A000925 -id:A000925 - cp's OEIS frontend

A347801 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^2.

0, 0, 1, 0, 0, 8, 0, 0, 16, 0, 18, 0, 0, 72, 0, 0, 0, 32, 81, 0, 128, 0, 0, 0, 0, 288, 50, 0, 0, 200, 0, 0, 256, 0, 450, 0, 0, 72, 0, 0, 288, 800, 0, 0, 0, 648, 0, 0, 0, 0, 723, 0, 1152, 392, 0, 0, 0, 0, 882, 0, 0, 1800, 0, 0, 0, 1696, 0, 0, 512, 0, 0, 0, 1296, 1152, 2450, 0, 0, 0, 0, 0, 2048, 0, 162, 0, 0, 4176, 0, 0, 0, 3200, 1458

Offset: 0

Seiichi Manyama, Sep 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000404, A000925, A037214, A347802, A347803.

a(n) = sum(i=1, n, sum(j=1, n, (i^2+j^2==n)*(i*j)^2));

my(N=99, x='x+O('x^N)); concat([0, 0], Vec(sum(k=0, sqrtint(N), k^2*x^k^2)^2))

a(n) is sum of i^2 * j^2 for positive integers i,j such that i^2+j^2=n.

A000592 Number of nonnegative solutions of x^2 + y^2 = z in first n shells.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 35, 37, 39, 41, 43, 45, 48, 50, 52, 54, 56, 58, 62, 64, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 86, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 112, 113, 117, 119, 121, 123, 127, 129, 131, 133, 135, 137

Offset: 0

N. J. A. Sloane

Cumulative totals of nonzero values in (or distinct values in cumulative totals of) A000925. - Franklin T. Adams-Watters, Jun 21 2006

Hansraj Gupta, A table of values of N_2(t), Res. Bull. East Panjab Univ. 1952, (1952). no. 20, 13-93.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..2749

Cf. A000925.

nn = 200; t = CoefficientList[Series[Sum[x^k^2, {k, 0, Sqrt[nn]}]^2, {x, 0, nn}], x]; Union[Accumulate[t]] (* Jean-François Alcover, Jul 20 2011, after T. D. Noe *)

N_2(t) = Sum_{j <= t} n_2(j) where n_2(j) is the number of nonnegative solutions (x,y) of x^2 + y^2 = j, the solution (x,y) being considered as different from (y,x) in case x != y.

More terms from Franklin T. Adams-Watters, Jun 21 2006

A362961 a(n) = Sum_{b=0..floor(sqrt(n)), n-b^2 is square} b.

Original entry on oeis.org

1, 1, 0, 2, 3, 0, 0, 2, 3, 4, 0, 0, 5, 0, 0, 4, 5, 3, 0, 6, 0, 0, 0, 0, 12, 6, 0, 0, 7, 0, 0, 4, 0, 8, 0, 6, 7, 0, 0, 8, 9, 0, 0, 0, 9, 0, 0, 0, 7, 13, 0, 10, 9, 0, 0, 0, 0, 10, 0, 0, 11, 0, 0, 8, 20, 0, 0, 10, 0, 0, 0, 6, 11, 12, 0, 0, 0, 0, 0, 12, 9, 10, 0

Offset: 1

Darío Clavijo, May 10 2023

a(n) = 0 if n in A022544.

a(n) > 0 if n in A001481.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A022544, A001481.

Cf. A143574 (sum of b^2), A000925.

a[n_]:=Sum[b Boole[IntegerQ[Sqrt[n-b^2]]],{b,0,Floor[Sqrt[n]]}]; Array[a,83] (* Stefano Spezia, May 15 2023 *)

a(n) = sum(b=0, sqrtint(n), if (issquare(n-b^2), b)); \\ Michel Marcus, May 16 2023

from gmpy2 import *
a = lambda n: sum([b for b in range(0, isqrt(n) + 1) if is_square(n - (b*b))])
print([a(n) for n in range(1, 84)])

from sympy import divisors
from sympy.solvers.diophantine.diophantine import cornacchia
def A362961(n):
    c = 0
    for d in divisors(n):
        if (k:=d**2)>n:
            break
        q, r = divmod(n,k)
        if not r:
            c += sum(d*(a[0]+(a[1] if a[0]!=a[1] else 0)) for a in cornacchia(1,1,q) or [])
    return c # Chai Wah Wu, May 15 2023

A217868 a(n) is the sum of total number of nonnegative integer solutions to each of a^2 + b^2 = n, a^2 + 2b^2 = n, a^2 + 3b^2 = n and a^2 + 7*b^2 = n. (Order matters for the equation a^2+b^2 = n).

Original entry on oeis.org

5, 2, 2, 6, 2, 1, 2, 3, 6, 2, 2, 3, 3, 0, 0, 7, 3, 3, 2, 2, 1, 1, 1, 1, 7, 2, 3, 4, 3, 0, 1, 4, 2, 3, 0, 7, 4, 1, 1, 2, 3, 0, 3, 2, 2, 0, 0, 3, 6, 4, 2, 5, 3, 2, 0, 1, 3, 2, 1, 0, 3, 0, 2, 8, 4, 2, 3, 3, 0, 0, 1, 4, 4, 2, 2, 4, 1, 0, 2, 2, 7, 3, 1, 3, 4, 1, 0, 3, 3, 2, 2, 1, 1, 0, 0, 1, 4, 2, 4, 8

Offset: 1

V. Raman, Oct 13 2012

nonn

Note: For the equation a^2 + b^2 = n, if there are two solutions (a,b) and (b,a), then they will be counted separately.
The sequences A216501 and A216671 give how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to.
1, 2, 3, 7 are the first four numbers with class number 1.
a(n) = A217462(n) when n is not the sum of two positive squares.
But when n is the sum of two positive squares, the ordered pairs for the equation x^2+y^2 = n count.
For example,
193 = 12^2 + 7^2.
193 = 7^2 + 12^2.
193 = 11^2 + 2*6^2.
193 = 1^2 + 3*8^2.
193 = 9^2 + 7*4^2.
So a(193) = 5. On the other hand, for the sequence A217462, the ordered pairs 12^2 + 7^2, 7^2 + 12^2 will be counted only once, so A217462(193) = 4.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Cf. A216501, A216671.
Cf. A217462 (related sequence of this when the order does not matter for the equation a^2 + b^2 = n).
Cf. A216501 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a > 0, b > 0).
Cf. A216671 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a >= 0, b >= 0).
Cf. A000925 (number of solutions to n = a^2+b^2 (when the solutions (a, b) and (b, a) are being counted differently) with a >= 0, b >= 0).
Cf. A216282 (number of solutions to n = a^2+2*b^2 with a >= 0, b >= 0).
Cf. A119395 (number of solutions to n = a^2+3*b^2 with a >= 0, b >= 0).
Cf. A216512 (number of solutions to n = a^2+7*b^2 with a >= 0, b >= 0).

for(n=1, 100, sol=0; for(x=0, 100, if(issquare(n-x*x)&&n-x*x>=0, sol++); if(issquare(n-2*x*x)&&n-2*x*x>=0, sol++); if(issquare(n-3*x*x)&&n-3*x*x>=0, sol++); if(issquare(n-7*x*x)&&n-7*x*x>=0, sol++)); printf(sol", "))

A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Number of ways to write n as an ordered sum of 2 squares > 1.

			G.f. = x^8 + 2*x^13 + x^18 + 2*x^20 + 2*x^25 + 2*x^29 + x^32 + 2*x^34 + 2*x^40 + ...
a(13) = 2 because we have [9, 4] and [4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A000925, A004018, A006456, A063725, A078134, A085989, A280542.

nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^2, {x, 0, nmax}], x]
CoefficientList[Series[(1 + 2 x - EllipticTheta[3, 0, x])^2/4, {x, 0, 105}], x]

G.f.: (Sum_{k>=2} x^(k^2))^2.

G.f.: (1/4)*(1 + 2*x - theta_3(0,x))^2, where theta_3 is the 3rd Jacobi theta function.

A305260 A linear mapping a(n) = x + y*n of pairs of nonnegative integers (x,y), where the pairs are enumerated first by radial coordinate r and in case of ties, by polar angle 0 <= phi <= Pi/2 in a polar coordinate system.

Original entry on oeis.org

0, 1, 2, 4, 2, 10, 8, 15, 18, 3, 30, 14, 37, 29, 44, 4, 64, 21, 73, 60, 44, 86, 5, 73, 99, 125, 31, 136, 61, 147, 124, 98, 163, 6, 204, 41, 217, 80, 230, 161, 204, 129, 255, 7, 308, 52, 235, 330, 198, 298, 107, 359, 163, 374, 276, 335, 8, 456, 66, 243, 424, 489, 132, 506, 390, 203, 531

Offset: 0

Hugo Pfoertner, Jun 15 2018

nonn

Secondary sorting by polar angle is equivalent to secondary sorting by y.

The sequence is an alternative solution to the riddle described in the comments of A304584.

			   y:
     |
   8 |  57  61  63  66  70
     |
   7 |  44  47  51  53  60  68
     |
   6 |  34  36  38  42  49  55  64
     |
   5 |  25  27  29  32  40  46  54  67
     |
   4 |  16  18  21  24  30  39  48  59  69
     |
   3 |  10  12  14  19  23  31  41  52  65
     |
   2 |   5   7   8  13  20  28  37  50  62
     |
   1 |   2   3   6  11  17  26  35  45  58
     |
   0 |   0   1   4   9  15  22  33  43  56  71
       _______________________________________
  x:     0   1   2   3   4   5   6   7   8   9
.
a(5) = x(5) + 5*y(5) = 0 + 5*2 = 10,
a(14) = x(14) + 14*y(14) = 2 + 14*3 = 44,
a(20) = x(20) + 20*y(20) = 4 + 20*2 = 44.

Cf. A000925, A283305, A283306, A304584, A304585.

n=-1;for(r2=0,81,for(y=0,round(sqrt(r2)),x2=r2-y^2;if(issquare(x2),print1(round(sqrt(x2))+y*(n++),", "))))

A363774 Expansion of 1/(Sum_{k>=0} x^(k^2))^2.

Original entry on oeis.org

1, -2, 3, -4, 3, 0, -5, 12, -18, 18, -9, -12, 44, -76, 93, -76, 5, 120, -273, 400, -414, 228, 200, -828, 1480, -1842, 1539, -268, -2004, 4824, -7168, 7568, -4518, -2784, 13577, -24900, 31563, -27236, 6816, 30308, -77010, 116844, -126018, 80180, 34140, -205932, 389275

Offset: 0

Seiichi Manyama, Jun 21 2023

Convolution inverse of A000925.
Column k=2 of A363778.
Cf. A162552, A274621.

my(N=50, x='x+O('x^N)); Vec(1/sum(k=0, sqrtint(N), x^k^2)^2)

a(0) = 1; a(n) = -(2/n) * Sum_{k=1..n} A162552(k) * a(n-k).

A214900 Number of ordered ways to represent n as the sum of three squares and one fourth power.

Original entry on oeis.org

1, 4, 6, 4, 4, 9, 9, 3, 3, 9, 12, 9, 4, 7, 12, 6, 4, 15, 18, 10, 12, 18, 12, 3, 6, 18, 27, 19, 5, 18, 24, 6, 6, 18, 21, 18, 18, 18, 18, 9, 9, 30, 33, 13, 6, 27, 24, 6, 4, 16, 33, 27, 18, 24, 33, 12, 12, 27, 18, 18, 12, 24, 30, 12, 4, 30, 45, 21, 18, 33, 30, 6, 12, 21, 33, 34, 10, 27, 30, 6, 9, 40, 39, 24, 25, 33, 39, 18, 9

Offset: 0

Joerg Arndt, Jul 29 2012

nonn

Different orderings of summands are counted, e.g., 1 = 1^2 + 0^2 + 0^4 + 0^4 = 0^2 + 1^2 + 0^4 + 0^4 = 0^2 + 0^2 + 1^4 + 0^4 = 0^2 + 0^2 + 0^4 + 1^4, so a(1)=4.

Conjecture: a(n) != 0, that is, all numbers are sums of three squares and one fourth power.

Cf. A000925 (two squares), A002102 (three squares).

N=10^3;  x='x+O('x^N);
S(e)=sum(j=0, ceil(N^(1/e)), x^(j^e));
v=Vec( S(4)^1 * S(2)^3 )

G.f.: (Sum_{j>=0} x^(j^2))^3 * (Sum_{j>=0} x^(j^4)) (see PARI code).

cp's OEIS Frontend

A347801 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^2.

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A000592 Number of nonnegative solutions of x^2 + y^2 = z in first n shells.

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A362961 a(n) = Sum_{b=0..floor(sqrt(n)), n-b^2 is square} b.

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A217868 a(n) is the sum of total number of nonnegative integer solutions to each of a^2 + b^2 = n, a^2 + 2*b^2 = n, a^2 + 3*b^2 = n and a^2 + 7*b^2 = n. (Order matters for the equation a^2+b^2 = n).

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A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

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A305260 A linear mapping a(n) = x + y*n of pairs of nonnegative integers (x,y), where the pairs are enumerated first by radial coordinate r and in case of ties, by polar angle 0 <= phi <= Pi/2 in a polar coordinate system.

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A363774 Expansion of 1/(Sum_{k>=0} x^(k^2))^2.

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A214900 Number of ordered ways to represent n as the sum of three squares and one fourth power.

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PARI

A217868 a(n) is the sum of total number of nonnegative integer solutions to each of a^2 + b^2 = n, a^2 + 2b^2 = n, a^2 + 3b^2 = n and a^2 + 7*b^2 = n. (Order matters for the equation a^2+b^2 = n).