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 1-8 of 8 results.

A000606 Number of nonnegative solutions to x^2 + y^2 + z^2 <= n.

Original entry on oeis.org

1, 4, 7, 8, 11, 17, 20, 20, 23, 29, 35, 38, 39, 45, 51, 51, 54, 63, 69, 72, 78, 84, 87, 87, 90, 99, 111, 115, 115, 127, 133, 133, 136, 142, 151, 157, 163, 169, 178, 178, 184, 199, 205, 208, 211, 223, 229, 229, 230, 239, 254, 260, 266, 278, 290, 290, 296
Offset: 0

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Author

N. J. A. Sloane

Keywords

References

  • H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
  • 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).

Links

Crossrefs

Cf. A000604, A117609.
Cf. A002102 (first differences).

Programs

  • Mathematica
    nn = 50; t = Table[0, {nn}]; Do[d = x^2 + y^2 + z^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}, {z, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)
  • Python
    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
            k += 1
  print(str(k), end=',')
# Alex Ratushnyak, Apr 01 2013

Formula

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^3. - Ilya Gutkovskiy, Mar 14 2017

Extensions

More terms from Sean A. Irvine, Dec 01 2010

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 1, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 2, 0, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 6, 2, 0, 0
Offset: 0

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Author

N. J. A. Sloane

Keywords

Examples

			Rows are
1,0,0,..;
1,1,0,0,1,0..;
1,2,1,0,2,2,..;
1,3,3,1,...

Links

Crossrefs

Rows k=0-10 give: A000007, A010052, A000925, A002102, A014110, A038671, A045848, A045849, A045850, A045851, A045852.
Diagonal gives A287617.
Antidiagonal sums give A302018.

Formula

i-th row is expansion of (1+x+x^4+x^9+...)^i.

Extensions

More terms from Erich Friedman

A321381 Expansion of 1/2 * Product_{i>=0, j>=0, k>=0} (1 + x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 3, 6, 11, 18, 33, 59, 96, 150, 230, 357, 552, 825, 1203, 1749, 2541, 3666, 5208, 7302, 10212, 14250, 19738, 27093, 36936, 50203, 68034, 91665, 122763, 163731, 217740, 288621, 380922, 500652, 655987, 857226, 1116798, 1450138, 1877232, 2423892, 3122094, 4010883, 5139132
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2018

Keywords

Links

Crossrefs

Cf. A002102, A033461, A321380.

Formula

G.f.: Product_{k>0} (1 + x^k)^A002102(k).

A295848 Number of nonnegative solutions to (x,y,z) = 1 and x^2 + y^2 + z^2 = n.

Original entry on oeis.org

0, 3, 3, 1, 0, 6, 3, 0, 0, 3, 6, 3, 0, 6, 6, 0, 0, 9, 3, 3, 0, 6, 3, 0, 0, 6, 12, 3, 0, 12, 6, 0, 0, 6, 9, 6, 0, 6, 9, 0, 0, 15, 6, 3, 0, 6, 6, 0, 0, 6, 12, 6, 0, 12, 9, 0, 0, 6, 6, 9, 0, 12, 12, 0, 0, 18, 12, 3, 0, 12, 6, 0, 0, 9, 18, 6, 0, 12, 6, 0, 0, 9, 9, 9
Offset: 0

Views

Author

Seiichi Manyama, Nov 29 2017

Keywords

Comments

a(n)=0 for n in A047536. - Robert Israel, Nov 30 2017

Examples

			a(1) = 3;
(1,0,0) = 1 and 1^2 + 0^2 + 0^2 = 1.
(0,1,0) = 1 and 0^2 + 1^2 + 0^2 = 1.
(0,0,1) = 1 and 0^2 + 0^2 + 1^2 = 1.
a(2) = 3;
(1,1,0) = 1 and 1^2 + 1^2 + 0^2 = 2.
(1,0,1) = 1 and 1^2 + 0^2 + 1^2 = 2.
(0,1,1) = 1 and 0^2 + 1^2 + 1^2 = 2.
a(3) = 1;
(1,1,1) = 1 and 1^2 + 1^2 + 1^2 = 3.
a(5) = 6;
(2,1,0) = 1 and 2^2 + 1^2 + 0^2 = 5.
(2,0,1) = 1 and 2^2 + 0^2 + 1^2 = 5.
(1,2,0) = 1 and 1^2 + 2^2 + 0^2 = 5.
(1,0,2) = 1 and 1^2 + 0^2 + 2^2 = 5.
(0,2,1) = 1 and 0^2 + 2^2 + 1^2 = 5.
(0,1,2) = 1 and 0^2 + 1^2 + 2^2 = 5.

Links

Crossrefs

Cf. A002102, A047536, A048240, A295819, A295849.

Programs

  • Maple
    N:= 100:
V:= Array(0..N):
for x from 0 to floor(sqrt(N/3)) do
  for y from x to floor(sqrt((N-x^2)/2)) do
    for z from y to floor(sqrt(N-x^2-y^2)) do
      if igcd(x,y,z) = 1 then
        r:= x^2 + y^2 + z^2;
        m:= nops({x,y,z});
        if m=3 then V[r]:= V[r]+6
        elif m=2 then V[r]:= V[r]+3
        else V[r]:= V[r]+1
        fi
      fi
od od od:
convert(V,list); # Robert Israel, Nov 30 2017
  • Mathematica
    f[n_] := Total[ Length@ Permutations@# & /@ Select[ PowersRepresentations[n, 3, 2], GCD[#[[1]], #[[2]], #[[3]]] == 1 &]]; Array[f, 90, 0] (* Robert G. Wilson v, Nov 30 2017 *)

A347802 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 12, 0, 0, 48, 0, 27, 64, 0, 216, 0, 0, 432, 48, 243, 0, 384, 972, 0, 768, 0, 864, 804, 0, 3456, 600, 0, 0, 1968, 3888, 1350, 3072, 0, 5508, 0, 0, 7776, 2400, 6075, 1728, 9600, 1944, 0, 4096, 7776, 21600, 2022, 0, 3456, 17424, 0, 13824, 21552, 0, 19521, 0, 31104, 15984, 0, 0, 21600, 34896, 11907
Offset: 0

Views

Author

Seiichi Manyama, Sep 14 2021

Keywords

Links

Crossrefs

Cf. A000408, A002102, A037215, A347801, A347803.

Programs

  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, (i^2+j^2+k^2==n)*(i*j*k)^2)));
  • PARI
    my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, sqrtint(N), k^2*x^k^2)^3))

Formula

a(n) is sum of i^2 * j^2 * k^2 for positive integers i,j,k such that i^2+j^2+k^2=n.

A281155 Expansion of (Sum_{k>=2} x^(k^2))^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 3, 0, 0, 1, 0, 6, 0, 0, 0, 3, 3, 0, 3, 0, 6, 0, 0, 3, 0, 3, 3, 6, 0, 0, 1, 6, 6, 0, 0, 0, 6, 0, 6, 6, 0, 3, 0, 6, 6, 0, 0, 6, 3, 3, 3, 6, 6, 0, 3, 0, 6, 1, 3, 12, 6, 0, 0, 6, 3, 6, 6, 0, 3, 0, 3, 15, 6, 0, 0, 6, 12, 0, 3, 3, 6, 6, 0, 12, 3, 0, 6, 6
Offset: 0

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Author

Ilya Gutkovskiy, Jan 16 2017

Keywords

Comments

Number of ways to write n as an ordered sum of 3 squares > 1.

Examples

			G.f. = x^12 + 3*x^17 + 3*x^22 + 3*x^24 + x^27 + 6*x^29 + 3*x^33 + 3*x^34 + 3*x^36 + ...
a(17) = 3 because we have [9, 4, 4], [4, 9, 4] and [4, 4, 9].

Links

Crossrefs

Cf. A000290, A005875, A002102, A006456, A063691, A078134, A280542.

Programs

  • Mathematica
    nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^3, {x, 0, nmax}], x]
CoefficientList[Series[(-1 - 2 x + EllipticTheta[3, 0, x])^3/8, {x, 0, 105}], x]

Formula

G.f.: (Sum_{k>=2} x^(k^2))^3.
G.f.: (1/8)*(-1 - 2*x + theta_3(0,x))^3, where theta_3 is the 3rd Jacobi theta function.

A363775 Expansion of 1/(Sum_{k>=0} x^(k^2))^3.

Original entry on oeis.org

1, -3, 6, -10, 12, -9, -2, 24, -54, 80, -84, 42, 66, -234, 420, -536, 450, -39, -740, 1770, -2688, 2898, -1722, -1320, 6078, -11349, 14736, -12992, 3084, 15999, -41212, 64032, -70788, 46020, 20778, -126132, 244120, -323421, 295410, -96848, -293868, 815829, -1297972
Offset: 0

Views

Author

Seiichi Manyama, Jun 21 2023

Keywords

Crossrefs

Convolution inverse of A002102.
Column k=3 of A363778.
Cf. A162552.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/sum(k=0, sqrtint(N), x^k^2)^3)

Formula

a(0) = 1; a(n) = -(3/n) * Sum_{k=1..n} A162552(k) * a(n-k).

A214900 Number of ordered ways to represent n as the sum of three squares and one fourth power.

Original entry on oeis.org

1, 4, 6, 4, 4, 9, 9, 3, 3, 9, 12, 9, 4, 7, 12, 6, 4, 15, 18, 10, 12, 18, 12, 3, 6, 18, 27, 19, 5, 18, 24, 6, 6, 18, 21, 18, 18, 18, 18, 9, 9, 30, 33, 13, 6, 27, 24, 6, 4, 16, 33, 27, 18, 24, 33, 12, 12, 27, 18, 18, 12, 24, 30, 12, 4, 30, 45, 21, 18, 33, 30, 6, 12, 21, 33, 34, 10, 27, 30, 6, 9, 40, 39, 24, 25, 33, 39, 18, 9
Offset: 0

Views

Author

Joerg Arndt, Jul 29 2012

Keywords

Comments

Different orderings of summands are counted, e.g., 1 = 1^2 + 0^2 + 0^4 + 0^4 = 0^2 + 1^2 + 0^4 + 0^4 = 0^2 + 0^2 + 1^4 + 0^4 = 0^2 + 0^2 + 0^4 + 1^4, so a(1)=4.
Conjecture: a(n) != 0, that is, all numbers are sums of three squares and one fourth power.

Crossrefs

Cf. A000925 (two squares), A002102 (three squares).

Programs

  • PARI
    N=10^3;  x='x+O('x^N);
S(e)=sum(j=0, ceil(N^(1/e)), x^(j^e));
v=Vec( S(4)^1 * S(2)^3 )

Formula

G.f.: (Sum_{j>=0} x^(j^2))^3 * (Sum_{j>=0} x^(j^4)) (see PARI code).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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