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A069894
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Centered square numbers: a(n) = 4*n^2 + 4*n + 2.
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18
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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
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OFFSET
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0,1
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COMMENTS
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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, Dec 21 2006
For n > 0, a(n-1) is the number of triples (w, x, y) having all terms in {0, ..., n} and min(|w - x|, |x - y|) = 1. - Clark Kimberling, Jun 12 2012
Consider the primitive Pythagorean triples (x(n), y(n), z(n) = y(n) + 1) with n >= 0, and x(n) = 2*n + 1, y(n) = 2*n*(n + 1), z(n) = 2*n*(n + 1) + 1. The sequence, a(n), is 2*z(n). - George F. Johnson, Oct 22 2012
Conference matrix orders (A000952) of the form n-1 is a perfect square are all in this sequence. All values less than 1000 are conference matrices except for 226 which is still an open question (Balonin & Seberry 2014). - Colin Hall, Nov 21 2018
For n > 0, a(n-1) is the number of maximum number of regions into which the plane can be divided using n convex quadrilaterals. Related: A077588 A077591. - Keyang Li, Jun 17 2022
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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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LINKS
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FORMULA
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(y*(2*x + 1))^2 + (y*(2*x^2 + 2*x))^2 = (y*(2*x^2 + 2*x + 1))^2, where y = 2. If a^2 + b^2 = c^2, then c^2 = y^2*(4*x^4 + 8*x^3 + 8*x^2 + 4*x + 1). Also 2*A001844.
G.f.: 2*(1 + x)^2/(1 - x)^3, a(0) = 2, a(1) = 10.
a(n+1) = a(n) + 4 + 4*sqrt(a(n) - 1).
a(n-1) * a(n+1) = (a(n)-4)^2 + 16.
a(n) - 1 = (2*n+1)^2 = A016754(n) for n > 0.
(a(n+1) - a(n-1))/8 = sqrt(a(n) - 1).
a(n+1) = 2*a(n) - a(n-1) + 8 for n > 2, a(0)=2, a(1)=10, a(2)=26.
a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2) for n > 3; a(0)=2, a(1)=10, a(2)=26, a(3)=50.
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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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MAPLE
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MATHEMATICA
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Table[4n(n + 1) + 2, {n, 0, 45}]
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PROG
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(PARI) vector(100, n, (2*n-1)^2+1); \\ Derek Orr, Jul 27 2014
(Sage) [(2*n+1)^2 + 1 for n in range(50)] # G. C. Greubel, Nov 21 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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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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Equation 4*n^2 + 4*n + 2 = n^2 + 1 edited by R. J. Mathar, Sep 16 2009
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STATUS
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approved
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