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
A diagonal in Ulam's spiral (SE corner). See the Wikipedia link. - Kival Ngaokrajang, Jul 25 2014
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
David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.
Iva Kodrnja, On Remarkable Properties of Number 2025, KoG, 29 (29), 74-80, 2025. See p. 79.
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.
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
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
(SageMath) [(2*n+1)^2 + 1 for n in range(50)] # G. C. Greubel, Nov 21 2018
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 10 2002
EXTENSIONS
Edited by Robert G. Wilson v, Apr 11 2002
Offset corrected by Charles R Greathouse IV, Jul 25 2010
STATUS
approved