A069133 -id:A069133 - cp's OEIS frontend

A239325 a(n) = 6n^2 + 8n + 1.

1, 15, 41, 79, 129, 191, 265, 351, 449, 559, 681, 815, 961, 1119, 1289, 1471, 1665, 1871, 2089, 2319, 2561, 2815, 3081, 3359, 3649, 3951, 4265, 4591, 4929, 5279, 5641, 6015, 6401, 6799, 7209, 7631, 8065, 8511, 8969, 9439, 9921, 10415, 10921, 11439, 11969

Offset: 0

Philippe Deléham, Mar 16 2014

Binomial transform of 1, 14, 12, 0, 0, 0 (0 continued).

Sum_{n>=0} 1/a(n) = (Psi(0,(4+sqrt(10))/6) - Psi(0,(4-sqrt(10))/6))/(2*sqrt(10)) = 1.14373625509612753878..., where Psi(n,k) is the n^th derivative of the digamma function. - Bruno Berselli, Mar 16 2014

			a(0) = 1*1 = 1;
a(1) = 1*1 + 14*1 = 15;
a(2) = 1*1 + 14*2 + 12*1 = 41;
a(3) = 1*1 + 14*3 + 12*3 = 79;
a(4) = 1*1 + 14*4 + 12*6 = 129; etc.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A069133, A100536, A139267.

Table[6 n^2 + 8 n + 1, {n, 0, 44}] (* or *)
CoefficientList[Series[(1 + 12 x - x^2)/(1 - x)^3, {x, 0, 44}], x] (* Michael De Vlieger, Oct 04 2016 *)
LinearRecurrence[{3,-3,1},{1,15,41},50] (* Harvey P. Dale, May 11 2019 *)

a(n)=6*n^2+8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (1 + 12*x - x^2)/(1-x)^3.
a(0) = 1, a(1) = 15, a(2) = 41; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = C(n,0) + 14*C(n,1) + 12*C(n,2).
a(n) = (A069133(n+1) + A100536(n+1) - A000290(n))/2.
a(n) = A139267(n+1) - 1. - Yuriy Sibirmovsky, Oct 04 2016

A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 13, 10, 1, 1, 9, 22, 25, 15, 1, 1, 11, 33, 46, 41, 21, 1, 1, 13, 46, 73, 79, 61, 28, 1, 1, 15, 61, 106, 129, 121, 85, 36, 1, 1, 17, 78, 145, 191, 201, 172, 113, 45, 1, 1, 19, 97, 190, 265, 301, 289, 232, 145, 55, 1, 1, 21

Offset: 0

Philippe Deléham, Mar 16 2014

			Square array begins:
n\k : 0......1......2......3......4......5......6......7......8......9
======================================================================
.0||  1......1......1......1......1......1......1......1......1......1
.1||  1......3......5......7......9.....11.....13.....15.....17.....19
.2||  1......6.....13.....22.....33.....46.....61.....78.....97....118
.3||  1.....10.....25.....46.....73....106....145....190....241....298
.4||  1.....15.....41.....79....129....191....265....351....449....559
.5||  1.....21.....61....121....201....301....421....561....721....901
.6||  1.....28.....85....172....289....436....613....820...1057...1324
.7||  1.....36....113....232....393....596....841...1128...1457...1828
.8||  1.....45....145....301....513....781...1105...1485...1921...2413
.9||  1.....55....181....379....649....991...1405...1891...2449...3079
10||  1.....66....221....466....801...1226...1741...2346...3041...3826
11||  1.....78....265....562....969...1486...2113...2850...3697...4654

Cf. Columns: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591, A081593.

Cf. Rows: A000012, A005408, A028872, A100536, A239325, A069133.

T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k).
T(n,k) = 3*T(n,k-1) - 3*T(n,k-2) + T(n,k-3).
T(n,k) = (T(n,k-1) + T(n,k+1))/2 - A161680(n).
T(n,k) = (T(n-1,k) + T(n+1,k) - A000290(n))/2.

A193518 T(n,k) = number of ways to place any number of 6X1 tiles of k distinguishable colors into an nX1 grid.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 1, 5, 7, 7, 5, 1, 1, 1, 1, 1, 6, 9, 10, 9, 6, 1, 1, 1, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 1, 1, 1, 8, 13, 16, 17, 16, 13, 9, 1, 1, 1, 1, 1, 9, 15, 19, 21, 21, 19, 19, 12, 1, 1, 1, 1, 1, 10, 17

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....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....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...18...19
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
..4..7..10..13..16..19..22..25..28...31...34...37...40...43...46...49...52...55
..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73
..6.11..16..21..26..31..36..41..46...51...56...61...66...71...76...81...86...91
..7.13..19..25..31..37..43..49..55...61...67...73...79...85...91...97..103..109
..9.19..31..45..61..79..99.121.145..171..199..229..261..295..331..369..409..451
.12.29..52..81.116.157.204.257.316..381..452..529..612..701..796..897.1004.1117
.16.43..82.133.196.271.358.457.568..691..826..973.1132.1303.1486.1681.1888.2107
.21.61.121.201.301.421.561.721.901.1101.1321.1561.1821.2101.2401.2721.3061.3421

			Some solutions for n=13 k=3; colors=1, 2, 3; empty=0
..0....0....0....0....0....3....0....0....0....0....2....0....0....2....2....1
..3....0....1....2....1....3....0....0....0....2....2....0....0....2....2....1
..3....0....1....2....1....3....0....0....0....2....2....0....0....2....2....1
..3....0....1....2....1....3....0....0....0....2....2....2....0....2....2....1
..3....0....1....2....1....3....0....0....3....2....2....2....0....2....2....1
..3....1....1....2....1....3....0....0....3....2....2....2....0....2....2....1
..3....1....1....2....1....0....3....0....3....2....3....2....0....0....0....2
..1....1....1....0....0....1....3....3....3....3....3....2....2....2....0....2
..1....1....1....0....0....1....3....3....3....3....3....2....2....2....0....2
..1....1....1....0....0....1....3....3....3....3....3....0....2....2....0....2
..1....1....1....0....0....1....3....3....0....3....3....0....2....2....0....2
..1....0....1....0....0....1....3....3....0....3....3....0....2....2....0....2
..1....0....1....0....0....1....0....3....0....3....0....0....2....2....0....0

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

Column 1 is A005708,
Column 2 is A143448(n-5),
Column 3 is A143456(n-5),
Row 12 is A190576(n+1),
Row 15 is A069133(n+1).

T:= proc(n, k) option remember;
      `if`(n<0, 0,
      `if`(n<6 or k=0, 1, k*T(n-6, 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

T[n_, k_] := T[n, k] = If[n<0, 0, If[n < 6 || k == 0, 1, k*T[n-6, k]+T[n-1, k]]]; Table[Table[T[n, d+1-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Mar 04 2014, after Alois P. Heinz *)

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/6]} (binomial(n-5*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

A264845 Centered 20-gonal (or icosagonal) primes.

Original entry on oeis.org

61, 421, 1321, 3061, 4201, 4621, 5521, 7561, 12601, 14821, 15601, 18061, 19801, 28621, 35401, 42901, 44221, 52561, 55501, 74821, 76561, 81901, 87421, 97021, 109201, 111301, 115561, 117721, 131101, 135721, 150061, 189061, 217561, 235621, 251221, 270601, 273901

Offset: 1

Ilya Gutkovskiy, Nov 26 2015

nonn

Primes of the form 10*k^2 - 10*k + 1.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Eric Weisstein's World of Mathematics, Prime Number

Cf. A000040, A069133.

Select[Table[10 n^2 - 10 n + 1, {n, 200}], PrimeQ] (* Vincenzo Librandi, Nov 27 2015 *)

for(n=1, 1e3, if(isprime(k=10*n^2-10*n+1), print1(k, ", "))) \\ Altug Alkan, Nov 26 2015

cp's OEIS Frontend

A239325 a(n) = 6*n^2 + 8*n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

Views

Author

Keywords

Examples

Crossrefs

Formula

A193518 T(n,k) = number of ways to place any number of 6X1 tiles of k distinguishable colors into an nX1 grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A264845 Centered 20-gonal (or icosagonal) primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A239325 a(n) = 6n^2 + 8n + 1.