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.

A047800 Number of different values of i^2 + j^2 for i,j in [0, n].

Original entry on oeis.org

1, 3, 6, 10, 15, 20, 27, 34, 42, 51, 61, 71, 83, 94, 106, 120, 135, 148, 165, 180, 198, 216, 235, 252, 273, 294, 315, 337, 360, 382, 408, 431, 457, 484, 508, 536, 567, 595, 624, 653, 687, 715, 749, 781, 813, 850, 884, 919, 957, 993, 1031, 1069, 1108, 1142
Offset: 0

Views

Author

Wouter Meeussen

Keywords

Comments

a(n-1) is the number of distinct distances on an n X n pegboard. What is its asymptotic growth? Can it be efficiently computed for large n? - Charles R Greathouse IV, Jun 13 2013
Conjecture (after Landau and Erdős): a(n) ~ c * n^2 / sqrt(log(n)), where c = 0.79... . - Vaclav Kotesovec, Mar 10 2016

Links

Crossrefs

Cf. A034966, A047801, A160663, A384797.

Programs

  • Haskell
    import Data.List (nub)
a047800 n = length $ nub [i^2 + j^2 | i <- [0..n], j <- [i..n]]
-- Reinhard Zumkeller, Oct 03 2012
  • Mathematica
    Table[ Length@Union[ Flatten[ Table[ i^2+j^2, {i, 0, n}, {j, 0, n} ] ] ], {n, 0, 49} ]
nmax = 100; sq = Table[i^2 + j^2, {i, 0, nmax}, {j, 0, nmax}]; Table[Length@Union[Flatten[Table[Take[sq[[j]], n + 1], {j, 1, n + 1}]]], {n, 0, nmax}] (* Vaclav Kotesovec, Mar 09 2016 *)
  • PARI
    a(n) = n++; #vecsort(vector(n^2, i, ((i-1)\n)^2+((i-1)%n)^2), , 8) \\ Charles R Greathouse IV, Jun 13 2013; edited by Michel Marcus, Jul 06 2025
  • PARI
    a(n) = #setbinop((i,j)->i^2+j^2, [0..n]); \\ Michel Marcus, Jul 07 2025
  • Python
    def A047800(n): return len(set(i**2+j**2 for i in range(n+1) for j in range(i+1))) # Chai Wah Wu, Jul 07 2025

A225273 T(n,k)=Number of distinct values of the sum of i^2 over n realizations of i in 0..k.

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 5, 10, 10, 5, 6, 15, 19, 14, 6, 7, 20, 32, 29, 18, 7, 8, 27, 45, 50, 38, 22, 8, 9, 34, 67, 74, 66, 47, 26, 9, 10, 42, 88, 111, 99, 82, 56, 30, 10, 11, 51, 116, 149, 147, 124, 98, 65, 34, 11, 12, 61, 145, 197, 201, 183, 149, 114, 74, 38, 12, 13, 71, 179, 247, 262
Offset: 1

Views

Author

R. H. Hardin May 04 2013

Keywords

Comments

Table starts
..2..3..4...5...6...7...8...9..10..11...12...13...14...15...16...17...18...19
..3..6.10..15..20..27..34..42..51..61...71...83...94..106..120..135..148..165
..4.10.19..32..45..67..88.116.145.179..212..260..300..347..402..464..517..592
..5.14.29..50..74.111.149.197.247.308..370..451..526..613..706..815..914.1037
..6.18.38..66..99.147.201.262.332.411..498..601..702..819..946.1078.1221.1375
..7.22.47..82.124.183.250.326.414.513..621..749..874.1018.1176.1338.1515.1706
..8.26.56..98.149.219.299.390.496.614..742..894.1045.1215.1404.1597.1807.2032
..9.30.65.114.174.255.348.454.577.715..863.1038.1216.1412.1630.1856.2098.2357
.10.34.74.130.199.291.397.518.658.815..984.1182.1385.1608.1855.2114.2388.2681
.11.38.83.146.224.327.446.582.739.915.1105.1326.1554.1804.2080.2370.2677.3005

Links

Crossrefs

Row 1 is A000027(n+1)
Row 2 is A047800
Row 3 is A034966
Row 4 is A047801
Row 5 is A132432(n+1)
Row 6 is A132438(n+1)

Formula

Empirical: column k is n*k^2 - A225277(k) for large n (n>36 suffices for k through 210)

A047801 Number of different values of i^2+j^2+k^2+l^2 for i,j,k,l in [ 0,n ].

Original entry on oeis.org

1, 5, 14, 29, 50, 74, 111, 149, 197, 247, 308, 370, 451, 526, 613, 706, 815, 914, 1037, 1146, 1284, 1416, 1565, 1698, 1876, 2030, 2209, 2380, 2578, 2757, 2972, 3168, 3401, 3612, 3850, 4071, 4339, 4575, 4843, 5089, 5388
Offset: 0

Views

Author

Wouter Meeussen

Keywords

Links

Crossrefs

Cf. A034966, A047800.
Partial sums of A122927.

Programs

  • Mathematica
    Table[ Length@Union@Flatten@Table[ i^2+j^2+k^2+l^2, {i, 0, n}, {j, i, n}, {k, j, n}, {l, k, n} ], {n, 0, 48} ]

Extensions

Definition corrected by Jonathan Vos Post, Nov 14 2007

A132432 Number of different values of i^2+j^2+k^2+l^2+m^2 for i,j,k,l,m in [0,n].

Original entry on oeis.org

1, 6, 18, 38, 66, 99, 147, 201, 262, 332, 411, 498, 601, 702, 819, 946, 1078, 1221, 1375, 1533, 1703, 1882, 2076, 2264, 2479, 2691, 2922, 3159, 3403, 3655, 3924, 4193, 4478, 4770, 5071, 5376, 5705, 6032, 6372, 6719, 7081, 7448, 7828, 8214, 8616, 9017, 9438
Offset: 0

Views

Author

Jonathan Vos Post, Nov 13 2007, Nov 14 2007

Keywords

Examples

			a(3) = 18 because the 18 different sums of 5 squares of integers from 0 to 2 are: {20, 17, 16, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0} by permutations of 2^2 + 2^2 + 2^2 + 2^2 + 2^2 = 20; 2^2 + 2^2 + 2^2 + 2^2 + 1^2 = 17; 2^2 + 2^2 + 2^2 + 2^2 + 0^2 = 16; 2^2 + 2^2 + 2^2 + 1^2 + 1^2 = 14; 2^2 + 2^2 + 2^2 + 1^2 + 0^2 = 13; 2^2 + 2^2 + 2^2 + 0^2 + 0^2 = 12; 2^2 + 2^2 + 1^2 + 1^2 + 1^2 = 11; 2^2 + 2^2 + 1^2 + 1^2 + 0^2 = 10; 2^2 + 2^2 + 1^2 + 0^2 + 0^2 = 9; 2^2 + 2^2 + 0^2 + 0^2 + 0^2 = 2^2 + 1^2 + 1^2 + 1^2 + 1^2 = 8; 2^2 + 1^2 + 1^2 + 1^2 + 0^2 = 7; 2^2 + 1^2 + 1^2 + 0^2 + 0^2 = 6; 2^2 + 1^2 + 0^2 + 0^2 + 0^2 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 = 5; 2^2 + 0^2 + 0^2 + 0^2 + 0^2 = 1^2 + 1^2 + 1^2 + 1^2 + 0^2 = 4; 1^2 + 1^2 + 1^2 + 0^2 + 0^2 = 3; 1^2 + 1^2 + 0^2 + 0^2 + 0^2 = 2; 1^2 + 0^2 + 0^2 + 0^2 + 0^2 = 1; 0^2 + 0^2 + 0^2 + 0^2 + 0^2 = 0.

Links

Crossrefs

Cf. A034966, A047800, A047801.

Programs

  • Maple
    S:= proc(k,n) option remember;
if k = 0 or n = 0 then {0} else
`union`(seq(map(`+`,procname(j,n-1),(k-j)*n^2),j=1..k-1),
{k*n^2},procname(k,n-1)) fi end proc:
seq(nops(S(5,n)),n=0..100); # Robert Israel, Jun 28 2018
  • Mathematica
    Table[Length@ Union@Flatten@ Table[i^2 + j^2 + k^2 + l^2 + m^2, {i, 0, n}, {j, i, n}, {k, j, n}, {l, k, n}, {m, l, n}], {n, 0, 49}]

Extensions

Offset corrected by Giovanni Resta, Jun 18 2016

A374710 Number of distinct sums i^3 + j^3 + k^3 for 0<=i<=j<=k<=n.

Original entry on oeis.org

1, 4, 10, 20, 35, 56, 82, 117, 162, 215, 279, 352, 426, 528, 645, 772, 899, 1057, 1235, 1429, 1647, 1883, 2133, 2415, 2694, 3022, 3374, 3721, 4130, 4564, 5011, 5503, 5986, 6531, 7045, 7664, 8273, 8945, 9659, 10383, 11127, 11949, 12809, 13713, 14649, 15633, 16644, 17715, 18735, 19916
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2024

Keywords

Crossrefs

Cf. A374707, A374711.
Cf. A034966.

Programs

  • PARI
    a(n) = my(v=vector(3*n^3+1)); for(i=0, n, for(j=i, n, for(k=j, n, v[i^3+j^3+k^3+1]+=1))); sum(i=1, #v, v[i]>0);

A132438 Number of different values of i^2+j^2+k^2+l^2+m^2+n^2 for i,j,k,l,m,n in [0,n].

Original entry on oeis.org

1, 7, 22, 47, 82, 124, 183, 250, 326, 414, 513, 621, 749, 874, 1018, 1176, 1338, 1515, 1706, 1899, 2110, 2331, 2568, 2806, 3066, 3324, 3612, 3903, 4201, 4513, 4841, 5173, 5523, 5882, 6248, 6626, 7026, 7433, 7842, 8271, 8715
Offset: 0

Views

Author

Jonathan Vos Post, Nov 13 2007, Nov 14 2007

Keywords

Comments

Number of distinct sums of 6 squares of integers from 0 through n.

Examples

			a(1) = 7 because the 7 distinct sums of squares from 0 through 1 are permutations of 1^2 + 1^1 + 1^2 + 1^2 + 1^2 + 1^2 = 6; 1^1 + 1^2 + 1^2 + 1^2 + 1^2 + 0^2 = 5; 1^1 + 1^2 + 1^2 + 1^2 + 0^2 + 0^2 = 4; 1^1 + 1^2 + 1^2 + 0^2 + 0^2 + 0^2 = 3; 1^1 + 1^2 + 0^2 + 0^2 + 0^2 + 0^2 = 2; 1^1 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 = 1; 0^2 + 0^1 + 0^2 + 0^2 + 0^2 + 0^2 = 0.

Crossrefs

Cf. A034966, A047800, A047801, A132432.

Programs

  • Mathematica
    Table[Length@ Union@Flatten@ Table[i^2 + j^2 + k^2 + l^2 + m^2 + n^2, {i, 0, p}, {j, i, p}, {k, j, p}, {l, k, p}, {m, l, p}, {n, m, p}], {p, 0, 40}]

Extensions

Offset corrected by Giovanni Resta, Jun 19 2016
Showing 1-6 of 6 results.

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