A046895 -id:A046895 - cp's OEIS frontend

A046896 Square elements of A046895.

1, 9, 3969, 5041, 9409, 89401, 35010889, 478953225, 741091729, 282110761881, 370672751241, 1930979822409, 5548799536921, 6445713623281, 81193957971361

Offset: 0

Vinay Vaishampayan and N. J. A. Sloane

No other terms in first 200000 terms of A046895, though sequence is believed to be infinite.

No other terms in first 8000000 terms of A046895. - T. D. Noe, Mar 11 2009

Cf. A046895.

s = 0; Reap[ Do[s = s + SquaresR[4, n]; If[IntegerQ[Sqrt[s]], Print[{n, s}]; Sow[s]], {n, 0, 5*10^6}]][[2, 1]] (* Jean-François Alcover, Jul 02 2012 *)

Extended by T. D. Noe, Mar 11 2009

A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 1, 7, 9, 3, 1, 9, 19, 9, 5, 1, 11, 33, 27, 13, 5, 1, 13, 51, 65, 33, 21, 5, 1, 15, 73, 131, 89, 57, 21, 5, 1, 17, 99, 233, 221, 137, 81, 21, 5, 1, 19, 129, 379, 485, 333, 233, 81, 25, 7, 1, 21, 163, 577, 953, 797, 573, 297, 93, 29, 7, 1, 23, 201, 835, 1713, 1793

Offset: 1

R. J. Mathar, Oct 29 2006, Oct 31 2006

Number of solutions to sum_(i=1,..,d) x[i]^2 <= n, x[i] in Z.

			T(2,2)=9 counts 1 pair (0,0) with sum 0, 4 pairs (-1,0),(1,0),(0,-1),(0,1) with sum 1 and 4 pairs (-1,-1),(-1,1),(1,1),(1,-1) with sum 2.
Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
  1  3   3    3    5     5     5     5      5      7      7
  1  5   9    9   13    21    21    21     25     29     37
  1  7  19   27   33    57    81    81     93    123    147
  1  9  33   65   89   137   233   297    321    425    569
  1 11  51  131  221   333   573   893   1093   1343   1903
  1 13  73  233  485   797  1341  2301   3321   4197   5757
  1 15  99  379  953  1793  3081  5449   8893  12435  16859
  1 17 129  577 1713  3729  6865 12369  21697  33809  47921
  1 19 163  835 2869  7189 14581 27253  49861  84663 129303
  1 21 201 1161 4541 12965 29285 58085 110105 198765 327829

Index entries for sequences related to sums of squares

Rows d=1..10 give A001650, A057655, A117609, A046895, A175360, A175361, A341396, A341397, A341398, A341399.

Cf. A005408 (column 1), A058331 (column 2), A161712 (column 3), A055426 (column 4), A055427 (column 9)

T := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+T(d-1,n-i^2) ; else cnts := cnts+1 ; fi ; fi ; od ; RETURN(cnts) ; end: for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",T(d,n)) ; od ; od;

t[d_, n_] := t[d, n] = t[d, n-1] + SquaresR[d, n]; t[d_, 0] = 1; Table[t[d-n, n], {d, 1, 12}, {n, 0, d-1}] // Flatten (* Jean-François Alcover, Jun 13 2013 *)

Recurrence along rows: T(d,n)=T(d,n-1)+A122141(d,n) for n>=1; T(d,n)=sum_{i=0..n} A122141(d,i). Recurrence along columns: cf. A123937.

A055410 Number of points in Z^4 of norm <= n.

Original entry on oeis.org

1, 9, 89, 425, 1281, 3121, 6577, 11833, 20185, 32633, 49689, 72465, 102353, 140945, 190121, 250553, 323721, 411913, 519025, 643441, 789905, 961721, 1156217, 1380729, 1638241, 1927297, 2257281, 2624417, 3035033, 3490601, 4000425

Offset: 0

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..500 from Andrew Howroyd)

Column k=4 of A302997.

Cf. A046895 (sizes of successive clusters in Z^4 lattice).

int A055410(int i)
{
    const int ring = i*i;
    int result = 0;
    for(int a = -i; a <= i; a++)
        for(int b = -i; b <= i; b++)
            for(int c = -i; c <= i; c++)
                for(int d = -i; d <= i; d++)
                    if ( ring >= a*a + b*b + c*c + d*d ) result++;
    return result;
} /* Oskar Wieland, Apr 08 2013 */

a[n_] := SeriesCoefficient[EllipticTheta[3, 0, x]^4/(1 - x), {x, 0, n^2}];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 23 2019, after Ilya Gutkovskiy *)

N=66;  q='q+O('q^(N^2));
t=Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4/(1-q)); /* A046895 */
vector(sqrtint(#t),n,t[(n-1)^2+1])
/* Joerg Arndt, Apr 08 2013 */

from math import isqrt
def A055410(n): return 1+((-(s:=n**2)*(n+1)+sum((q:=s//k)*((k<<1)+q+1) for k in range(1,n+1))&-1)<<2)+(((t:=isqrt(m:=s>>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 24 2024

a(n) = A046895(n^2). - Joerg Arndt, Apr 08 2013

a(n) = [x^(n^2)] theta_3(x)^4/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018

A175360 Partial sums of A000132.

Original entry on oeis.org

1, 11, 51, 131, 221, 333, 573, 893, 1093, 1343, 1903, 2463, 2863, 3423, 4223, 5183, 5913, 6393, 7633, 9153, 9905, 11025, 12865, 14465, 15665, 16875, 18875, 21115, 22715, 24395, 27115, 30315, 31795, 33235, 36915, 39955, 42205, 45005, 48285

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

The 5th row of A122510.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000132, A055411, A122510.

# uses Python code for A046895
from math import isqrt
def A175360(n): return A046895(n)+(sum(A046895(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

a(n^2) = A055411(n).

G.f.: theta_3(x)^5 / (1 - x). - Ilya Gutkovskiy, Feb 13 2021

A224213 Number of nonnegative solutions to x^2 + y^2 + z^2 + u^2 <= n.

Original entry on oeis.org

1, 5, 11, 15, 20, 32, 44, 48, 54, 70, 88, 100, 108, 124, 148, 160, 165, 189, 219, 235, 253, 281, 305, 317, 329, 357, 399, 427, 439, 475, 523, 539, 545, 581, 623, 659, 688, 716, 764, 792, 810, 858, 918, 946, 970, 1030, 1078, 1102, 1110, 1154, 1226, 1274, 1304, 1352

Offset: 0

Alex Ratushnyak, Apr 01 2013

nonn

Cf. A014110 (first differences).
Cf. A224212 (number of nonnegative solutions to x^2 + y^2 <= n).
Cf. A000606 (number of nonnegative solutions to x^2 + y^2 + z^2 <= n).
Cf. A046895 (number of integer solutions to x^2 + y^2 + z^2 + u^2 <= n).

nn = 50; t = Table[0, {nn}]; Do[d = x^2 + y^2 + z^2 + u^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}, {z, 0, nn}, {u, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)

for n in range(99):
  k = 0
  for x in range(99):
    s = x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y
        if sy>n: break
        for z in range(99):
            sz = sy + z*z
            if sz>n: break
            for u in range(99):
              su = sz + u*u
              if su>n: break
              k+=1
  print(str(k), end=', ')

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^4. - Ilya Gutkovskiy, Mar 14 2017

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)).

A055411 Number of points in Z^5 of norm <= n.

Original entry on oeis.org

1, 11, 221, 1343, 5913, 16875, 42205, 89527, 176377, 313259, 532509, 853399, 1322921, 1961211, 2846933, 4005143, 5554265, 7491355, 9977557, 13065527, 16907817, 21524019, 27179909, 33921671, 42036401, 51452803, 62664773

Offset: 0

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..500 from Andrew Howroyd)

Column k=5 of A302997.

Cf. A122510.

t[d_, n_] := t[d, n] = t[d, n - 1] + SquaresR[d, n]; t[d_, 0] = 1;
a[n_] := t[5, n^2];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 27 2019, after R. J. Mathar *)

# uses Python code for A046895
def A055411(n): return A046895(m:=n**2)+(sum(A046895(m-k**2) for k in range(1,n+1))<<1) # Chai Wah Wu, Jun 23 2024

a(n) = A122510(5,n^2). - R. J. Mathar, Apr 21 2010

a(n) = [x^(n^2)] theta_3(x)^5/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018

A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

Original entry on oeis.org

1, 17, 129, 577, 1713, 3729, 6865, 12369, 21697, 33809, 47921, 69233, 101041, 136209, 174737, 231185, 306049, 384673, 469457, 579217, 722353, 876465, 1025649, 1220337, 1481521, 1733537, 1979713, 2306753, 2697537, 3087777, 3482913, 3959585, 4558737, 5155473

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A000143.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A000122, A000143, A001650, A046895, A055407, A055414, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 8)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[EllipticTheta[3, 0, x]^8/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[8, n], {n, 0, 33}] // Accumulate

from math import prod
from sympy import factorint
def A341397(n): return (sum((prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(m).items()) for m in range(1,n+1)))<<4)+1 # Chai Wah Wu, Jun 21 2024

G.f.: theta_3(x)^8 / (1 - x).

a(n^2) = A055414(n).

A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 9, 27, 89, 333, 1341, 5449, 21697, 84663, 327829, 1275739, 5020457, 19964623, 79883141, 320317827, 1284656385, 5152761033, 20686311261, 83182322509, 335110196569, 1352277390001, 5463873556381, 22097867887045, 89441286136465, 362277846495883, 1468465431530457

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

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

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

Main diagonal of A122510.

Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

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

a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - Vaclav Kotesovec, Apr 14 2018

A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 15, 99, 379, 953, 1793, 3081, 5449, 8893, 12435, 16859, 24419, 33659, 42115, 53203, 69779, 88273, 106081, 125821, 153541, 187981, 217437, 248741, 298469, 351277, 394691, 446939, 515259, 589307, 657683, 728803, 828259, 939223, 1029159, 1124023, 1260103

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A008451.

Index entries for sequences related to sums of squares

Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 10 2021

nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[7, n], {n, 0, 35}] // Accumulate

my(q='q+O('q^(55))); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^7/(1-q)) \\ Joerg Arndt, Jun 21 2024

G.f.: theta_3(x)^7 / (1 - x).

a(n^2) = A055413(n).

cp's OEIS Frontend

A046896 Square elements of A046895.

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A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

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A055410 Number of points in Z^4 of norm <= n.

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A175360 Partial sums of A000132.

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A224213 Number of nonnegative solutions to x^2 + y^2 + z^2 + u^2 <= n.

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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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A055411 Number of points in Z^5 of norm <= n.

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A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

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