A003154 -id:A003154 - cp's OEIS frontend

A332243 Starhex honeycomb numbers: a(n) = 13 + 60n + 60n^2.

13, 133, 373, 733, 1213, 1813, 2533, 3373, 4333, 5413, 6613, 7933, 9373, 10933, 12613, 14413, 16333, 18373, 20533, 22813, 25213, 27733, 30373, 33133, 36013, 39013, 42133, 45373, 48733, 52213, 55813, 59533, 63373, 67333, 71413, 75613, 79933, 84373

Offset: 0

John Elias, Feb 07 2020

			Example: a(2) = 13 + 60*2 + 60*2^2 = 373.
Illustration of initial terms:
.                               0
.                            0 0 0 0
.                             0 0 0
.                      0     0 0 0 0     0
.                   0 0 0 0 * * 0 * * 0 0 0 0
.                    0 0 0 * * * * * * 0 0 0
.                   0 0 0 0 * * 0 * * 0 0 0 0
.                      0 * * 0 0 0 0 * * 0
.                       * * * 0 0 0 * * *
.                      0 * * 0 0 0 0 * * 0
.                   0 0 0 0 * * 0 * * 0 0 0 0
.                    0 0 0 * * * * * * 0 0 0
.    0              0 0 0 0 * * 0 * * 0 0 0 0
. 0 * * 0              0     0 0 0 0     0
.  * 0 *                      0 0 0
. 0 * * 0                    0 0 0 0
.    0                          0
.
.    13                         133

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.

John Elias, Illustration of Initial Terms
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003154, A003215, A062786.

Subsequence of A082369: cf. formula.

Array[12 (5 #^2 + 5 # + 1) + 1 &, 38, 0] (* Michael De Vlieger, Feb 07 2020 *)
LinearRecurrence[{3,-3,1},{13,133,373},40] (* Harvey P. Dale, Nov 18 2023 *)

A332243(n)=n*(n+1)*60+13 \\ M. F. Hasler, Jun 09 2023

a(n) = 12*(5*n*(n + 1) + 1) + 1.
From Stefano Spezia, Feb 07 2020: (Start)
O.g.f.: (13 + 94*x + 13*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(13 + 120*x + 60*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-2) for n > 2. (End)
a(n) = A082369(A001844(n)). - M. F. Hasler, Jun 09 2023

A083576 Least n-digit prime star number.

Original entry on oeis.org

13, 181, 1093, 10333, 100621, 1040833, 10085473, 100000837, 1000705861, 10003758337, 100012479337, 1000001026513, 10000000794181, 100000078156441, 1000000776555301, 10000014919168441, 100000008716411701

Offset: 2

Jason Earls, Jun 13 2003

Robert Israel, Table of n, a(n) for n = 2..999

Cf. A003154 (star numbers).

f:= proc(n) local i,t,s;
    t:= ceil(1/2 + sqrt(3+6*10^(n-1))/6);
    for i from t do
      s:= 6*i*(i-1)+1;
      if isprime(s) then return s fi
    od
end proc:
map(f, [$2..40]); # Robert Israel, Jun 12 2017

With[{psn=Select[Table[6n(n-1)+1,{n,130*10^6}],PrimeQ]},Table[ SelectFirst[ psn,IntegerLength[#]==k&],{k,2,18}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 12 2020 *)

star(n) = 6*n*(n-1)+1; L(n)=length(Str(n)); { stl(m)=print1(0","); k=1; for(n=2,m, while(!isprime(star(k)) || L(star(k))!=n, k++); print1(star(k)",")) }

More terms from David Wasserman, Nov 23 2004

A083749 Brilliant star numbers: semiprimes of the form 6n^2 - 6n + 1 such that both prime factors have the same number of decimal digits.

Original entry on oeis.org

121, 253, 793, 1261, 1633, 2773, 3901, 11881, 19153, 24961, 49141, 50233, 59401, 90037, 128773, 162361, 186913, 197653, 268393, 294373, 422941, 492493, 1425937, 1515037, 1663213, 1811701, 2092141, 2185273, 2601733, 3020341, 3063061

Offset: 1

Jason Earls, Jun 17 2003

Brilliant numbers are A078972; "star numbers" here refers to centered 12-gonal numbers A003154, but there exist other definitions for this name. - M. F. Hasler, Nov 01 2012

Intersection of A003154 and A078972, or of A218172 and A078972.

Cf. A063773.

Definition clarified by Charles R Greathouse IV, Nov 01 2012

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

Original entry on oeis.org

1, 4, 11, 26, 55, 105, 184, 301, 466, 690, 985, 1364, 1841, 2431, 3150, 4015, 5044, 6256, 7671, 9310, 11195, 13349, 15796, 18561, 21670, 25150, 29029, 33336, 38101, 43355, 49130, 55459, 62376, 69916, 78115, 87010, 96639, 107041, 118256, 130325

Offset: 1

Dan Graybill (clopen(AT)comcast.net), Aug 12 2008

Consider the array of triangular, square and centered polygonal numbers (irregular variant of A086272 and A086273):
 3   6  10  15  21  28  36  45  55  A000217
 4   9  16  25  36  49  64  81 100  A000290
 6  16  31  51  76 106 141 181 226  A005891
 7  19  37  61  91 127 169 217 271  A003215
 8  22  43  71 106 148 197 253 316  A069099
 9  25  49  81 121 169 225 289 361  A016754
10  28  55  91 136 190 253 325 406  A060544
11  31  61 101 151 211 281 361 451  A062786
12  34  67 111 166 232 309 397 496  A069125
13  37  73 121 181 253 337 433 541  A003154
14  40  79 131 196 274 365 469 586  A069126
15  43  85 141 211 295 393 505 631  A069127
etc. The sequence contains the antidiagonal sums of this array. - R. J. Mathar, Jun 05 2011
For comparison, the antidiagonal sums of A086270 are essentially A006522 starting at the 4th term. - R. J. Mathar, Sep 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217.

a(n) = (n-1)*(n^3+11*n^2-38*n+120)/24, n>1. - R. J. Mathar, Sep 12 2008

G.f.: x*(1-x+x^2+x^3-x^5)/(1-x)^5. - Alexander R. Povolotsky, Jun 06 2011

A193515 T(n,k) = number of ways to place any number of 3X1 tiles of k distinguishable colors into an nX1 grid.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 7, 6, 1, 1, 6, 9, 10, 13, 9, 1, 1, 7, 11, 13, 22, 23, 13, 1, 1, 8, 13, 16, 33, 43, 37, 19, 1, 1, 9, 15, 19, 46, 69, 73, 63, 28, 1, 1, 10, 17, 22, 61, 101, 121, 139, 109, 41, 1, 1, 11, 19, 25, 78, 139, 181, 253, 268, 183, 60, 1, 1, 12

Offset: 1

R. H. Hardin, with proof and formula from Robert Israel in the Sequence Fans Mailing List, Jul 29 2011

Table starts:
..1...1...1...1...1....1....1....1....1....1....1....1.....1.....1.....1.....1
..1...1...1...1...1....1....1....1....1....1....1....1.....1.....1.....1.....1
..2...3...4...5...6....7....8....9...10...11...12...13....14....15....16....17
..3...5...7...9..11...13...15...17...19...21...23...25....27....29....31....33
..4...7..10..13..16...19...22...25...28...31...34...37....40....43....46....49
..6..13..22..33..46...61...78...97..118..141..166..193...222...253...286...321
..9..23..43..69.101..139..183..233..289..351..419..493...573...659...751...849
.13..37..73.121.181..253..337..433..541..661..793..937..1093..1261..1441..1633
.19..63.139.253.411..619..883.1209.1603.2071.2619.3253..3979..4803..5731..6769
.28.109.268.529.916.1453.2164.3073.4204.5581.7228.9169.11428.14029.16996.20353

			Some solutions for n=7 k=3; colors=1,2,3 and empty=0
..3....0....0....2....0....1....3....0....0....0....1....0....3....1....0....0
..3....0....0....2....2....1....3....2....1....0....1....3....3....1....0....0
..3....1....0....2....2....1....3....2....1....2....1....3....3....1....0....3
..1....1....3....0....2....0....0....2....1....2....3....3....0....2....0....3
..1....1....3....0....0....2....2....2....1....2....3....2....1....2....1....3
..1....0....3....0....0....2....2....2....1....0....3....2....1....2....1....0
..0....0....0....0....0....2....2....2....1....0....0....2....1....0....1....0

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000930,
Column 2 is A003229(n-1),
Column 3 is A084386,
Column 4 is A089977,
Column 10 is A178205,
Row 6 is A028872(n+2),
Row 7 is A144390(n+1),
Row 8 is A003154(n+1).

T:= proc(n, k) option remember;
      `if`(n<0, 0,
      `if`(n<3 or k=0, 1, k*T(n-3, k) +T(n-1, k)))
    end:
seq(seq(T(n, d+1-n), n=1..d), d=1..13); # Alois P. Heinz, Jul 29 2011

nmax = 13; t[?Negative, ] = 0; t[n_, k_] /; (n < 3 || k == 0) = 1; t[n_, k_] := t[n, k] = k*t[n-3, k] + t[n-1, k]; Flatten[ Table[ t[n-k+1, k], {n , 1, nmax}, {k, n, 1, -1}]](* Jean-François Alcover, Nov 28 2011, after Maple *)

With z X 1 tiles of k colors on an n X 1 grid (with n >= z), either there is a tile (of any of the k colors) on the first spot, followed by any configuration on the remaining (n-z) X 1 grid, or the first spot is vacant, followed by any configuration on the remaining (n-1) X 1. So T(n,k) = T(n-1,k) + k*T(n-z,k), with T(n,k) = 1 for n=0,1,...,z-1. The solution is T(n,k) = sum_r r^(-n-1)/(1 + z k r^(z-1)) where the sum is over the roots of the polynomial k x^z + x - 1.
T(n,k) = sum {s=0..[n/3]} (binomial(n-2*s,s)*k^s).
For z X 1 tiles, T(n,k,z) = sum{s=0..[n/z]} (binomial(n-(z-1)*s,s)*k^s). - R. H. Hardin, Jul 31 2011

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 7, 1, 1, 5, 10, 13, 11, 1, 1, 6, 13, 19, 21, 16, 1, 1, 7, 16, 25, 31, 31, 22, 1, 1, 8, 19, 31, 41, 46, 43, 29, 1, 1, 9, 22, 37, 51, 61, 64, 57, 37, 1, 1, 10, 25, 43, 61, 76, 85, 85, 73, 46, 1, 1, 11, 28, 49, 71, 91, 106, 113, 109, 91

Offset: 0

Kival Ngaokrajang, Jul 07 2014

T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.
It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.
The triangle under the main diagonal is A121722.
Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0, ...). Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - Gary W. Adamson, Oct 01 2015

			Table begins:
       C0  C1  C2  C3  C4  C5
  n/k  0   1   2   3   4   5   ...
R0 0   1   1   1   1   1   1   ...
R1 1   1   2   3   4   5   6   ...
R2 2   1   4   7   10  13  16  ...
R3 3   1   7   13  19  25  31  ...
R4 4   1   11  21  31  41  51  ...
R5 5   1   16  31  46  61  76  ...
R6 6   1   22  43  64  85  106 ...
R7 7   1   29  57  85  113 141 ...
R8 8   1   37  73  109 145 181 ...
R9 9   1   46  91  136 181 226 ...
  ...  ... ... ... ... ... ... ...
C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12  =  A003154.
R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.

Kival Ngaokrajang, Illustration for n = 0..3, k = 1..4

Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.

T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

A358053 a(n) = 14*n - 1.

Original entry on oeis.org

13, 27, 41, 55, 69, 83, 97, 111, 125, 139, 153, 167, 181, 195, 209, 223, 237, 251, 265, 279, 293, 307, 321, 335, 349, 363, 377, 391, 405, 419, 433, 447, 461, 475, 489, 503, 517, 531, 545, 559, 573, 587, 601, 615, 629, 643, 657, 671, 685, 699, 713, 727, 741, 755, 769, 783, 797

Offset: 1

Leo Tavares, Oct 27 2022

This sequence can be illustrated by a star outline with a central row of counters.

Subsequence of primes is A045473. - Bernard Schott, Jan 25 2023

Paolo Xausa, Table of n, a(n) for n = 1..10000
Leo Tavares, Illustration: Mid-line Stars.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008596, A003154, A017053, A045473, A045944, A202804.

14*Range[100] - 1 (* Paolo Xausa, Oct 04 2024 *)

a(n) = 14*n - 1.
a(n) = A003154(n+1) - A202804(n-1).
a(n) = A003154(n+1) - 2*A045944(n-1).
From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: x*(13 + x)/(x - 1)^2.
E.g.f.: exp(x)*(14*x - 1) + 1.
a(n) = A017053(2*n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

Original entry on oeis.org

1, 7, 13, 25, 43, 61, 85, 115, 145, 181, 223, 265, 313, 367, 421, 481, 547, 613, 685, 763, 841, 925, 1015, 1105, 1201, 1303, 1405, 1513, 1627, 1741, 1861, 1987, 2113, 2245, 2383, 2521, 2665, 2815, 2965, 3121, 3283, 3445, 3613, 3787, 3961, 4141, 4327, 4513, 4705, 4903, 5101, 5305, 5515, 5725

Offset: 0

Aaron David Fairbanks, May 17 2023

In contrast, A003154 (the star numbers) counts the hexagonal lattice points within a hexagram centered at a lattice point and with the vertices of the central hexagon at the six lattice points a given number of steps outward from the central point.
Besides the first term, the first differences are given by six times A004396.
A005448 (the centered triangular numbers) counts just the lattice points within one of the two triangles that make up the hexagram.

			Illustration of initial terms:
.
.                                 o     o
.                   o   o          o o o
.         o o        o o          o o o o
.   o    o o o    o o o o o    o o o o o o o
.         o o        o o          o o o o
.                   o   o          o o o
.                                 o     o
.
.   1      7          13             25
.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A003154, A004396, A005448, A007980, A081272.

Table[6*Ceiling[n*(n + 1)/3] + 1, {n, 0, 60}] (* Amiram Eldar, Jul 28 2023 *)

a(n) = 6*ceil(n*(n+1)/3) + 1; \\ Michel Marcus, Jun 14 2024

a(n) = 6*ceiling(n*(n+1)/3) + 1.
a(n) = 6*A007980(n-1) + 1 for n >= 1.
a(n+1) - a(n) = 6*A004396(n+1).
a(3n) = A081272(n).
G.f.: (1 + 5*x + 5*x^3 + x^4)/((1 - x)^3*(1 + x + x^2)). - Stefano Spezia, Feb 06 2025

A374584 Numbers k such that 7*k + 2 is a square.

Original entry on oeis.org

1, 2, 14, 17, 41, 46, 82, 89, 137, 146, 206, 217, 289, 302, 386, 401, 497, 514, 622, 641, 761, 782, 914, 937, 1081, 1106, 1262, 1289, 1457, 1486, 1666, 1697, 1889, 1922, 2126, 2161, 2377, 2414, 2642, 2681, 2921, 2962, 3214, 3257, 3521, 3566, 3842, 3889

Offset: 1

Juri-Stepan Gerasimov, Aug 12 2024

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

The numbers k such that (m + (9-m)*k) is a square: A000217 (m = 1), this sequence (m = 2), A003154 (m = 3), A195162 (m = 4), A028387 (m = 5), A100536 (m = 6), A059993 (m = 7), A028884 (m = 8).

Cf. A047341.

[k: k in [0..4000] | IsSquare(7*k + 2)];

((Table[7*n + {3, 4}, {n, 0, 23}] // Flatten)^2 - 2)/7 (* Amiram Eldar, Aug 12 2024 *)

a(n) = (A047341(n)^2 - 2)/7. - Amiram Eldar, Aug 12 2024

cp's OEIS Frontend

A332243 Starhex honeycomb numbers: a(n) = 13 + 60*n + 60*n^2.

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A083576 Least n-digit prime star number.

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A083749 Brilliant star numbers: semiprimes of the form 6n^2 - 6n + 1 such that both prime factors have the same number of decimal digits.

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A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

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A193515 T(n,k) = number of ways to place any number of 3X1 tiles of k distinguishable colors into an nX1 grid.

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Formula

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

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A358053 a(n) = 14*n - 1.

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A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

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Mathematica

A332243 Starhex honeycomb numbers: a(n) = 13 + 60n + 60n^2.