A047800 -id:A047800 - cp's OEIS frontend

A384797 a(n) = A047800(n) - A047800(n-1).

2, 3, 4, 5, 5, 7, 7, 8, 9, 10, 10, 12, 11, 12, 14, 15, 13, 17, 15, 18, 18, 19, 17, 21, 21, 21, 22, 23, 22, 26, 23, 26, 27, 24, 28, 31, 28, 29, 29, 34, 28, 34, 32, 32, 37, 34, 35, 38, 36, 38, 38, 39, 34, 40, 42, 44, 43, 44, 42, 49, 42, 42, 46, 45, 51, 50, 46, 47, 50

Offset: 1

Hugo Pfoertner, Jun 17 2025

This is the number of distinct positive distances of the [0,n] X [0,n] points of a lattice square from the origin that are added when the size of the square is increased from n-1 to n. Among the newly added squared distances n^2, n^2+1^2, n^2+2^2, ..., n^2+n^2, some may already have occurred in smaller lattice point squares and are therefore not counted.

			a(1) = 2: Newly added squared distances are 1 and 2.
a(5) = 5: The squared distance 25=5^2+0^2 already occurs as 4^2+3^2.
a(13) = 11: There are 3 already represented squared distances, 13^2+0^2=12^2+5^2, 13^2+1^2=11^2+7^2, 13^2+4^2=11^2+8^2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A047800, A160663, A384798, A384799.

def A384797(n):
    s = set(i**2+j**2 for i in range(n) for j in range(i+1))
    return n+1-sum(1 for i in range(n+1) if n**2+i**2 in s) # 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

R. H. Hardin May 04 2013

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

R. H. Hardin, Table of n, a(n) for n = 1..1325

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)

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

A034966 Number of different values of i^2 + j^2 + k^2 for i,j,k in [ 0,n ] (or [ -n,n ]).

Original entry on oeis.org

1, 4, 10, 19, 32, 45, 67, 88, 116, 145, 179, 212, 260, 300, 347, 402, 464, 517, 592, 649, 727, 803, 886, 953, 1057, 1146, 1243, 1343, 1453, 1547, 1680, 1784, 1914, 2041, 2165, 2288, 2454, 2578, 2723, 2866, 3037, 3179, 3363, 3516, 3696, 3868, 4041, 4205

Offset: 0

Wouter Meeussen

Number of distinct lengths of main diagonals of all i X j X k boxes with edge lengths i,j,k in [0,n]

Robin Visser, Table of n, a(n) for n = 0..10000 (terms 0..200 from T. D. Noe).

Cf. A047800, A047801.

Table[ Length@Union[ Flatten[ Table[ i^2+j^2+k^2, {i, 0, n}, {j, 0, n}, {k, 0, n} ] ] ], {n, 0, 49} ]

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

Wouter Meeussen

Robin Visser, Table of n, a(n) for n = 0..5000 (terms n = 0..100 from T. D. Noe).

Cf. A034966, A047800.

Partial sums of A122927.

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} ]

Definition corrected by Jonathan Vos Post, Nov 14 2007

A319476 a(n) is the minimum number of distinct distances between n non-attacking rooks on an n X n chessboard.

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 5, 6, 5, 7, 9, 7, 8, 11, 13, 9, 11, 14, 16, 17, 19, 21, 21, 14, 14

Offset: 1

Peter Kagey, Oct 12 2018

a(n) <= n - 1, which is the number of distinct distances the rooks are placed along a diagonal.

Conjecture: a(n^2) = A047800(n-1) - 1. - Peter Kagey, Nov 02 2018

			For n = 7 a board with a(7) = 5 distinct distances is
  +---+---+---+---+---+---+---+
7 |   |   | * |   |   |   |   |
  +---+---+---+---+---+---+---+
6 |   |   |   |   |   | * |   |
  +---+---+---+---+---+---+---+
5 | * |   |   |   |   |   |   |
  +---+---+---+---+---+---+---+
4 |   |   |   | * |   |   |   |
  +---+---+---+---+---+---+---+.
3 |   |   |   |   |   |   | * |
  +---+---+---+---+---+---+---+
2 |   | * |   |   |   |   |   |
  +---+---+---+---+---+---+---+
1 |   |   |   |   | * |   |   |
  +---+---+---+---+---+---+---+
    A   B   C   D   E   F   G
The distances between pairs of points are:
1)   sqrt(10) (e.g., A5 to B2),
2) 2*sqrt(2)  (e.g., A5 to C7),
3) 4*sqrt(2)  (e.g., B2 to F6),
4) 2*sqrt(10) (e.g., A5 to G3), and
5)   sqrt(26) (e.g., A5 to F6).

Giovanni Resta, Illustration of a(3)-a(14)

Cf. A008404, A319476, A320575, A320576, A320448, A320573, A320574.

a(11)-a(14) from Giovanni Resta, Oct 17 2018

a(15)-a(25) from Bert Dobbelaere, Dec 30 2018

A160663 Number of distinct sums that one can obtain by adding two squares among the n first ones.

Original entry on oeis.org

2, 5, 9, 14, 19, 26, 33, 41, 50, 60, 70, 82, 93, 105, 119, 134, 147, 164, 179, 197, 215, 234, 251, 272, 293, 314, 336, 359, 381, 407, 430, 456, 483, 507, 535, 566, 594, 623, 652, 686, 714, 748, 780, 812, 849, 883, 918, 956, 992, 1030, 1068, 1107, 1141, 1181

Offset: 1

Romain CARRE (romain.carre.2008(AT)enseirb.fr), May 22 2009

nonn

Essentially the same as A047800: a(n) = A047800(n) - 1.
Let A be the set of the n first squares (1,4,9,...,n^2). Let A+A be the corresponding sumset (= {a,b,a+b where (a,b) in A^2}). That very sequence describes the number of elements of A+A, relatively to n.
a(n-1) is the number of distinct positive 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
An upper bound is a(n) <= A102548(2n^2) << n^2/log n. - Charles R Greathouse IV, Jan 16 2023

			For n = 3, A = {1,4,9}, A+A = {1,4,9} U {2,5,10,8,13,18} thus A+A = {1,2,4,5,8,9,10,13,18}, and hence card(A+A) = 9; a(3) = 9.

Melvyn B. Nathanson (1996). "Additive Number Theory: the Classical Bases" Graduate Texts in Mathematics. 164. Springer-Verlag. p. 192. ISBN 0-387-94656-X.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
L. G. Schnirelmann, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933) 649-690.
L. G. Schnirelmann, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933) 649-690. doi:10.1007/BF01448914.
Samuel S. Wagstaff, Jr., The Schnirelmann density of the sums of three squares, Proc. Amer. Math. Soc. 52 (1975), 1-7.
Wikipedia, Additive number theory
Wikipedia, Schnirelmann density
Wikipedia, Edmund Landau

a:= proc(n) local A, i, j; A:= [i^2$i=1..n]; nops([{A[], seq (seq (A[i]+A[j], j=1..i), i=1..nops(A))}[]]) end: seq (a(n), n=1..60); # Alois P. Heinz, Jun 16 2009

a[n_] := (Table[i^2 + j^2, {i, 0, n}, {j, i, n}] // Flatten // Union // Length) - 1; Array[a, 60] (* Jean-François Alcover, May 25 2018 *)

a(n)=n++; #vecsort(vector(n^2,i,((i-1)\n)^2+((i-1)%n)^2),,8)-1 \\ Charles R Greathouse IV, Jun 13 2013

a(n)=my(u=vector(n,i,i^2),v=List(u)); for(i=1,n, for(j=1,i, listput(v,u[i]+u[j]))); u=0; #Set(v) \\ Charles R Greathouse IV, Nov 18 2022

first(n)=my(v=vector(n),u=[]); for(k=1,n, my(k2=k^2,w=vector(k,i,i^2+k2)); w=setunion(w,[k2]); u=setunion(u,w); v[k]=#u); v \\ Charles R Greathouse IV, Nov 18 2022

def a(n):
    SUM, SQR = set(), set(x**2 for x in range(1, n + 1))
    for i in SQR:
        SUM.add(i)
        for j in SQR: SUM.add(i + j)
    return len(SUM)
# Romain CARRE (romain.carre.2008(AT)enseirb.fr), Apr 16 2010

a(n) = card(A+A) where A={k^2} k=1..n and A+A = {a,b,a+b where (a,b) in A^2}.
Trivially 2n <= a(n) <= n(n+1)/2. - Charles R Greathouse IV, Oct 30 2015
a(n) << n^2/sqrt(log n) [see A000404]. - Charles R Greathouse IV, Oct 30 2015

More terms from Alois P. Heinz, Jun 16 2009

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

Jonathan Vos Post, Nov 13 2007, Nov 14 2007

			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.

Robert Israel, Table of n, a(n) for n = 0..500

Cf. A034966, A047800, A047801.

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

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}]

Offset corrected by Giovanni Resta, Jun 18 2016

A374707 Number of distinct sums i^3 + j^3 for 0<=i<=j<=n.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 90, 104, 119, 135, 151, 169, 188, 208, 229, 251, 274, 298, 322, 348, 375, 402, 431, 461, 492, 524, 556, 590, 623, 659, 695, 733, 772, 811, 851, 893, 936, 980, 1025, 1071, 1118, 1166, 1213, 1263, 1314, 1365, 1418, 1471, 1525, 1580, 1637, 1695, 1753

Offset: 0

Seiichi Manyama, Jul 17 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A374710, A374711.

Cf. A047800, A061791.

a(n) = my(v=vector(2*n^3+1)); for(i=0, n, for(j=i, n, v[i^3+j^3+1]+=1)); sum(i=1, #v, v[i]>0);

A384798 Records in A384797.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 31, 34, 37, 38, 39, 40, 42, 44, 49, 51, 56, 58, 64, 69, 71, 73, 77, 79, 82, 92, 94, 101, 104, 108, 109, 111, 116, 118, 120, 129, 136, 140, 141, 155, 160, 172, 174, 181, 190, 193, 194, 208, 209

Offset: 1

Hugo Pfoertner, Jun 17 2025

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A047800, A384797, A384799.

from itertools import count, islice
def A384798_gen(): # generator of terms
    s, m = {0}, 1
    for n in count(1):
        c = n+1
        for i in range(n+1):
            if (k:=n**2+i**2) in s:
                c -= 1
            else:
                s.add(k)
        if c>m:
            yield c
            m = c
A384798_list = list(islice(A384798_gen(),30)) # Chai Wah Wu, Jul 07 2025

A384799 Positions of records in A384797.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 22, 24, 27, 28, 30, 33, 35, 36, 40, 45, 48, 52, 54, 55, 56, 60, 65, 70, 78, 80, 90, 95, 100, 105, 108, 110, 120, 130, 135, 140, 150, 155, 156, 160, 165, 168, 180, 190, 200, 205, 210, 225, 240, 255, 260, 270, 280, 285, 300

Offset: 1

Hugo Pfoertner, Jun 17 2025

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A047800, A384797, A384798.

from itertools import count, islice
def A384799_gen(): # generator of terms
    s, m = {0}, 1
    for n in count(1):
        c = n+1
        for i in range(n+1):
            if (k:=n**2+i**2) in s:
                c -= 1
            else:
                s.add(k)
        if c>m:
            yield n
            m = c
A384799_list = list(islice(A384799_gen(),30)) # Chai Wah Wu, Jul 07 2025

cp's OEIS Frontend

A384797 a(n) = A047800(n) - A047800(n-1).

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

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A034966 Number of different values of i^2 + j^2 + k^2 for i,j,k in [ 0,n ] (or [ -n,n ]).

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

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A319476 a(n) is the minimum number of distinct distances between n non-attacking rooks on an n X n chessboard.

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A160663 Number of distinct sums that one can obtain by adding two squares among the n first ones.

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Maple

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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].

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A374707 Number of distinct sums i^3 + j^3 for 0<=i<=j<=n.

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