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Search: id:A001033
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| A001033 |
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Numbers n such that the sum of the squares of n consecutive 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)
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+0
4
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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 (noe(AT)sspectra.com), Oct 24 2007
A134419 shows how A001032 and this sequence are related. - T. D. Noe, Nov 04 2007
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 numbers, Math. Mag. 41 (1968), 255-258.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..100
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FORMULA
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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 (njas(AT)research.att.com).
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 (noe(AT)sspectra.com), Oct 24 2007
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EXAMPLE
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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.
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CROSSREFS
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Cf. A056131, A056132.
Sequence in context: A095409 A111026 A124186 this_sequence A100647 A030666 A030676
Adjacent sequences: A001030 A001031 A001032 this_sequence A001034 A001035 A001036
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com)
Corrected and extended by T. D. Noe (noe(AT)sspectra.com), Oct 24 2007
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