A144562 -id:A144562 - cp's OEIS frontend

A144640 Row sums from A144562.

3, 17, 48, 102, 185, 303, 462, 668, 927, 1245, 1628, 2082, 2613, 3227, 3930, 4728, 5627, 6633, 7752, 8990, 10353, 11847, 13478, 15252, 17175, 19253, 21492, 23898, 26477, 29235, 32178, 35312, 38643, 42177, 45920, 49878, 54057, 58463, 63102, 67980, 73103

Offset: 1

Vincenzo Librandi, Jan 21 2009, Jun 29 2009

Row 2 of the convolution array A213833. - Clark Kimberling, Jul 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A144562, A213833.

I:=[3, 17, 48, 102]; [n le 4 select I[n] else 4*Self(n-1) -6*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jul 06 2012

A144640:= n-> n*(2*n^2 +5*n -1)/2; seq(A144640(n), n=1..40); # G. C. Greubel, Mar 01 2021

CoefficientList[Series[(3+5*x-2*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)

[n*(2*n^2 +5*n -1)/2 for n in (1..40)] # G. C. Greubel, Mar 01 2021

a(n) = n*(2*n^2 + 5*n - 1)/2. - Jon E. Schoenfield, Jun 24 2010
G.f.: x*(3+5*x-2*x^2)/(1-x)^4. - Vincenzo Librandi, Jul 06 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 06 2012
E.g.f.: x*(6 + 11*x + 2*x^2)*exp(x)/2. - G. C. Greubel, Mar 01 2021

A142463 a(n) = 2n^2 + 2n - 1.

Original entry on oeis.org

-1, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511, 4703, 4899, 5099

Offset: 0

Roger L. Bagula, Sep 19 2008

Essentially the same as A132209.
From Vincenzo Librandi, Nov 25 2010: (Start)
Numbers k such that 2*k + 3 is a square.
First diagonal of A144562. (End)
The terms a(n) give the values for c of indefinite binary quadratic forms [a, b, c] = [2, 4n+2, a(n)] of discriminant D = 12, where a and c can be switched. The positive numbers represented by these forms are given in A084917. - Klaus Purath, Aug 31 2023

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Leo Tavares, Illustration: Hexagonic Diamonds.
Leo Tavares, Illustration: Hexagonic Rectangles.
Leo Tavares, Illustration: Hexagonic Crosses.
Leo Tavares, Illustration: Hexagonic Columns.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A005408, A132209, A144562, A188653.

Cf. A005563, A016945, A001105, A000290, A004767, A046092, A002378, A028387.

Magma
```
[2*n^2+2*n-1: n in [0..100]];
```

A142463:= n-> 2*n^2 +2*n -1; seq(A142463(n), n=0..50); # G. C. Greubel, Mar 01 2021

Array[ -#*(2-#*2)-1&,5!,1] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
Table[2n^2+2n-1,{n,0,50}] (* Harvey P. Dale, Feb 29 2024 *)

a(n)=2*n^2+2*n-1 \\ Charles R Greathouse IV, Sep 24 2015

[2*n^2 +2*n -1 for n in (0..50)] # G. C. Greubel, Mar 01 2021

a(n) = a(n-1) + 4*n.
From Paul Barry, Nov 03 2009: (Start)
G.f.: (1 - 6*x + x^2)/(1-x)^3.
a(n) = 4*C(n+1,2) - 1. (End)
a(n) = -A188653(2*n+1). - Reinhard Zumkeller, Apr 13 2011
a(n) = 3*( Sum_{k=1..n} k^5 )/( Sum_{k=1..n} k^3 ), n > 0. - Gary Detlefs, Oct 18 2011
a(n) = (A005408(n)^2 - 3)/2. - Zhandos Mambetaliyev, Feb 11 2017
E.g.f.: (-1 + 4*x + 2*x^2)*exp(x). - G. C. Greubel, Mar 01 2021
From Leo Tavares, Nov 22 2021: (Start)
a(n) = 2*A005563(n) - A005408(n). See Hexagonic Diamonds illustration.
a(n) = A016945(n-1) + A001105(n-1). See Hexagonic Rectangles illustration.
a(n) = A004767(n-1) + A046092(n-1). See Hexagonic Crosses illustration.
a(n) = A002378(n) + A028387(n-1). See Hexagonic Columns illustration. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Dec 03 2021
Sum_{n>=0} 1/a(n) = tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022

Edited by the Associate Editors of the OEIS, Sep 02 2009

A153238 Numbers k such that 2*k + 3 is composite.

Original entry on oeis.org

3, 6, 9, 11, 12, 15, 16, 18, 21, 23, 24, 26, 27, 30, 31, 33, 36, 37, 39, 41, 42, 44, 45, 46, 48, 51, 54, 56, 57, 58, 59, 60, 61, 63, 65, 66, 69, 70, 71, 72, 75, 76, 78, 79, 81, 83, 84, 86, 87, 90, 91, 92, 93, 96, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 114

Offset: 1

Vincenzo Librandi, Dec 21 2008

One less than the associated value in A047845. - R. J. Mathar, Jan 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Marc Wolf, François Wolf, Representation theorem of composite odd numbers indices, SCIREA Journal of Mathematics (2018) 3.3, 106-117.

Cf. A067076, A047845, A144562.

[ n: n in [0..120] | not IsPrime(2*n+3) ]; // Klaus Brockhaus, Dec 19 2010

lst={};Do[If[ !PrimeQ[2*n+3],AppendTo[lst,n]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Select[Range[120], CompositeQ[2 # + 3] &] (* Michael De Vlieger, Oct 17 2018 *)

for(n=1,115,if(!isprime(2*n+3),print1(n,", "))) \\ Joerg Arndt, Dec 19 2010

a(n) ~ n. - Charles R Greathouse IV, Dec 28 2011

Edited by Ray Chandler, Jan 07 2009

A152811 a(n) = 2(n^2 + 2n - 2).

Original entry on oeis.org

2, 12, 26, 44, 66, 92, 122, 156, 194, 236, 282, 332, 386, 444, 506, 572, 642, 716, 794, 876, 962, 1052, 1146, 1244, 1346, 1452, 1562, 1676, 1794, 1916, 2042, 2172, 2306, 2444, 2586, 2732, 2882, 3036, 3194, 3356, 3522, 3692, 3866, 4044, 4226, 4412, 4602, 4796, 4994

Offset: 1

Vincenzo Librandi, Dec 17 2008

Positive numbers k such that 2*k + 12 is a square. [Comment simplified by Zak Seidov, Jan 14 2009]
Sequence gives positive x values of solutions (x,y) to the Diophantine equation 2*x^3 + 12*x^2 = y^2. Corresponding y values are 8*A154560. There are three other solutions: (0,0), (-4,8) and (-6,0).
From a(2) onwards, third subdiagonal of triangle defined in A144562.
Also, nonnegative numbers of the form (m+sqrt(-3))^2 + (m-sqrt(-3))^2. - Bruno Berselli, Mar 13 2015
a(n-1) is the maximum Zagreb index over all maximal 2-degenerate graphs with n vertices.  The extremal graphs are 2-stars, so the bound also applies to 2-trees.  (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024

			a(4) = 2*(4^2 + 2*4 - 2) = 44 = 2*22 = 2*A028872(5); 2*44^3 + 12*44^2 = 193600 = 440^2 is a square.
The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs, Vol. 0(1) (2024), Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin., Vol. 89(1) (2024), pp. 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J. Comb. Optim., Vol. 27 (2014), pp. 271-291.
I. Gutman and K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., No. 50 (2004), pp. 83-92.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028872 (n^2-3), A154560 ((n+3)^2*n/2+1), A144562 (triangle T(m,n) = 2m*n+m+n-1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).

Magma
```
[ 2*(n^2+2*n-2) : n in [1..47] ];
```

Table[2*n*(n + 2) - 4, {n, 50}] (* Paolo Xausa, Aug 08 2024 *)

{m=4700; for(n=1, m, if(issquare(2*n^3+12*n^2), print1(n, ",")))}

G.f.: 2*x*(1 + 3*x - 2*x^2)/(1-x)^3. [corrected by Elmo R. Oliveira, Nov 17 2024]
a(n) = 2*A028872(n+1).
a(n) = a(n-1) + 4*n + 2 for n > 1, a(1)=2.
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/3 - cot(sqrt(3)*Pi)*Pi/(4*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = -(2 + sqrt(3)*Pi*cosec(sqrt(3)*Pi))/12. (End)
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: 2*(exp(x)*(x^2 + 3*x - 2) + 2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Edited and extended by Klaus Brockhaus, Jan 12 2009

A102732 Primes of the form 13n+5.

Original entry on oeis.org

5, 31, 83, 109, 239, 317, 421, 499, 577, 733, 811, 863, 941, 967, 1019, 1097, 1123, 1201, 1279, 1409, 1487, 1669, 1721, 1747, 1877, 2111, 2137, 2267, 2293, 2371, 2423, 2579, 2657, 2683, 2917, 2969, 3203, 3229, 3307, 3359, 3463, 3541, 3593, 3671, 3697

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 07 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A144562. - Vincenzo Librandi, Jan 17 2009

[n: n in PrimesUpTo(3800) | IsDivisibleBy(n-5,13)];  // Bruno Berselli, Apr 05 2011

[ a: n in [0..350] | IsPrime(a) where a is 13*n +5 ]; // Vincenzo Librandi, Apr 06 2011

lst={};Do[p=13*n+5;If[PrimeQ[p],AppendTo[lst,p]],{n,0,4*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[13n+5,{n,0,1000}],PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

A139606 a(n) = 15*n + 6.

Original entry on oeis.org

6, 21, 36, 51, 66, 81, 96, 111, 126, 141, 156, 171, 186, 201, 216, 231, 246, 261, 276, 291, 306, 321, 336, 351, 366, 381, 396, 411, 426, 441, 456, 471, 486, 501, 516, 531, 546, 561, 576, 591, 606, 621, 636, 651, 666, 681, 696, 711, 726, 741, 756, 771, 786

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 6th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

6th transversal numbers (or 6-transversal numbers): (A000217(6)-6)*n + 6.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A067076, A144562, A153238.

Cf. A000217, A008597, A016873, A057145, A139600.

[15*n+6 : n in [0..60]]; // Vincenzo Librandi, Oct 06 2011

Range[6, 1000, 15] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
15*Range[0,60]+6 (* or *) LinearRecurrence[{2,-1},{6,21},60] (* Harvey P. Dale, Aug 09 2019 *)

a(n)=15*n+6 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,6).
G.f.: 3*(2+3*x)/(x-1)^2 . - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 12 2024: (Start)
E.g.f.: 3*exp(x)*(2 + 5*x).
a(n) = 3*A016873(n) = A008597(n) + 6.
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A111199 Numbers k such that 4k + 9 is prime.

Original entry on oeis.org

1, 2, 5, 7, 8, 11, 13, 16, 20, 22, 23, 25, 26, 32, 35, 37, 41, 43, 46, 47, 55, 56, 58, 62, 65, 67, 68, 71, 76, 77, 82, 85, 86, 91, 95, 97, 98, 100, 103, 106, 110, 112, 113, 125, 128, 133, 137, 140, 142, 146, 148, 151, 152, 158, 161, 163, 166, 167, 173, 175, 181, 187

Offset: 1

Parthasarathy Nambi, Oct 24 2005

nonn

			For k=98, 4*k + 9 = 401 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A005098, A095278, A089986, A153238, A144562.

is(n)=isprime(4*n+9) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A005098(n+1) - 2. - R. J. Mathar, Sep 23 2009

More terms from R. J. Mathar, Sep 23 2009

A141851 Primes congruent to 4 mod 11.

Original entry on oeis.org

37, 59, 103, 191, 257, 367, 389, 433, 499, 521, 587, 631, 653, 719, 829, 983, 1049, 1093, 1181, 1291, 1423, 1489, 1511, 1621, 1709, 1753, 1907, 1951, 1973, 2017, 2039, 2083, 2237, 2281, 2347, 2633, 2677, 2699, 2897, 2963, 3271, 3359, 3469, 3491, 3557, 3623

Offset: 1

N. J. A. Sloane, Jul 11 2008

Primes congruent to 15 mod 22. - Chai Wah Wu, Apr 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A144562, A141849, A090187, A141850.

[ p: p in PrimesUpTo(5000) | p mod 11 eq 4 ]; // Vincenzo Librandi, Apr 19 2011

Select[Range[4,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==4 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A138629 Primes of form 17*n+7.

Original entry on oeis.org

7, 41, 109, 211, 313, 347, 449, 619, 653, 823, 857, 1061, 1129, 1163, 1231, 1367, 1571, 1741, 1877, 1979, 2081, 2251, 2557, 2591, 2659, 2693, 2897, 2999, 3067, 3169, 3203, 3271, 3373, 3407, 3917, 4019, 4597, 4733, 4801, 4903, 4937, 5039, 5107, 5209, 5413

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 14 2008

			17*0+7=7, 17*2+7=41, 17*6+7=109, 17*18+7=211, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A144562.

[a: n in [0..450] | IsPrime(a) where a is 17*n+7 ]; // Vincenzo Librandi, Sep 01 2016

a={};Do[x=17*n+7;If[PrimeQ[x],AppendTo[a,x]],{n,10^2}];a
Select[17Range[0, 500] + 7, PrimeQ] (* Vincenzo Librandi, Sep 01 2016 *)

More terms from N. J. A. Sloane, Jul 11 2008

A153642 a(n) = 4n^2 + 24n + 8.

Original entry on oeis.org

36, 72, 116, 168, 228, 296, 372, 456, 548, 648, 756, 872, 996, 1128, 1268, 1416, 1572, 1736, 1908, 2088, 2276, 2472, 2676, 2888, 3108, 3336, 3572, 3816, 4068, 4328, 4596, 4872, 5156, 5448, 5748, 6056, 6372, 6696, 7028, 7368, 7716, 8072, 8436, 8808, 9188

Offset: 1

Vincenzo Librandi, Dec 30 2008

2*(fifth subdiagonal of triangle A144562).
Sequence gives values x of solutions (x, y) to the Diophantine equation x^3+28*x^2 = y^2. For a more comprehensive list of solutions see A155135.
For n >= 3, a(n - 1) is the number of checkmate positions with white queen and white king against black king on an n X n board. Reason: The black king can only be on the edge. There are 4*(4*n + 1) checkmate positions where the black king is in the corner, 4*(2*n + 4) checkmate positions where the black king is immediately adjacent to the corner square, and there are 4*(n - 4)*(n + 2) checkmate positions where the black king is on another edge square. That's a total of 4*n^2 + 16*n - 12 = a(n - 1) checkmate positions. - Felix Huber, Oct 29 2023

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028881, A144562, A155135, A155136,

Magma
```
[ 4*(n+3)^2-28: n in [1..45] ];
```

LinearRecurrence[{3, -3, 1}, {36, 72, 116}, 50] (* Vincenzo Librandi, Feb 25 2012 *)

a(n)=4*n*(n+6)+8 \\ Charles R Greathouse IV, Dec 28 2011

a(n) = A155135(2n+8) = A155136(2n+7).
a(n) = 4*A028881(n+3).
G.f.: 4*(3 - x)*(3 - 2*x)/(1-x)^3.
a(n)= 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: 4*(-2 + (2 + 7*x + x^2)*exp(x)). - G. C. Greubel, Aug 23 2016
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/56 - cot(sqrt(7)*Pi)*Pi/(8*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 31/168 - cosec(sqrt(7)*Pi)*Pi/(8*sqrt(7)). (End)

Edited and extended by Klaus Brockhaus, Jan 21 2009

cp's OEIS Frontend

A144640 Row sums from A144562.

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A142463 a(n) = 2*n^2 + 2*n - 1.

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A153238 Numbers k such that 2*k + 3 is composite.

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A152811 a(n) = 2*(n^2 + 2*n - 2).

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A102732 Primes of the form 13n+5.

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Magma

Magma

Mathematica

A139606 a(n) = 15*n + 6.

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Magma

Mathematica

PARI

Formula

A111199 Numbers k such that 4k + 9 is prime.

A142463 a(n) = 2n^2 + 2n - 1.

A152811 a(n) = 2(n^2 + 2n - 2).

A153642 a(n) = 4n^2 + 24n + 8.