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 A069894 Centered square numbers: a(n) = 4*n^2 + 4*n + 2. 13
 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; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 Ulam's spiral (SE corner). See the Wikipedia link. - Kival Ngaokrajang, Jul 25 2014 Conference matrix orders (A000952) of the form n - 1 is a perfect square are all in this series. 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 REFERENCES L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176. LINKS Ivan Panchenko, Table of n, a(n) for n = 0..1000 N. A. Balonin, Jennifer Seberry, A Review and New Symmetric Conference Matrices, Research Online, Faculty of Engineering and Information Sciences, University of Wollongong, 2014. Wikipedia, Ulam Spiral Construction. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA (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. a(n) = (2*n + 1)^2 + 1. - Vladimir Joseph Stephan Orlovsky, Nov 10 2008 [Corrected by R. J. Mathar, Sep 16 2009] a(n) = 8*n + a(n-1) for n > 0, a(0)=2. - Vincenzo Librandi, Aug 08 2010 From George F. Johnson, Oct 22 2012: (Start) 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. a(n) = A033996(n) + 2 = A002522(2n + 1). a(n)^2 = A033996(n)^2 + A016825(n)^2. (End) a(n) = A001105(n) + A001105(n+1). - Bruno Berselli, Jul 03 2017 E.g.f.: 2*(1 + 4*x + 2*x^2)*exp(x). - G. C. Greubel, Nov 21 2018 EXAMPLE 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. MAPLE A069894:=n->4*n^2+4*n+2: seq(A069894(n), n=0..50); # Wesley Ivan Hurt, Jul 26 2014 MATHEMATICA Table[4n(n + 1) + 2, {n, 0, 45}] PROG (MAGMA) [4*n^2+4*n+2 : n in [0..50]]; // Wesley Ivan Hurt, Jul 26 2014 (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 CROSSREFS Cf. A001844. Sequence in context: A167386 A027719 A254709 * A045605 A294871 A212969 Adjacent sequences:  A069891 A069892 A069893 * A069895 A069896 A069897 KEYWORD nonn,easy AUTHOR Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 10 2002 EXTENSIONS Edited by Robert G. Wilson v, Apr 11 2002 Equation 4*n^2 + 4*n + 2 = n^2 + 1 edited by R. J. Mathar, Sep 16 2009 Offset corrected by Charles R Greathouse IV, Jul 25 2010 STATUS approved

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Last modified April 22 12:53 EDT 2021. Contains 343177 sequences. (Running on oeis4.)