A304163 -id:A304163 - cp's OEIS frontend

A069131 Centered 18-gonal numbers.

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations.
Lamine Ngom, An origin of A069131 (illustration).
Leo Tavares, Illustration: Tri-Hexagons.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,19,55},50] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A304164 a(n) = 27n^2 - 21n + 6 (n>=1).

Original entry on oeis.org

12, 72, 186, 354, 576, 852, 1182, 1566, 2004, 2496, 3042, 3642, 4296, 5004, 5766, 6582, 7452, 8376, 9354, 10386, 11472, 12612, 13806, 15054, 16356, 17712, 19122, 20586, 22104, 23676, 25302, 26982, 28716, 30504, 32346, 34242, 36192, 38196, 40254, 42366

Offset: 1

Emeric Deutsch, May 09 2018

a(n) is the number of edges in the HcDN1(n) network (see Fig. 3 in the Hayat et al. manuscript).

Colin Barker, Table of n, a(n) for n = 1..1000
S. Hayat, M. A. Malik, and M. Imran, Computing topological indices of honeycomb derived networks, Romanian J. of Information Science and Technology, 18, No. 2, 2015, 144-165.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A304163.

List([1..40],n->27*n^2-21*n+6); # Muniru A Asiru, May 10 2018

Maple
```
seq(27*n^2-21*n+6, n = 1 .. 40);
```

Table[27n^2-21n+6,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{12,72,186},40] (* Harvey P. Dale, Aug 31 2024 *)

a(n) = 27*n^2-21*n+6; \\ Altug Alkan, May 09 2018

Vec(6*x*(2 + 6*x + x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, May 10 2018

From Colin Barker, May 10 2018: (Start)
G.f.: 6*x*(2 + 6*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
E.g.f.: 3*exp(x)*(2 + 2*x + 9*x^2) - 6. - Stefano Spezia, Apr 15 2023

A304165 a(n) = 324n^2 - 336n + 102 (n >= 1).

Original entry on oeis.org

90, 726, 2010, 3942, 6522, 9750, 13626, 18150, 23322, 29142, 35610, 42726, 50490, 58902, 67962, 77670, 88026, 99030, 110682, 122982, 135930, 149526, 163770, 178662, 194202, 210390, 227226, 244710, 262842, 281622, 301050, 321126, 341850, 363222, 385242, 407910, 431226, 455190, 479802, 505062

Offset: 1

Emeric Deutsch, May 09 2018

a(n) is the first Zagreb index of the HcDN1(n) network (see Fig. 3 in the Hayat et al. manuscript).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of HcDN1(n) is M(HcDN1(n);x,y) = 6x^3*y^3 + 12(n-1)x^3*y^5 + 6nx^3*y^6 + 18(n-1)x^5*y^6 + (27n^2 -57n +30)x^6*y^6. - Emeric Deutsch, May 11 2018

Colin Barker, Table of n, a(n) for n = 1..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
S. Hayat, M. A. Malik, and M. Imran, Computing topological indices of honeycomb derived networks, Romanian J. of Information Science and Technology, 18, No. 2, 2015, 144-165.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A304163, A304164, A304166.

List([1..40],n->324*n^2-336*n+102); # Muniru A Asiru, May 10 2018

Maple
```
seq(324*n^2-336*n+102,n=1..40);
```

Table[324n^2-336n+102,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{90,726,2010},40] (* Harvey P. Dale, Apr 12 2020 *)

a(n) = 324*n^2-336*n+102; \\ Altug Alkan, May 09 2018

Vec(6*x*(15 + 76*x + 17*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, May 10 2018

From Colin Barker, May 10 2018: (Start)
G.f.: 6*x*(15 + 76*x + 17*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
E.g.f.: 6*(exp(x)*(17 - 2*x + 54*x^2) - 17). - Stefano Spezia, Apr 15 2023

A304166 a(n) = 972n^2 - 1224n + 414 with n > 0.

Original entry on oeis.org

162, 1854, 5490, 11070, 18594, 28062, 39474, 52830, 68130, 85374, 104562, 125694, 148770, 173790, 200754, 229662, 260514, 293310, 328050, 364734, 403362, 443934, 486450, 530910, 577314, 625662, 675954, 728190, 782370, 838494, 896562, 956574, 1018530, 1082430, 1148274, 1216062, 1285794, 1357470, 1431090, 1506654

Offset: 1

Emeric Deutsch, May 09 2018

a(n) provides the second Zagreb index of the HcDN1(n) network (see Fig. 3 in the Hayat et al. paper).
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of HcDN1(n) is M(HcDN1(n); x,y) = 6x^3*y^3 + 12(n-1)x^3*y^5 + 6nx^3*y^6 + 18(n-1)x^5*y^6 + (27n^2 - 57n + 30)x^6*y^6. - Emeric Deutsch, May 11 2018

Colin Barker, Table of n, a(n) for n = 1..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
S. Hayat, M. A. Malik, and M. Imran, Computing topological indices of honeycomb derived networks, Romanian J. of Information Science and Technology, 18, No. 2, 2015, 144-165.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A304163, A304164, A304165.

Maple
```
seq(972*n^2-1224*n+414, n = 1 .. 40);
```

a(n) = 972*n^2-1224*n+414; \\ Altug Alkan, May 09 2018

Vec(18*x*(9 + 76*x + 23*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, May 10 2018

From Colin Barker, May 10 2018: (Start)
G.f.: 18*x*(9 + 76*x + 23*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
E.g.f.: 18*(exp(x)*(23 - 14*x + 54*x^2) - 23). - Stefano Spezia, Apr 15 2023

cp's OEIS Frontend

A069131 Centered 18-gonal numbers.

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A304164 a(n) = 27*n^2 - 21*n + 6 (n>=1).

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A304165 a(n) = 324*n^2 - 336*n + 102 (n >= 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

PARI

Formula

A304166 a(n) = 972*n^2 - 1224*n + 414 with n > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

PARI

Formula

A304164 a(n) = 27n^2 - 21n + 6 (n>=1).

A304165 a(n) = 324n^2 - 336n + 102 (n >= 1).

A304166 a(n) = 972n^2 - 1224n + 414 with n > 0.