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
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
KEYWORD
nonn,nice,easy
AUTHOR
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