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.

Showing 1-6 of 6 results.

A302862 a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 8, 20, 57, 160, 422, 1076, 2780, 7449, 20462, 56348, 153909, 418268, 1139703, 3126068, 8618611, 23801146, 65708424, 181391905, 501296216, 1387834518, 3848187985, 10680579812, 29660831057, 82415406493, 229156296047, 637659848888, 1775648562970, 4947475298595
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 14 2018

Keywords

Comments

a(n) = number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n.

Links

Crossrefs

Cf. A000606, A003059, A010052, A224212, A224213, A287617, A302860, A302863.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n}], {n, 0, 30}]

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A000143.

Links

Crossrefs

Cf. A000122, A000143, A001650, A046895, A055407, A055414, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341398, 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, 8)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 33; CoefficientList[Series[EllipticTheta[3, 0, x]^8/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[8, n], {n, 0, 33}] // Accumulate
  • Python
    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

Formula

G.f.: theta_3(x)^8 / (1 - x).
a(n^2) = A055414(n).

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A008451.

Links

Crossrefs

Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, 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, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[7, n], {n, 0, 35}] // Accumulate
  • PARI
    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

Formula

G.f.: theta_3(x)^7 / (1 - x).
a(n^2) = A055413(n).

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).

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

Views

Author

Ilya Gutkovskiy, Apr 14 2018

Keywords

Comments

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

Links

Crossrefs

Main diagonal of A302997.
Cf. A000122, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416, A122510, A232173, A302860.

Programs

  • Mathematica
    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}]

Formula

a(n) = A122510(n,n^2).
Showing 1-6 of 6 results.

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