This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372276 #23 Jun 04 2025 00:25:21 %S A372276 9,4,9,1,0,7,9,1,2,3,4,2,7,5,8,5,2,4,5,2,6,1,8,9,6,8,4,0,4,7,8,5,1,2, %T A372276 6,2,4,0,0,7,7,0,9,3,7,6,7,0,6,1,7,7,8,3,5,4,8,7,6,9,1,0,3,9,1,3,0,6, %U A372276 3,3,3,0,3,5,4,8,4,0,1,4,0,8,0,5,7,3,0 %N A372276 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7. %H A372276 Paolo Xausa, <a href="/A372276/b372276.txt">Table of n, a(n) for n = 0..10000</a> %H A372276 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7. %H A372276 A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a> %H A372276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>. %H A372276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>. %H A372276 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>. %H A372276 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>. %F A372276 Largest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0. %e A372276 0.949107912342758524526189684047851262400770937670617783548769... %t A372276 First[RealDigits[Root[LegendreP[7, #] &, 7], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *) %o A372276 (PARI) solve (x = 0.8, 1.0, 429*x^6 - 693*x^4 + 315*x^ - 35) \\ _A.H.M. Smeets_, May 31 2025 %Y A372276 Cf. A008316, A100258. %Y A372276 There are floor(k/2) positive zeros of the Legendre polynomial of degree k: %Y A372276 k | zeros | corresponding weights for Legendre-Gauss quadrature %Y A372276 ---+---------------------------------+---------------------------------------------------- %Y A372276 2 | A020760 | A000007*10 %Y A372276 3 | A010513/10 | A010716 %Y A372276 4 | A372267, A372268 | A382103, A382104 %Y A372276 5 | A372269, A372270 | A382105, A382106 %Y A372276 6 | A372271, A372272, A372273 | A382107, A382686, A382687 %Y A372276 7 | A372274, A372275, this sequence | A382688, A382689, A382690 %K A372276 nonn,cons %O A372276 0,1 %A A372276 _Pontus von Brömssen_, Apr 25 2024