A163756 -id:A163756 - 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

A069127 Centered 14-gonal numbers.

Original entry on oeis.org

1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201, 4551, 4915, 5293, 5685, 6091, 6511, 6945, 7393, 7855, 8331, 8821, 9325, 9843, 10375, 10921, 11481, 12055, 12643, 13245, 13861, 14491

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Binomial transform of [1, 14, 14, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 14, 0, 0, 0, ...]. - Gary W. Adamson, Jul 29 2011

Centered tetradecagonal numbers or centered tetrakaidecagonal numbers. - Omar E. Pol, Oct 03 2011

			a(5) = 141 because 7*5^2 - 7*5 + 1 = 175 - 35 + 1 = 141.
a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
From _Bruno Berselli_, Oct 27 2017: (Start)
1   =         -(1) + (2).
15  =       -(1+2) + (3+4+5+6).
43  =     -(1+2+3) + (4+5+6+7+8+9+10).
85  =   -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14).
141 = -(1+2+3+4+5) + (6+7+8+9+10+11+12+13+14+15+16+17+18). (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Heptagonal Stars.
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. A005448, A001844, A005891, A003215, A069099.

Cf. A163756, A193053.

FoldList[#1 + #2 &, 1, 14 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[14*Range[0,50]]+1 (* Harvey P. Dale, Apr 09 2012 *)

a(n)=7*n^2-7*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 7*n^2 - 7*n + 1.
a(n) = 14*n+a(n-1)-14 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+12*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
a(n) = A163756(n-1) + 1. - Omar E. Pol, Oct 03 2011
a(n) = a(-n+1) = A193053(2n-2) + A193053(2n-3). - Bruno Berselli, Oct 21 2011
Sum_{n>=1} 1/a(n) = Pi * tan(sqrt(3/7)*Pi/2) / sqrt(21). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 8*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 8/e - 1. (End)
a(n) = A069099(n) + 7*A000217(n-1). - Leo Tavares, Jul 09 2021
E.g.f.: exp(x)*(1 + 7*x^2) - 1. - Stefano Spezia, Aug 01 2024

A262221 a(n) = 25n(n + 1)/2 + 1.

Original entry on oeis.org

1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651

Offset: 0

Bruno Berselli, Sep 15 2015

Also centered 25-gonal (or icosipentagonal) numbers.
This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
For k=2*n, the formula shown above gives A011379.
Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.
Cf. centered polygonal numbers listed in A069190.
Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.

Magma
```
[25*n*(n+1)/2+1: n in [0..50]];
```

Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
25*Accumulate[Range[0,50]]+1 (* or *) LinearRecurrence[{3,-3,1},{1,26,76},50] (* Harvey P. Dale, Jan 29 2023 *)

PARI
```
vector(50, n, n--; 25*n*(n+1)/2+1)
```
Sage
```
[25*n*(n+1)/2+1 for n in (0..50)]
```

G.f.: (1 + 23*x + x^2)/(1 - x)^3.
a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A123296(n) + 1.
a(n) = A000217(5*n+2) - 2.
a(n) = A034856(5*n+1).
a(n) = A186349(10*n+1).
a(n) = A054254(5*n+2) with n>0, a(0)=1.
a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.
Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=0} a(n)/n! = 77*e/2.
Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A194715 15 times triangular numbers.

Original entry on oeis.org

0, 15, 45, 90, 150, 225, 315, 420, 540, 675, 825, 990, 1170, 1365, 1575, 1800, 2040, 2295, 2565, 2850, 3150, 3465, 3795, 4140, 4500, 4875, 5265, 5670, 6090, 6525, 6975, 7440, 7920, 8415, 8925, 9450, 9990, 10545, 11115, 11700, 12300, 12915, 13545, 14190, 14850, 15525

Offset: 0

Omar E. Pol, Oct 03 2011

Sequence found by reading the line from 0, in the direction 0, 15, ... and the same line from 0, in the direction 0, 45, ..., in the square spiral whose vertices are the generalized 17-gonal numbers.
Sum of the numbers from 7*n to 8*n. - Wesley Ivan Hurt, Dec 23 2015
Also the number of 4-cycles in the (n+6)-triangular honeycomb obtuse knight graph. - Eric W. Weisstein, Jul 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A028895, A035008, A045943, A069128, A163756.

Cf. A001105 (3-cycles in the triangular honeycomb obtuse knight graph), A290391 (5-cycles), A290392 (6-cycles). - Eric W. Weisstein, Jul 29 2017

[15*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, Oct 04 2011

A194715:=n->15*n*(n+1)/2: seq(A194715(n), n=0..60); # Wesley Ivan Hurt, Dec 23 2015

15*Accumulate[Range[0, 60]] (* Harvey P. Dale, Feb 12 2012 *)
Table[15 n (n + 1)/2, {n, 0, 60}] (* Wesley Ivan Hurt, Dec 23 2015 *)
15 Binomial[Range[20], 2] (* Eric W. Weisstein, Jul 28 2017 *)
15 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 28 2017 *)

a(n)=15*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 15*n*(n+1)/2 = 15*A000217(n) = 5*A045943(n) = 3*A028895(n) = A069128(n+1) - 1.
From Wesley Ivan Hurt, Dec 23 2015: (Start)
G.f.: 15*x/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = Sum_{i=7*n..8*n} i. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Sum_{n>=1} 1/a(n) = 2/15.
Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/15.
Product_{n>=1} (1 - 1/a(n)) = -(15/(2*Pi))*cos(sqrt(23/15)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (15/(2*Pi))*cos(sqrt(7/15)*Pi/2). (End)
E.g.f.: 15*exp(x)*x*(2 + x)/2. - Elmo R. Oliveira, Dec 25 2024

A281944 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with n colors such that exactly two balls are of a color seen previously in the sequence.

Original entry on oeis.org

1, 2, 14, 3, 42, 150, 4, 84, 600, 1560, 5, 140, 1500, 7800, 16800, 6, 210, 3000, 23400, 100800, 191520, 7, 294, 5250, 54600, 352800, 1340640, 2328480, 8, 392, 8400, 109200, 940800, 5362560, 18627840, 30240000, 9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400, 10, 630, 18000, 327600, 4233600, 40219200, 279417600, 1360800000, 4191264000, 6187104000

Offset: 1

Jeremy Dover, Feb 02 2017

			n=1 => AAA -> T(1,3)=1
n=2 => AAA,BBB -> T(2,3)=2
   AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB,AABB,ABAB,ABBA,BAAB,BABA,BBAA -> T(2,4)=14
Triangle starts:
   1
   2,  14
   3,  42,   150
   4,  84,   600,   1560
   5, 140,  1500,   7800,   16800
   6, 210,  3000,  23400,  100800,   191520
   7, 294,  5250,  54600,  352800,  1340640,  2328480
   8, 392,  8400, 109200,  940800,  5362560, 18627840,  30240000
   9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400

Jeremy Dover, Table of n, a(n) for n = 1..1035
Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

Columns of table: T(n,3) = A000027(n), T(n,4) = A163756(n).

Other sequences in table: T(n,n+2) = A037960(n).

Table[(Binomial[k, 3] + 3 Binomial[k, 4]) n!/(n + 2 - k)!, {n, 12}, {k, 3, n + 2}] // Flatten (* Michael De Vlieger, Feb 05 2017 *)

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Feb 04 2017

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!.

T(n, k) = n*T(n-1,k-1) + (k-2)*A281881(n,k-1).

cp's OEIS Frontend

A069131 Centered 18-gonal numbers.

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A069127 Centered 14-gonal numbers.

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A262221 a(n) = 25*n*(n + 1)/2 + 1.

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A194715 15 times triangular numbers.

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A281944 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with n colors such that exactly two balls are of a color seen previously in the sequence.

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A262221 a(n) = 25n(n + 1)/2 + 1.