A233733 -id:A233733 - cp's OEIS frontend

A223732 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly one way.

3, 6, 9, 11, 14, 17, 18, 19, 21, 22, 26, 27, 29, 30, 34, 35, 42, 43, 45, 46, 49, 50, 53, 61, 65, 67, 70, 73, 75, 78, 82, 91, 93, 97, 106, 109, 115, 133, 142, 145, 147, 157, 163, 169, 190, 193, 202, 205, 235, 253, 265, 277, 298, 397, 403, 427, 442, 445, 505, 793

Offset: 1

Wolfdieter Lang, Apr 05 2013

nonn

These are the increasingly ordered numbers a(n) for which A233730(a(n)) = 1. See also A233731. These are the numbers n with exactly one representation as a primitive sum of three nonzero squares (not taking into account the order of the three terms, and the number to be squared for each term is taken positive).

Conjecture: 793 = 6^2 + 9^2 + 26^2 is the largest element of this sequence. - Alois P. Heinz, Apr 06 2013

			a(1) = 3 because there is no solution for m = 1 and 2 as a primitive sum of three nonzero squares, and m = 3 = 1^2 + 1^2 + 1^2 is the only solution with [a,b,c] = [1,1,1].
a(5) = 14 because 14 is the fifth largest member of the set S1, and [a,b,c] = [1,2,3] denotes this unique representation for m = 14.

Eugen J. Ionascu, Ehrhart polynomial for lattice squares, cubes and hypercubes, arXiv:1508.03643 [math.NT], 2015.

Cf. A233730, A233731, A233733, A233734.

threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[800], threeSquaresCount[#] == 1 &] (* Jean-François Alcover, Jun 21 2013 *)

This sequence lists the increasingly ordered members of the set S1 := {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, gcd(a,b,c) = 1, with only one such solution for this m}.

A233730 Number of (n+1)X(5+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

7976, 115972, 1851404, 32240288, 578163016, 10577526052, 195088215888, 3616933627344, 67204709367712, 1250516988640176, 23282612726211812, 433669103881115132, 8078852328328338404, 150521113304511790216

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

Column 5 of A233733

			Some solutions for n=2
..3..0..2..3..1..0....3..2..3..2..0..2....1..0..2..3..1..3....2..3..1..0..1..3
..2..1..0..2..0..2....1..0..1..3..2..3....0..2..3..1..0..1....0..1..2..3..2..1
..1..3..1..3..1..3....0..2..0..1..0..2....1..0..1..0..2..3....2..3..1..0..1..3

R. H. Hardin, Table of n, a(n) for n = 1..210

A233731 Number of (n+1)X(6+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

30260, 631628, 14802604, 389230764, 10577526052, 297173770244, 8382657501872, 239905202721744, 6836811878013164, 196682109947646076, 5621738544272692176, 161957852533354058144, 4633563231512519521944, 133535514248052233170788

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

Column 6 of A233733

			Some solutions for n=1
..1..0..2..0..2..0..2....3..2..3..2..0..3..0....1..0..1..0..1..3..1
..3..2..1..2..3..1..3....2..0..1..0..1..2..1....0..2..3..1..3..2..3

R. H. Hardin, Table of n, a(n) for n = 1..210

A233726 Number of (n+1) X (1+1) 0..3 arrays with every 2 X 2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

40, 148, 556, 2104, 7976, 30260, 114820, 435720, 1653488, 6274804, 23812172, 90364680, 342924296, 1301361476, 4938528804, 18741194360, 71120848176, 269896091748, 1024227097996, 3886833426776, 14750121431720, 55975149559572

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

			Some solutions for n=5:
..0..2....3..1....3..2....0..2....2..3....0..1....0..1....1..3....0..2....3..0
..2..1....2..0....2..0....1..3....1..0....1..3....1..3....0..2....1..3....2..1
..0..2....1..2....0..1....0..1....0..2....2..1....0..1....2..1....3..2....3..0
..2..3....0..3....2..3....3..2....2..1....0..2....2..0....1..3....0..1....2..1
..1..0....1..2....3..1....0..1....0..2....2..1....0..1....0..2....2..0....0..2
..3..2....2..0....1..0....1..3....1..0....1..3....1..3....1..3....0..1....2..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 1 of A233733.

Empirical: a(n) = 3*a(n-1) + 5*a(n-2) - 7*a(n-3) - 2*a(n-4).

Empirical g.f.: 4*x*(10 + 7*x - 22*x^2 - 6*x^3) / (1 - 3*x - 5*x^2 + 7*x^3 + 2*x^4). - Colin Barker, Oct 11 2018

A233727 Number of (n+1)X(2+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

148, 756, 3972, 21432, 115972, 631628, 3435216, 18733728, 102007004, 556289404, 3030290308, 16522951748, 90024199928, 490800907380, 2674421936852, 14579265732508, 79450066179340, 433084651112312, 2360226967287744

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

Column 2 of A233733

			Some solutions for n=5
..0..2..0....1..2..0....1..0..1....3..2..3....1..0..2....1..2..3....3..2..0
..1..0..1....2..0..1....3..2..0....2..0..1....3..1..3....3..1..0....1..3..1
..3..1..3....3..2..3....2..0..1....3..2..3....1..0..1....2..0..2....3..2..3
..2..0..1....1..0..1....0..1..3....0..1..0....2..3..2....3..1..0....2..0..1
..1..2..0....2..3..2....2..0..2....1..3..1....3..1..3....2..0..2....3..2..0
..2..0..1....3..1..3....3..2..3....0..1..2....1..0..1....3..1..0....1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 3*a(n-1) +30*a(n-2) -71*a(n-3) -192*a(n-4) +415*a(n-5) +410*a(n-6) -847*a(n-7) -269*a(n-8) +544*a(n-9) +38*a(n-10) -42*a(n-11) -4*a(n-12)

A233728 Number of (n+1)X(3+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

556, 3972, 29816, 233432, 1851404, 14802604, 118695412, 953471940, 7663964828, 61629254552, 495650497344, 3986689001988, 32067041076200, 257939925202452, 2074814466960036, 16689518674136900, 134248084171273808

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

Column 3 of A233733

			Some solutions for n=3
..3..2..0..3....1..3..1..0....3..2..3..1....1..3..1..3....0..1..0..1
..2..0..1..2....3..2..0..2....1..3..1..0....0..1..0..2....1..3..1..3
..3..1..3..1....0..1..2..3....0..2..3..2....2..3..2..1....0..2..0..2
..1..2..1..2....3..2..0..2....1..0..1..0....0..1..3..0....1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 9*a(n-1) +46*a(n-2) -508*a(n-3) -481*a(n-4) +10680*a(n-5) -3945*a(n-6) -112548*a(n-7) +115263*a(n-8) +657688*a(n-9) -958275*a(n-10) -2192200*a(n-11) +4058993*a(n-12) +4067658*a(n-13) -9776097*a(n-14) -3651898*a(n-15) +13759514*a(n-16) +407041*a(n-17) -11221093*a(n-18) +1862871*a(n-19) +5154086*a(n-20) -1437504*a(n-21) -1269561*a(n-22) +444011*a(n-23) +147392*a(n-24) -60418*a(n-25) -4972*a(n-26) +2810*a(n-27) -123*a(n-28)

A233729 Number of (n+1)X(4+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

2104, 21432, 233432, 2723072, 32240288, 389230764, 4696134192, 57174669348, 693181060760, 8455863542400, 102666251086924, 1252787231370072, 15219081421614496, 185695021438104644, 2256549296014445884

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

Column 4 of A233733

			Some solutions for n=2
..0..1..3..1..2....2..0..2..0..2....2..0..1..2..0....2..0..3..1..3
..2..3..2..0..3....3..2..1..2..1....3..2..0..3..2....0..1..2..0..2
..0..1..0..1..2....2..0..2..0..2....1..3..1..2..0....2..0..3..2..1

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 84

Empirical recurrence of order 84 (see link above)

A233732 Number of (n+1)X(7+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

114820, 3435216, 118695412, 4696134192, 195088215888, 8382657501872, 365411698279744, 16077504208896432, 710340588106387612, 31469577641576479052, 1395852241413401555300, 61965217435895823239576

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

Column 7 of A233733

			Some solutions for n=1
..3..1..0..2..1..0..1..0....1..0..2..1..0..2..1..2....1..0..1..3..1..0..1..2
..1..0..2..1..3..1..3..2....0..2..1..3..2..1..3..1....3..2..0..1..0..2..3..1

R. H. Hardin, Table of n, a(n) for n = 1..210

A233725 Number of (n+1)X(n+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

Original entry on oeis.org

40, 756, 29816, 2723072, 578163016, 297173770244, 365411698279744, 1098248392267801764, 7928574480658025834472, 139889299008340046298542520

Offset: 1

R. H. Hardin, Dec 15 2013

nonn

Diagonal of A233733

			Some solutions for n=3
..1..3..2..3....3..1..3..2....0..1..3..1....0..1..3..2....2..0..2..0
..0..1..0..1....2..0..1..3....2..0..2..0....2..0..2..0....3..2..3..2
..3..2..3..2....3..1..3..2....3..1..3..1....3..1..3..1....2..0..2..0
..0..1..0..1....1..0..1..0....1..0..2..3....1..0..2..0....0..1..0..1

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A223732 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly one way.

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A233730 Number of (n+1)X(5+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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A233731 Number of (n+1)X(6+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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A233726 Number of (n+1) X (1+1) 0..3 arrays with every 2 X 2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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A233727 Number of (n+1)X(2+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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A233728 Number of (n+1)X(3+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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A233729 Number of (n+1)X(4+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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A233732 Number of (n+1)X(7+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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A233725 Number of (n+1)X(n+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of the edge differences equal to 6 and no adjacent elements equal.

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