cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A373882 Number of lattice points inside or on the 4-dimensional hypersphere x^2 + y^2 + z^2 + u^2 = 10^n.

Original entry on oeis.org

9, 569, 49689, 4937225, 493490641, 49348095737, 4934805110729, 493480252693889, 49348022079085897, 4934802199975704129, 493480220066583590433, 49348022005552308828457, 4934802200546833521392241, 493480220054489318828539601, 49348022005446802425711456713, 4934802200544679211736756034457
Offset: 0

Views

Author

Seiichi Manyama, Jun 21 2024

Keywords

Links

Crossrefs

Cf. A068785, A373881, A373883, A373884, A373885.
Cf. A024916, A046895.

Programs

  • PARI
    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(4, 10^n);
  • Python
    from math import isqrt
def A373882(n): return 1+((-(s:=isqrt(a:=10**n))**2*(s+1)+sum((q:=a//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=a>>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

Formula

a(n) = A046895(10^n).
a(n) == 1 (mod 8).
Limit_{n->oo} a(n) = Pi^2*100^n/2. - Hugo Pfoertner, Jun 21 2024

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A008452.

Links

Crossrefs

Cf. A000122, A001650, A008452, A046895, A055408, A055415, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341399.

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A000144.

Links

Crossrefs

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

Programs

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

Formula

G.f.: theta_3(x)^10 / (1 - x).
a(n^2) = A055416(n).

A175369 Partial sums of A175368.

Original entry on oeis.org

1, 9, 33, 65, 81, 81, 81, 81, 89, 137, 233, 297, 297, 297, 297, 297, 321, 417, 513, 513, 513, 513, 513, 513, 545, 609, 609, 617, 665, 761, 825, 825, 841, 841, 841, 889, 1081, 1273, 1273, 1273, 1273, 1273, 1273, 1369, 1561, 1561, 1561, 1561, 1561, 1561, 1561
Offset: 0

Views

Author

R. J. Mathar, Apr 24 2010

Keywords

Comments

Number of integer 4-tuples (x,y,z,u) satisfying |x|^3+|y|^3+|z|^3+|u|^3 <= n, -n <= x,y,z,u <= n. A variant of A046895 with cubes instead of squares.

A350740 Number of integer points (x, y, z, w) at distance <= 1/2 from 3-sphere of radius n.

Original entry on oeis.org

1, 32, 200, 528, 1280, 2744, 4272, 6592, 10144, 15048, 19824, 25824, 34744, 43520, 55184, 64680, 80864, 99184, 115616, 135144, 157344, 185872, 207304, 239600, 272960, 310240, 351096, 385392, 433040, 485528, 531728, 583696, 646056, 714800, 779488, 842928
Offset: 0

Views

Author

Jeongseop Lee, Jan 12 2022

Keywords

Links

Crossrefs

A 4-dimensional version of A016728.
Cf. A046895.

Programs

  • Maple
    N:= 40: # for a(0)..a(N)
V:= Array(0..N):
for x from 0 to N do
  for y from x to N do
    for z from y to N do
      for w from z to N do
        S:= {x,y,z,w};
        L:= [x,y,z,w];
        m:= round(sqrt(x^2 + y^2 + z^2 + w^2));
        if m > N then next fi;
        f:= 4!/mul(numboccur(s,L)!, s = S) * 2^(4 - numboccur(0,[x,y,z,w]));
        V[m]:= V[m] + f;
od od od od;
convert(V,list); # Robert Israel, Mar 08 2024
  • Python
    from itertools import product
for R in range(100):
    c = 0
    for s in product(range(2*R + 1), repeat = 4):
        if (2*R - 1)**2 <= 4*sum((i - R)**2 for i in s) <= (2*R + 1)**2: c += 1
    print(c if R != 0 else 1, end = ', ')
  • Python
    from itertools import combinations_with_replacement
from math import prod
from collections import Counter
def A350740(n):
    if n == 0: return 1
    x, y = (2*n-1)**2, (2*n+1)**2
    return sum(24//prod((1,1,2,6,24)[d] for d in q.values())<<4-q[0] for q in map(Counter,combinations_with_replacement(range(n+1),4)) if x <= sum(b*a**2 for a, b in q.items())<<2 <= y) # Chai Wah Wu, Jun 20 2024
  • Python
    # Uses Python code in A046895
def A350740(n): return A046895(n*(n+1))-A046895(n*(n-1)) if n else 1 # Chai Wah Wu, Jun 21 2024

Formula

a(n) = A046895(n^2+n)-A046895(n^2-n) for n > 0. - Chai Wah Wu, Jun 21 2024

A372613 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n, where x, y, z, w are positive odd integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 5, 5, 5, 5, 11, 11, 11, 11, 11, 11, 11, 11, 19, 19, 19, 19, 19, 19, 19, 19, 32, 32, 32, 32, 32, 32, 32, 32, 44, 44, 44, 44, 44, 44, 44, 44, 58, 58, 58, 58, 58, 58, 58, 58, 82, 82, 82, 82, 82, 82, 82, 82, 100, 100, 100, 100, 100, 100, 100, 100
Offset: 0

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Links

Crossrefs

Cf. A008438, A046895, A224213, A349611, A372511, A372512.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[EllipticTheta[2, 0, x^4]^4/(16 (1 - x)), {x, 0, nmax}], x]
Previous Showing 11-16 of 16 results.

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