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Search: id:A069894
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| A069894 |
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Centered square numbers: 4*n^2 + 4*n + 2. |
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+0
4
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| 2, 10, 26, 50, 82, 122, 170, 226, 290, 362, 442, 530, 626, 730, 842, 962, 1090, 1226, 1370, 1522, 1682, 1850, 2026, 2210, 2402, 2602, 2810, 3026, 3250, 3482, 3722, 3970, 4226, 4490, 4762, 5042, 5330, 5626, 5930, 6242, 6562, 6890, 7226, 7570, 7922, 8282
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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Any number may be substituted for y to yield similar sequences. The number set used determines values given (i.e.- integer yields integer). All centered square integers in the set of integers may be found by this formula.
1/2 + 1/10 + 1/26 +...= (Pi/4)*tanh(Pi/2) [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2006
Except for the first term, a(n)=8*n+a(n-1), (with a(1)=10) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 24 2009]
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REFERENCES
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L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.
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FORMULA
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[y(2x + 1)]^2 + [y(2x^2 + 2x)]^2 = [y(2x^2 + 2x +1)]^2 where y = 2. If a^2 + b^2 = c^2, then c^2 = y^2(4x^4 + 8x^3 + 8x^2 + 4x + 1). Also 2*A001844.
a(n) = (2n+1)^2+1. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 10 2008, corrected R. J. Mathar, Sep 16 2009]
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EXAMPLE
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If y = 3, then 81 + 144 = 225; if y = 4, then 12^2 + 16^2 = 20^2; 7^2 + 24^2 = 25^2 = 15^2 + 20^2.
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MATHEMATICA
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Table[4n(n + 1) + 2, {n, 0, 45}]
lst={}; Do[AppendTo[lst, n^2+1], {n, 1, 2*4!, 2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 10 2008]
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CROSSREFS
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Cf. A001844.
Adjacent sequences: A069891 A069892 A069893 this_sequence A069895 A069896 A069897
Sequence in context: A119062 A058373 A027719 this_sequence A045605 A009307 A131130
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KEYWORD
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nonn,new
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AUTHOR
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Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 10 2002
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EXTENSIONS
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Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 11 2002
Edited the equation 4n^2+4n+2=n^2+1 - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 16 2009
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