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 A069894 Centered square numbers: a(n) = 4*n^2 + 4*n + 2. 18
 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 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 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 and Jennifer Seberry, A Review and New Symmetric Conference Matrices, Research Online, Faculty of Engineering and Information Sciences, University of Wollongong, 2014. Keyang Li, figure for n=1,2,3,4,5 Tintarn, n convex quadrilaterals in the plane 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 a(n) = A261327(4*n+2). - Paul Curtz, Dec 23 2021 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 25 01:35 EDT 2024. Contains 371964 sequences. (Running on oeis4.)