A117609 -id:A117609 - cp's OEIS frontend

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

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).

A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pin^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).

Original entry on oeis.org

1, 7, 19, 57, 81, 251, 437, 691, 739, 1743, 3695, 6619, 8217, 9771, 14771, 15155, 16831, 18805, 26745, 30551, 41755, 46297, 54339, 72359, 86407, 96969, 131059, 344859, 395231, 519963, 607141, 677397, 741509, 893019, 917217, 1288415, 1406811, 1789599, 1827927, 3085785, 3216051, 3444439, 3524869

Offset: 0

N. J. A. Sloane

nonn

The initial value a(0) = 1 corresponds to an initial A000092(0) = 0 which is the index of a record in the sense that the value P(0) = 0 is larger than all preceding values, because there are none. - M. F. Hasler, May 04 2022

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).

W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.

Cf. A000323, A000036, A000092, A000099, A000223.

Cf. A117609 (A(n) in name).

P[n_] := (s = Sum[SquaresR[3, k], {k, 0, n}]) - Round[(4/3)*Pi*n^(3/2)]; record = 0; A000092 = Reap[For[n = 0, n <= 10^4, n++, If[(p = Abs[P[n]]) > record, record = p; Print[s]; Sow[s]]]][[2, 1]] (* Jean-François Alcover, Feb 08 2016, after M. F. Hasler in A000092 *)

a(n) = A117609(A000092(n)), considering A000092(0) = 0. - M. F. Hasler, May 04 2022

Revised Jun 28 2005

a(37)-a(42) from Vincenzo Librandi, Aug 21 2016

A341398 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

Original entry on oeis.org

1, 19, 163, 835, 2869, 7189, 14581, 27253, 49861, 84663, 129303, 190071, 284055, 409335, 550455, 732855, 995241, 1312617, 1656153, 2077497, 2634777, 3300057, 4003641, 4804281, 5872665, 7129227, 8363307, 9784491, 11635755, 13670475, 15727755, 18066315, 20950491

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A008452.

Index entries for sequences related to sums of squares

Cf. A000122, A001650, A008452, A046895, A055408, A055415, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, 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, 9)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 10 2021

nmax = 32; CoefficientList[Series[EllipticTheta[3, 0, x]^9/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[9, n], {n, 0, 32}] // Accumulate

G.f.: theta_3(x)^9 / (1 - x).

a(n^2) = A055415(n).

A341399 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Original entry on oeis.org

1, 21, 201, 1161, 4541, 12965, 29285, 58085, 110105, 198765, 327829, 503509, 765589, 1152509, 1642109, 2243069, 3083569, 4221529, 5551949, 7115789, 9166133, 11777333, 14763893, 18121973, 22316213, 27634481, 33512921, 39812441, 47674841, 57294401, 67510721, 78592961

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A000144.

Index entries for sequences related to sums of squares

Cf. A000122, A000144, A001650, A046895, A055409, A055416, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341398.

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, 10)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..31);  # Alois P. Heinz, Feb 10 2021

nmax = 31; CoefficientList[Series[EllipticTheta[3, 0, x]^10/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[10, n], {n, 0, 31}] // Accumulate

G.f.: theta_3(x)^10 / (1 - x).

a(n^2) = A055416(n).

A373881 Number of lattice points inside the ball x^2 + y^2 + z^2 <= 10^n.

Original entry on oeis.org

7, 147, 4169, 132451, 4187857, 132459677, 4188781437, 132461190717, 4188790061109, 132461176423805, 4188790203273025, 132461176878317635

Offset: 0

Seiichi Manyama, Jun 21 2024

Cf. A068785, A373882, A373883, A373884, A373885, A373896.

Cf. A000605, A117609.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(3, 10^n);

a(n) = A117609(10^n).

Limit_{n->oo} a(n) = (4*Pi/3)*(10^n)^(3/2). - Hugo Pfoertner, Jun 21 2024

a(7)-a(10) from Hugo Pfoertner, Jun 21 2024

a(11) from Chai Wah Wu, Jun 24 2024

A122699 Least integer that can be written as a sum of 3 squares in n nontrivial ways (ignoring order and signs).

Original entry on oeis.org

7, 0, 9, 41, 81, 146, 194, 306, 369, 425, 594, 689, 866, 1109, 1161, 1154, 1361, 1634, 1781, 1889, 2141, 2729, 2609, 3626, 3366, 3566, 3449, 3506, 4241, 4289, 4826, 5066, 5381, 7034, 5561, 6254, 7229, 7829, 8186, 8069, 8126, 8609, 8921, 8774, 10386

Offset: 0

Wouter Meeussen, Oct 21 2006

nonn

Essentially the same as A095809. [From R. J. Mathar, Aug 02 2008]

			a(4)=81 since it can be written as 0^2+0^2+9^2, 1^2+4^2+8^2, 3^2+6^2+6^2, 4^2+4^2+7^2.

Wouter Meeussen, Table of n, a(n) for n = 0..112

Cf. A117609.

Needs["NumberTheory`NumberTheoryFunctions`"]; Flatten[Position[Table[Length@SumOfSquaresRepresentations[3,n], {n,0,65000}],#,1,1]-1]& /@ Range[0,120]

A175366 Partial sums of A175365.

Original entry on oeis.org

1, 7, 19, 27, 27, 27, 27, 27, 33, 57, 81, 81, 81, 81, 81, 81, 93, 117, 117, 117, 117, 117, 117, 117, 125, 125, 125, 131, 155, 179, 179, 179, 179, 179, 179, 203, 251, 251, 251, 251, 251, 251, 251, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 287, 311

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

Number of integer triples (x,y,z) satisfying |x|^3+|y|^3+|z|^3 <= n, -n <= x,y,z <= n. A variant of A117609 with cubes instead of squares.

A175376 Partial sums of A175375.

Original entry on oeis.org

1, 7, 19, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 33, 57, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 93, 117, 117, 117, 117, 117, 117, 117, 117, 117, 117, 117, 117, 117, 117, 117, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

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

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

Cf. A117609, A175366, A175375.

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

G.f.: (1 + 2*Sum_{j>0} x^(j^4))^3/(1-x). - Robert Israel, May 01 2019

A372512 Number of solutions to x^2 + y^2 + z^2 <= n, where x, y, z are positive odd integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 17, 17, 20, 20, 20, 20, 20, 20, 20, 20, 26, 26, 26, 26, 26, 26, 26, 26, 35, 35, 35, 35, 35, 35, 35, 35, 38, 38, 38, 38, 38, 38, 38, 38, 45, 45, 45, 45, 45, 45

Offset: 0

Ilya Gutkovskiy, May 04 2024

nonn

Index entries for sequences related to sums of squares

Cf. A000606, A008437, A085121, A117609, A211639, A349610, A372511.

nmax = 80; CoefficientList[Series[EllipticTheta[2, 0, x^4]^3/(8 (1 - x)), {x, 0, nmax}], x]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)*Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A341398 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A341399 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A373881 Number of lattice points inside the ball x^2 + y^2 + z^2 <= 10^n.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

Extensions

A122699 Least integer that can be written as a sum of 3 squares in n nontrivial ways (ignoring order and signs).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A175366 Partial sums of A175365.

Views

Author

Keywords

A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pin^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).