A055410 -id:A055410 - cp's OEIS frontend

A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 33, 29, 9, 1, 1, 11, 89, 123, 49, 11, 1, 1, 13, 221, 425, 257, 81, 13, 1, 1, 15, 485, 1343, 1281, 515, 113, 15, 1, 1, 17, 953, 4197, 5913, 3121, 925, 149, 17, 1, 1, 19, 1713, 12435, 23793, 16875, 6577, 1419, 197, 19, 1, 1, 21, 2869, 33809, 88273, 84769, 42205, 11833, 2109, 253, 21, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n.

			Square array begins:
  1,   1,   1,    1,     1,      1,  ...
  1,   3,   5,    7,     9,     11,  ...
  1,   5,  13,   33,    89,    221,  ...
  1,   7,  29,  123,   425,   1343,  ...
  1,   9,  49,  257,  1281,   5913,  ...
  1,  11,  81,  515,  3121,  16875,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..10 give A000012, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.
Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434.
Main diagonal gives A302861.
Cf. A000122, A122510, A302996, A302998.

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k/(1 - x), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

T(n,k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j=-infinity..infinity} x^(j^2))^k.

A046895 Sizes of successive clusters in Z^4 lattice.

Original entry on oeis.org

1, 9, 33, 65, 89, 137, 233, 297, 321, 425, 569, 665, 761, 873, 1065, 1257, 1281, 1425, 1737, 1897, 2041, 2297, 2585, 2777, 2873, 3121, 3457, 3777, 3969, 4209, 4785, 5041, 5065, 5449, 5881, 6265, 6577, 6881, 7361, 7809, 7953, 8289, 9057

Offset: 0

N. J. A. Sloane

Number of lattice points inside or on the 4-sphere x^2 + y^2 + z^2 + u^2 = n. - T. D. Noe, Mar 14 2009

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms up to 1000 from Noe)
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.

Partial sums of A000118.

Cf. A117609

Accumulate[ Table[ SquaresR[4, n], {n, 0, 42}]] (* Jean-François Alcover, May 11 2012 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]^2*QP[q^4]^2))^4/(1-q) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Joerg Arndt *)

q='q+O('q^66);
Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4/(1-q))
/* Joerg Arndt, Apr 08 2013 */

from math import isqrt
def A046895(n): return 1+((-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=n>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # Chai Wah Wu, Jun 21 2024

a(n) = A122510(4,n). a(n^2) = A055410(n). - R. J. Mathar, Apr 21 2010
G.f.: T3(q)^4/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - Joerg Arndt, Apr 08 2013
Pi^2/2 * (sqrt(n)-1)^4 < a(n) < Pi^2/2 * (sqrt(n)+1)^4 for n > 0. - Charles R Greathouse IV, Feb 17 2015
a(n) = Pi^2/2 * n^2 + O(n (log n)^(2/3)) using a result of Walfisz. - Charles R Greathouse IV, Feb 18 2015
a(n) = 1 + 8*A024916(n) - 32*A024916(floor(n/4)) by Jacobi's four-square theorem. - Peter J. Taylor, Jun 03 2020

A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

Original entry on oeis.org

1, 5, 32, 94, 219, 437, 804, 1362, 2177, 3271, 4768, 6708, 9227, 12381, 16254, 20954, 26707, 33461, 41480, 50884, 61703, 74183, 88606, 104862, 123481, 144241, 167604, 193648, 222799, 254731, 290244, 329512, 372545, 419661, 470822, 526646, 587481, 653505

Offset: 2

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Index entries for sequences related to sums of squares

Cf. A000122, A001182, A046895, A055403, A055410, A063730, A224213, A253663, A302995, A341424, A341425, A341426, A341427, A341428, A341429.

b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 4):
seq(a(n), n=2..39);  # Alois P. Heinz, Feb 11 2021

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^4/(16 (1 - x)), {x, 0, n^2}], {n, 2, 39}]

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^4 / (16 * (1 - x)).

A302861 a(n) = [x^(n^2)] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 13, 123, 1281, 16875, 252673, 4031123, 70554353, 1318315075, 26107328109, 549772933959, 12147113355505, 280978137279483, 6780378828922333, 169829490474843659, 4409771551548703649, 118361723203178140163, 3277041835527134201777, 93455465161026267454527

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius n.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Main diagonal of A302997.

Cf. A000122, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416, A122510, A232173, A302860.

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n^2}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n^2}], {n, 0, 19}]

a(n) = A122510(n,n^2).

A349611 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n^2, where x, y, z, w are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 32, 44, 82, 120, 207, 277, 405, 541, 768, 966, 1272, 1592, 2087, 2489, 3103, 3719, 4588, 5348, 6386, 7522, 8891, 10175, 11909, 13623, 15818, 17742, 20278, 22720, 25923, 28917, 32361, 36031, 40368, 44488, 49400, 54358, 60377, 65835, 72341

Offset: 0

Ilya Gutkovskiy, Nov 23 2021

nonn

			a(4) = 5 since there are solutions (1,1,1,1), (3,1,1,1), (1,3,1,1), (1,1,3,1), (1,1,1,3).

Robert Israel, Table of n, a(n) for n = 0..2500
Index entries for sequences related to sums of squares

Cf. A055403, A055410, A341423, A349609, A349610.

N:= 100: # for a(0) .. a(N)
F:= add(x^(k^2),k = 1 ... N,2):
F:= expand(F^4):
L:= ListTools:-PartialSums([seq](coeff(F,x,n),n=0..N^2)):
L[[seq(n^2+1,n=0..N)]]; # Robert Israel, Dec 21 2023

Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^4/(16 (1 - x)), {x, 0, n^2}], {n, 0, 44}]

a(n) = [x^(n^2)] theta_2(x^4)^4 / (16 * (1 - x)).

A175370 a(n) = A175369(n^2).

Original entry on oeis.org

1, 9, 81, 137, 321, 609, 1081, 1561, 2137, 3193, 4393, 4889, 6617, 8425, 9529, 11905, 14169, 17097, 18953, 21953, 25105, 29177, 32681, 35937, 41105, 46321, 50529, 55217, 62177, 67665, 74849, 80817, 87185, 96449, 104945, 112609, 120881, 130137

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

Number of integer 4-tuples (x,y,z,u) satisfying |x|^3+|y|^3+|z|^3+|u|^3 <= n^2, -n <= x,y,z <= n. If the cubes were replaced by squares, A055410 would result.

A175377 a(n) = A175376(n^2).

Original entry on oeis.org

1, 7, 27, 27, 33, 81, 117, 125, 125, 131, 251, 275, 275, 311, 335, 335, 349, 493, 493, 613, 613, 637, 637, 697, 697, 727, 871, 991, 999, 1023, 1143, 1143, 1191, 1191, 1215, 1215, 1281, 1449, 1569, 1617, 1737, 1785, 1785, 1809, 1889, 1985, 1985, 2033, 2033

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

Number of integer triples (x,y,z) satisfying x^4+y^4+z^4 <= n^2, -n <= x,y,z <= n.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A055410, A175370.

N:= 100: # to get a(1)..a(N)
A:= Array(0..N):
for i from 0 while i^4 <= N^2 do
  if i=0 then ai:= 1 else ai:= 2 fi;
  for j from 0 while i^4 + j^4 <= N^2 do
    if j=0 then aj:= 1 else aj:= 2 fi;
    for k from 0 do
      v:= ceil((i^4 + j^4 + k^4)^(1/2));
      if v > N then break fi;
      if k = 0 then ak:= 1 else ak:= 2 fi;
      A[v..N]:= map(`+`,A[v..N] ,ai*aj*ak);
od od od:
convert(A,list); # Robert Israel, May 01 2019

A380832 Number of points in Z^4 of norm <= n where the sum of the four entries is even.

Original entry on oeis.org

1, 1, 49, 169, 625, 1465, 3337, 5689, 10009, 15937, 24865, 35761, 51265, 69817, 94849, 124009, 161497, 204529, 260137, 320497, 394705, 478705, 577489, 687913, 819313, 960457, 1127785, 1309153, 1517161, 1742497, 2001505, 2273473, 2585905, 2920009, 3297337, 3700153, 4144105, 4618657, 5145865, 5703073

Offset: 0

Steven Lu, Feb 05 2025

nonn

Points in Z^4 with even sum of entries forms the D_4 lattice. That is to say, the sequence is the "ball" pattern on D_4 lattice.

a(n) == 1 (mod 24).

			a(2) = 49, because in the ball with radius 2, there is 1 point (0,0,0,0), 8 points similar to (0,0,0,2), 24 points similar to (0,0,1,1), and 16 points similar to (1,1,1,1).

Steven Lu, Python program

Cf. A055410.

a(n) = sum(x=-n, n, sum(y=-n, n, sum(z=-n, n, sum(t=-n, n, (((x+y+z+t) % 2)==0) && (x^2+y^2+z^2+t^2 <=n^2))))); \\ Michel Marcus, Feb 09 2025

Python
```
# See Steven Lu's link
```

A319617 Number of Integer solutions to w^2 + x^2 + y^2 + z^2 < n^2; number of lattice points inside a 4-sphere of radius n.

Original entry on oeis.org

0, 1, 65, 321, 1257, 2873, 6265, 11377, 20161, 31665, 48945, 71401, 102041, 139481, 188753, 247329, 323697, 409457, 516121, 640393, 789161, 955793, 1153025, 1376305, 1637929, 1921049, 2252889, 2615673, 3033665, 3483633, 3990753, 4547945, 5173145, 5840393, 6589945, 7395921, 8287297, 9238001, 10281977, 11402457, 12633145, 13929377

Offset: 0

Brian J. Harrild, Sep 24 2018

			For n=2 there are 65 lattice points in Z^4 such that w^2+x^2+y^2+x^2 < 4

a(n) = A055410(n) - A267326(n).

for n in range (0,51):
    NumPoints=0
    for w in range (-n,n+1):
        for x in range (-n,n+1):
            for y in range (-n,n+1):
                for z in range (-n,n+1):
                    if w**2+x**2+y**2+z**2

cp's OEIS Frontend

A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

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A046895 Sizes of successive clusters in Z^4 lattice.

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A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

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A302861 a(n) = [x^(n^2)] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

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A349611 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n^2, where x, y, z, w are positive odd integers.

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A175370 a(n) = A175369(n^2).

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A175377 a(n) = A175376(n^2).

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Maple

A380832 Number of points in Z^4 of norm <= n where the sum of the four entries is even.

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PARI