A339048 -id:A339048 - cp's OEIS frontend

A338432 Triangle read by rows: T(n, k) = (n - k + 1)^2 + 2*k^2, for n >= 1, and k = 1, 2, ..., n.

3, 6, 9, 11, 12, 19, 18, 17, 22, 33, 27, 24, 27, 36, 51, 38, 33, 34, 41, 54, 73, 51, 44, 43, 48, 59, 76, 99, 66, 57, 54, 57, 66, 81, 102, 129, 83, 72, 67, 68, 75, 88, 107, 132, 163, 102, 89, 82, 81, 86, 97, 114, 137, 166, 201

Offset: 1

Wolfdieter Lang, Dec 09 2020

This triangle is obtained from the array A(m, k) = m^2 + 2*k^2, for k and m >= 1, read by upwards antidiagonals. This array A is of interest for representing numbers as a sum of three non-vanishing squares with two squares coinciding.
For the numbers represented this way, see A154777. To find the actual values for m and k (taken positive), given a representable number from A154777, one can also use the number triangle T(n, k) = A(n-k+1, k).
To find the number of representations of value N (from A154777), it is sufficient to consider the rows n >= 1 not exceeding n_{max} = Floor(N, Min), where the sequence Min gives the minima of the numbers in each row: Min = {min(n)}_{n>=1} with min(n) = min(T(n, 1), T(n, 2), ..., T(n, n)) and Floor(N, Min) is the greatest member of Min not exceeding N.
Conjecture: min(n) = T(n, ceiling(n/3)), n >= 1. This is the sequence (n+1)^2 - ceiling(n/3)*(2*(n+1) - 3*ceiling(n/3)) = A071619(n+1) = ceiling((2/3)*(n+1)^2) = (n+1)^2 - floor((1/3)*(n+1)^2) = 3, 6, 11, 17, 24, 33, 43, .... (Proof of these identities by considering the three n (mod 3) cases.)
For the multiplicities of the representable values A154777(n), see A339047.
The author met this representation problem in connection with special triples of integer curvatures in the Descartes-Steiner five circle problem.

			The triangle T(n, k) begins:
n \ k  1   2   3   4   5   6   7   8   9  10  11  12 ...
1:     3
2:     6   9
3:    11  12  19
4:    18  17  22  33
5:    27  24  27  36  51
6:    38  33  34  41  54  73
7:    51  44  43  48  59  76  99
8:    66  57  54  57  66  81 102 129
9:    83  72  67  68  75  88 107 132 163
10:  102  89  82  81  86  97 114 137 166 201
11:  123 108  99  96  99 108 123 144 171 204 243
12:  146 129 118 113 114 121 134 153 178 209 246 289
...
----------------------------------------------------
T(5, 1) = 5^2 + 2*1^2 = 27 = T(5, 3) = 3^2 + 2*3^2. A338433(11) = 2 for A154777(11) = 27.
T(4, 4) = 1^2 + 2*4^2 = 33 = T(6, 2) = 5^2 + 2*2^2. A338433(12) = 2 for A154777(12) = 33.
T(5, 5) = 1^2 + 2*5^2 = 51 = T(7, 1) = 7^2 + 2*1^2. A338433(20) = 2 for A154777(20) = 51.
T(7, 7) = 1^1 - 2*7^2 = 99 = T(11, 3) = 9^2 + 2*3^2 = 99 = T(11, 5) = 7^2 + 2*5^2. A338433(39) = 3 for A154777(39) = 99.
The first multiplicity 4 appears for 297.

Cf. A154777, A339047.
Cf. Columns k = 1..3: A059100, A189833, A241848.
Cf. Diagonals m = 1..4: A058331, A255843, A339048, A255847.

T(n, k) = A(n - k + 1, k), with the array A(m, k) = m^2 + 2*k^2, for n >= 1 and k = 1, 2, ..., n, and 0 otherwise.
G.f. of T and A column k (offset 0): G(k, x) = (1 + x + 2*(1 - x)^2*k^2)/(1-x)^3, for k >= 1.
G.f. of T diagonal m (A row m) (offset 0): D(m, x) = ((2*(1+x) + (1-x)^2*m^2)/(1-x)^3), for m >= 1.
G.f. of row polynomials in x (that is, g.f. of the triangle): G(z,x) = (3 - 3*z + (2 - 6*x + x^2)*z^2 + (2 + x)*x*z^3)*x*z / ((1 - z)*(1 - x*z))^3.

A111040 Numbers n such that 2*n^2 + 9 is prime.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 20, 22, 25, 26, 31, 35, 38, 40, 41, 44, 46, 50, 52, 58, 59, 61, 68, 71, 73, 77, 79, 80, 82, 88, 92, 94, 95, 101, 103, 106, 107, 112, 115, 116, 125, 130, 137, 139, 140, 146, 154, 157, 158, 163, 167, 169, 170, 176, 178, 179, 193, 205

Offset: 1

Parthasarathy Nambi, Oct 05 2005

			If n=26 then (2*n^2) + 9 = 1361 (prime).

Cf. A339048.

a:=proc(n) if isprime(2*n^2+9)=true then n else fi end: seq(a(n),n=0..280); # Emeric Deutsch, Oct 06 2005

Select[Range[205], PrimeQ[2#^2+9] &] (* Stefano Spezia, Aug 07 2024 *)

is(n)=isprime(2*n^2+9) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Emeric Deutsch, Oct 06 2005

A201476 Primes of the form 2*k^2 + 9.

Original entry on oeis.org

11, 17, 41, 59, 107, 137, 251, 347, 401, 521, 587, 809, 977, 1259, 1361, 1931, 2459, 2897, 3209, 3371, 3881, 4241, 5009, 5417, 6737, 6971, 7451, 9257, 10091, 10667, 11867, 12491, 12809, 13457, 15497, 16937, 17681, 18059, 20411, 21227, 22481, 22907, 25097, 26459

Offset: 1

Vincenzo Librandi, Dec 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A111040, A339048.

[a: n in [0..400] | IsPrime(a) where a is 2*n^2+9];

Mathematica
```
Select[Table[2n^2+9,{n,0,700}],PrimeQ]
```

a(n) = A339048(A111040(n)). - Elmo R. Oliveira, Apr 21 2025

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A338432 Triangle read by rows: T(n, k) = (n - k + 1)^2 + 2*k^2, for n >= 1, and k = 1, 2, ..., n.

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A111040 Numbers n such that 2*n^2 + 9 is prime.

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A201476 Primes of the form 2*k^2 + 9.

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