A274470 -id:A274470 - cp's OEIS frontend

A001033 Numbers n such that the sum of the squares of n consecutive positive odd numbers x^2 + (x+2)^2 + ... + (x+2n-2)^2 = k^2 for some integer k. The least values of x and k for each n are in A056131 and A056132, respectively.

1, 16, 25, 33, 49, 52, 64, 73, 97, 100, 121, 148, 169, 177, 193, 196, 241, 244, 249, 256, 276, 289, 292, 297, 313, 337, 361, 388, 393, 400, 409, 457, 481, 484, 528, 529, 537, 577, 592, 625, 628, 649, 673, 676, 708, 724, 753, 772, 784, 793, 832, 841, 852, 897, 913, 961, 964, 976, 996

Offset: 1

N. J. A. Sloane

Papers by Sollfrey, Hunter and Makowski correct and extend the work of Alfred. However, they do not consider n = 97, 241, 244, 276, 528 and 832, which are in this sequence. I have verified that there are no other n < 1000. - T. D. Noe, Oct 24 2007
A134419 shows how A001032 and this sequence are related. - T. D. Noe, Nov 04 2007
The number 4 is not in this sequence due to the requirement that the odd integers be positive, otherwise 6^2 = (-1)^2 + 1^2 + 3^2 + 5^2.

			a(1) = 1 from 1^2.
a(2) = 16 from 27^2 + 29^2 + ... + 55^2 + 57^2 = 172^2.
a(4) = 33 from 91^2 + 93^2 + ... + 153^2 + 155^2 = 715^2.

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).

Christopher E. Thompson, Table of n, a(n) for n = 1..7103 (up to 250000, extending first 100 terms computed by T. D. Noe).
U. Alfred, Sums of squares of consecutive odd integers, Math. Mag., 40 (1967), 194-199.
J. A. H. Hunter, A note on sums of squares of consecutive odd numbers, Math. Mag. 42 (1969), 145.
Andrzej Makowski, Remark on the paper "Sums of squares of consecutive odd numbers", Math. Mag. 43 (1970), 212-213.
William Sollfrey, Note on sums of squares of consecutive odd integers, Math. Mag. 41 (1968), 255-258.

Cf. A056131, A056132, A274470.

r[1] = {True, {1, 1}}; r[n_] := (rn = Reduce[x > 0 && k > 0 && Sum[(x + 2*j)^2, {j, 0, n - 1}] == k^2, {x, k}, Integers]; srn = Simplify[(rn /. C[1] -> 0) || (rn /. C[1] -> 1) || (rn /. C[1] -> 2)]; rnOdd = Which[rn === False, False, srn[[0]] === And, srn, True, Select[srn, OddQ[x /. ToRules[#1]] & ]]; If[ rnOdd === False, {False, {0, 0}}, {True, {x, k} /. Flatten[{ToRules[rnOdd]}]}]); A001033 = Reap[Do[rn = r[n]; {x0, k0} = rn[[2]]; If[rn[[1]] && OddQ[x0], Print[{n, x0, k0}]; Sow[n]], {n, 1, 1000}]][[2, 1]] (* Jean-François Alcover, Mar 14 2012 *)

We must solve m*(3*x^2 + 6*m*x - 6*x + 4*m^2 - 6*m + 2)/3 = k^2 in integers (x, m, k). - N. J. A. Sloane

For a given n, we must determine whether the generalized Pell equation 4n*y^2 + 4y*n^2 + n(4n^2-1)/3 = k^2 has any integer solutions with y >= 0. Note that x = 2y+1 will be the first odd number being squared. If there are solutions then n is in this sequence. - T. D. Noe, Oct 24 2007

More terms from Robert G. Wilson v
Corrected and extended by T. D. Noe, Oct 24 2007
1024 was missing from b-file. - Christopher E. Thompson, Feb 05 2016

A274471 Numbers missing from A134419 despite satisfying the necessary congruence conditions (see comments).

Original entry on oeis.org

564, 842, 1284, 2306, 2308, 2402, 2459, 3602, 3650, 3803, 6242, 6338, 6779, 7044, 7058, 7319, 7643, 8088, 8354, 8363, 8402, 8543, 8628, 9122, 9168, 9412, 10607, 10826, 10852, 11257, 11378, 11447, 12203, 12436, 12458, 12722, 12984, 13682, 14162, 14388, 14424, 14639

Offset: 1

Christopher E. Thompson, Jun 24 2016

nonn

A134419 consists of those n where x^2 - n*y^2 = n(n-1)(n+1)/3 has integer solutions for x and y. There are easily verified necessary congruence conditions for that to occur:
  (defining x||y to mean x|y and x and y/x are coprime)
  if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);
  if p^e||n with p=5 or 7 (mod 12), then e is even;
  if 3^e||(n+1) with e>0, then e is odd;
  if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.
However, these conditions are not sufficient. This sequence consists of the numbers n satisfying the congruence conditions but for which the Pellian equation has no integer solutions.
If n = k^2*m where m is squarefree, then a necessary (but not sufficient) condition for n to occur in this sequence is that the narrow class group of quadratic forms of discriminant 4*m has more than one class per genus, or equivalently that the narrow class group is not an elementary 2-group.

Christopher E. Thompson, Table of n, a(n) for n = 1..799 [values up to 250000]

Cf. A134419, A274469, A274470.

cp's OEIS Frontend

A001033 Numbers n such that the sum of the squares of n consecutive positive odd numbers x^2 + (x+2)^2 + ... + (x+2n-2)^2 = k^2 for some integer k. The least values of x and k for each n are in A056131 and A056132, respectively.

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