A181786 -id:A181786 - cp's OEIS frontend

A181787 Number of solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

0, 0, 0, 3, 0, 0, 3, 6, 0, 12, 0, 9, 3, 6, 6, 15, 0, 9, 12, 15, 0, 33, 9, 18, 3, 12, 6, 39, 6, 18, 15, 24, 0, 48, 9, 30, 12, 24, 15, 45, 0, 27, 33, 33, 9, 60, 18, 36, 3, 48, 12, 60, 6, 36, 39, 45, 6, 78, 18, 45, 15, 42, 24, 114, 0, 36, 48, 51, 9, 93, 30, 54, 12, 51, 24, 87, 15, 87, 45, 60, 0, 120, 27, 63, 33, 51, 33, 105, 9, 63, 60, 84, 18, 123, 36, 75, 3, 69, 48, 165, 12

Offset: 0

T. D. Noe, Nov 12 2010

Note that a(n)=0 for n=0 and the n in A094958.

			a(3)=3 because 3^2 = 1^2+2^2+2^2 = 2^2+1^2+2^2 = 2^2+2^2+1^2. - _Robert Israel_, Aug 02 2019

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A016725, A016727, A181786, A181788.

N:= 200: # for a(0)..a(N)
A:= Array(0..N):
mults:= [1,3,6]:
for a from 1 while 3*a^2 <= N^2 do
  if a::odd then b0:= a+1; db:= 2 else b0:= a; db:= 1 fi;
  for b from b0 by db while a^2 + 2*b^2 <= N^2 do
    if (a+b)::odd then c0:= b + (b mod 2); dc:= 2 else c0:= b; dc:= 1 fi;
    for c from c0 by dc do
      v:= a^2 + b^2 + c^2;
      if v > N^2 then break fi;
      if issqr(v) then
        w:= sqrt(v);
        A[w]:= A[w]+ mults[nops({a,b,c})];
      fi
od od od:
convert(A,list); # Robert Israel, Aug 02 2019

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,nn}, {b,nn}, {c,nn}]; Prepend[t,0]

a(n) = A063691(n^2). - Michel Marcus, Apr 25 2015

a(2*n) = a(n). - Robert Israel, Aug 02 2019

A181788 Number of solutions to n^2 = a^2 + b^2 + c^2 with nonnegative a, b, c.

Original entry on oeis.org

1, 3, 3, 6, 3, 9, 6, 9, 3, 15, 9, 12, 6, 15, 9, 24, 3, 18, 15, 18, 9, 36, 12, 21, 6, 27, 15, 42, 9, 27, 24, 27, 3, 51, 18, 39, 15, 33, 18, 54, 9, 36, 36, 36, 12, 69, 21, 39, 6, 51, 27, 69, 15, 45, 42, 54, 9, 81, 27, 48, 24, 51, 27, 117, 3, 63, 51, 54, 18, 96, 39, 57, 15, 60, 33, 102, 18, 90, 54, 63, 9, 123, 36, 66, 36, 78, 36, 114, 12, 72, 69, 93, 21, 126, 39, 84, 6, 78, 51, 168, 27

Offset: 0

T. D. Noe, Nov 12 2010

nonn

Paul D. Hanna and Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Cf. A016725, A016727, A181786, A181787.

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,0,nn}, {b,0,nn}, {c,0,nn}]; Prepend[t,1]

{a(n)=local(G=sum(k=0,n,x^(k^2)+x*O(x^(n^2))));polcoeff(G^3,n^2)} /* Paul D. Hanna */

A(n)=my(G=sum(k=0,n,x^(k^2),x*O(x^(n^2)))^3); vector(n+1, k, polcoeff(G,(k-1)^2)) \\ Charles R Greathouse IV, Apr 20 2012

G.f.: [x^(n^2)] G(x)^3 where G(x) = Sum_{k>=0} x^(k^2); the notation [x^(n^2)] G(x)^3 denotes the coefficient of x^(n^2) in G(x)^3. [From Paul D. Hanna, Apr 20 2012]

A225206 Number of Pythagorean quadruples (a, b, c, d) with d < 10^n.

Original entry on oeis.org

6, 571, 56268, 5614390, 561232920, 56120665334, 5612026652893, 561202243017532, 56120219419339591

Offset: 1

Arkadiusz Wesolowski, May 01 2013

a(n) ~ Pi*A225207(n)/(1+G), where G is Catalan's constant (A006752).

			a(1) = 6 because there are six solutions (a, b, c, d) as follows: (1, 2, 2, 3), (2, 4, 4, 6), (2, 3, 6, 7), (1, 4, 8, 9), (3, 6, 6, 9), (4, 4, 7, 9) with d < 10.

Wikipedia, Pythagorean quadruple

Cf. A169580, A181786, A225207.

a(n) = Sum_{k=1..10^n-1} A181786(k). - Max Alekseyev, Feb 28 2023

a(4) from Giovanni Resta, May 01 2013

a(5)-a(9) from Max Alekseyev, Feb 28 2023

A208882 Number of representations of square of prime(n) as a^2 + b^2 + c^2 with 0 < a <= b <= c.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 8, 7, 9, 9, 9, 10, 11, 11, 12, 12, 13, 14, 13, 14, 16, 17, 17, 18, 18, 19, 19, 21, 21, 21, 23, 22, 24, 24, 24, 25, 27, 28, 29, 28, 29, 30, 30, 32, 32, 33, 33, 34, 34, 35, 36, 36, 39, 39, 39, 39, 42, 42, 44, 43, 44

Offset: 1

Zak Seidov, Mar 02 2012

nonn

Almost monotonically increasing sequence, only rarely a(n) <= a(n-1), contrary to case of n instead of prime(n) (A181786).

			a(2)=1 because prime(2)=3 and 3^2 = 1^2 + 2^2 + 2^2,
a(4)=1 because prime(4)=7 and 7^2 = 2^2 + 3^2 + 6^2,
a(5)=2 because prime(5)=11 and 11^2 = 2^2 + 6^2 + 9^2 = 6^2 + 6^2 + 7^2.

Samuel Harkness, Table of n, a(n) for n = 1..1229

Cf. A002144, A181786.

Table[Length[FindInstance[{Prime[n]^2==a^2+b^2+c^2,0Harvey P. Dale, Mar 06 2020 *)

A210311 Primes that can be represented exactly in one way as a^2 + b^2 + c^2, 0 < a <= b <= c.

Original entry on oeis.org

3, 11, 17, 19, 29, 43, 53, 61, 67, 73, 97, 109, 157, 163, 193, 277, 397

Offset: 1

Zak Seidov, Mar 20 2012

nonn

Note that there are no primes = 7 mod 8.
This sequence is probably complete. Is there a proof?
There are no more terms < 10^7. - Donovan Johnson, Mar 22 2012

			{p,a,b,c}: {3,1,1,1}, {11,1,1,3}, {17,2,2,3}, {19,1,3,3}, {29,2,3,4}, {43,3,3,5}, {53,1,4,6}, {61,3,4,6}, {67,3,3,7}, {73,1,6,6}, {97,5,6,6}, {109,3,6,8}, {157,2,3,12}, {163,1,9,9}, {193,6,6,11}, {277,4,6,15}, {397,3,8,18}.

Cf. A181786, A210338.

A210338 Primes that can be represented exactly in two ways as a^2 + b^2 + c^2, 0 < a <= b <= c.

Original entry on oeis.org

41, 59, 83, 107, 113, 137, 139, 181, 197, 211, 229, 283, 307, 313, 317, 331, 337, 373, 379, 421, 457, 499, 541, 547, 577, 613, 643, 709, 757, 853, 877, 883, 907, 1093, 1213

Offset: 1

Zak Seidov, Mar 20 2012

nonn

This sequence is probably complete. Is there a proof of this?

There are no more terms < 10^7. - Donovan Johnson, Mar 22 2012

			{p,a,b,c}:
{41,1,2,6}, {41,3,4,4}
{59,1,3,7}, {59,3,5,5}
{83,1,1,9}, {83,3,5,7}
{107,1,5,9}, {107,3,7,7}.

Cf. A181786, A210311.

cp's OEIS Frontend

A181787 Number of solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

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A181788 Number of solutions to n^2 = a^2 + b^2 + c^2 with nonnegative a, b, c.

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A225206 Number of Pythagorean quadruples (a, b, c, d) with d < 10^n.

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A208882 Number of representations of square of prime(n) as a^2 + b^2 + c^2 with 0 < a <= b <= c.

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A210311 Primes that can be represented exactly in one way as a^2 + b^2 + c^2, 0 < a <= b <= c.

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A210338 Primes that can be represented exactly in two ways as a^2 + b^2 + c^2, 0 < a <= b <= c.

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