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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 inA056131andA056132,respectively.
(Formerly M4999 N2152)
5
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 (list; graph; refs; listen; history; text; internal format)
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
A134419shows howA001032and 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
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms fromRobert G. Wilson v
Corrected and extended byT. D. Noe,Oct 24 2007
1024 was missing from b-file. -Christopher E. Thompson,Feb 05 2016
STATUS
approved

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Last modified September 2 08:20 EDT 2024. Contains 375604 sequences. (Running on oeis4.)