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. (Formerly M4999 N2152)

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

1,2

Comments

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.

References

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).

Links

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.

Formula

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

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A056131, A056132, A274470.

Sequence in context: A124186 A274240 A176512 * A100647 A245371 A235717

Adjacent sequences: A001030 A001031 A001032 * A001034 A001035 A001036

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved