A339047 -id:A339047 - cp's OEIS frontend

A154777 Numbers of the form x^2 + 2*y^2 with positive integers x and y.

3, 6, 9, 11, 12, 17, 18, 19, 22, 24, 27, 33, 34, 36, 38, 41, 43, 44, 48, 51, 54, 57, 59, 66, 67, 68, 72, 73, 75, 76, 81, 82, 83, 86, 88, 89, 96, 97, 99, 102, 107, 108, 113, 114, 118, 121, 123, 129, 131, 132, 134, 136, 137, 139, 144, 146, 147, 150, 152, 153, 162, 163

Offset: 1

M. F. Hasler, Jan 24 2009

Subsequence of A002479 (which allows for x=0 and/or y=0). See there for further references. See A155560 cf for intersection of sequences of type (x^2 + k*y^2).
Also, subsequence of A000408 (with 2*y^2 = y^2 + z^2).
If m and n are terms also n*m is (in particular any power of term is also a term). - Zak Seidov, Nov 30 2011
If m is a term, 2*m is also. - Zak Seidov, Nov 30 2011
Select terms that are multiples of 25: 75, 150, 225, 275, 300, 425, 450, 475, 550, 600, 675, 825, 850, 900, 950, 1025, 1075, 1100, ... Divide them by 25: 3, 6, 9, 11, 12, 17, 18, 19, 22, 24, 27, 33, 34, 36, 38, 41, 43, 44, 48, 51, 54, 57, 59, 66, 67, 68, 72, ... and we get the original sequence. - Zak Seidov, Dec 01 2011
This sequence is closed under multiplication because A002479 is. - Jerzy R Borysowicz, Jun 13 2020

			a(1) = 3 = 1^2 + 2*1^2 is the least number that can be written as A + 2B where A, B are positive squares.
a(2) = 6 = 2^2 + 2*1^2 is the second smallest number that can be written in this way.

Zak Seidov, Table of n, a(n) for n = 1..10000

Subsequence of A002479 and hence of A000408.

Cf. A155560, A338432 (triangle version of array), A339047 (multiplicities).

f[upto_]:=Module[{max=Ceiling[Sqrt[upto-1]]},Select[Union[ First[#]^2+ 2Last[#]^2&/@Tuples[Range[13],{2}]],#<=upto&]]; f[200] (* Harvey P. Dale, Jun 17 2011 *)

isA154777(n,/* use optional 2nd arg to get other analogous sequences */c=2) = { for( b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1,200, isA154777(n) & print1(n","))

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A154777 Numbers of the form x^2 + 2*y^2 with positive integers x and y.

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