) is the first Zagreb index of the triangular grid graph T[n] (see the West reference, p. 390). The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 10 2016
The M-polynomial of the triangular grid graph T[n] is M(T[n], x, y) = 6*x^2*y^4 + 3*(n-1)*x^4*y^4 +6*(n-2)*x^4*y^6+3*(n-2)*(n-3)*x^6*y^6/2. - Emeric Deutsch, May 09 2018
This is the number of overlapping six sphinx tiled shapes in the sphinx tessellated hexagon described in A291582. - Craig Knecht, Sep 13 2017
a(n) is the number of words of length 3n over the alphabet {a,b,c}, where the number of b's plus the number of c's is 2. - Juan Camacho, Mar 03 2021
Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Mar 12 2021

References

  • D. B. West, Introduction to Graph Theory, 2nd edition, Prentice-Hall, 2001.

Links

Crossrefs

Cf. A000326, A049450, A062741, A033579, A152743, A153449, A153793.
Cf. A008588, A195160, A195321.

Programs

Formula

a(n) = 18*n^2 - 6*n = 12*A000326(n) = 6*A049450(n) = 4*A062741(n) = 3*A033579(n) = 2*A152743(n).
a(n) = 36*n + a(n-1) - 24 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
G.f.: 12*x*(1 + 2*x)/(1-x)^3. - Colin Barker, Feb 14 2012
a(0)=0, a(1)=12, a(2)=60; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 11 2012
E.g.f.: 6*x*(2 + 3*x)*exp(x). - G. C. Greubel, Aug 29 2016
a(n) = A291582(n) - A195321(n) for n > 0. - Craig Knecht, Sep 13 2017
a(n) = A195321(n) - A008588(n). - Omar E. Pol, Mar 12 2021
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = log(3)/4 - Pi/(12*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) - log(2)/3. (End)

A153780 10 times pentagonal numbers: a(n) = 5*n*(3*n-1).

Original entry on oeis.org

0, 10, 50, 120, 220, 350, 510, 700, 920, 1170, 1450, 1760, 2100, 2470, 2870, 3300, 3760, 4250, 4770, 5320, 5900, 6510, 7150, 7820, 8520, 9250, 10010, 10800, 11620, 12470, 13350, 14260, 15200, 16170, 17170, 18200, 19260, 20350
Offset: 0

Views

Author

Omar E. Pol, Jan 01 2009

Keywords

Links

Crossrefs

Cf. A000326, A049450, A152734, A152996, A153449.

Programs

  • Mathematica
    Table[5*n*(3*n - 1), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,10,50},25] (* G. C. Greubel, Aug 28 2016 *)
  • PARI
    a(n) = 5*n*(3*n-1); \\ Michel Marcus, Aug 28 2016

Formula

a(n) = 15*n^2 - 5*n = 10*A000326(n) = 5*A049450(n) = 2*A152734(n).
a(n) = 30*n + a(n-1) - 20 for n>0, a(0) = 0. - Vincenzo Librandi, Aug 03 2010
G.f.: 10*x*(1+2*x)/(1-x)^3. - Colin Barker, Feb 14 2012
From G. C. Greubel, Aug 28 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: 5*x*(2 + 3*x)*exp(x). (End)
Showing 1-2 of 2 results.

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