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 A372275 #24 Jun 04 2025 00:25:25 %S A372275 7,4,1,5,3,1,1,8,5,5,9,9,3,9,4,4,3,9,8,6,3,8,6,4,7,7,3,2,8,0,7,8,8,4, %T A372275 0,7,0,7,4,1,4,7,6,4,7,1,4,1,3,9,0,2,6,0,1,1,9,9,5,5,3,5,1,9,6,7,4,2, %U A372275 9,8,7,4,6,7,2,1,8,0,5,1,3,7,9,2,8,2,6 %N A372275 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7. %H A372275 Paolo Xausa, <a href="/A372275/b372275.txt">Table of n, a(n) for n = 0..10000</a> %H A372275 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 A372275 A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a> %H A372275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>. %H A372275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>. %H A372275 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>. %H A372275 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>. %F A372275 Middle positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0. %e A372275 0.741531185599394439863864773280788407074147647141390260119955... %t A372275 First[RealDigits[Root[LegendreP[7, #] &, 6], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *) %o A372275 (PARI) solve (x = 0.6, 0.8, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ _A.H.M. Smeets_, May 31 2025 %Y A372275 Cf. A008316, A100258. %Y A372275 There are floor(k/2) positive zeros of the Legendre polynomial of degree k: %Y A372275 k | zeros | corresponding weights for Legendre-Gauss quadrature %Y A372275 ---+---------------------------------+---------------------------------------------------- %Y A372275 2 | A020760 | A000007*10 %Y A372275 3 | A010513/10 | A010716 %Y A372275 4 | A372267, A372268 | A382103, A382104 %Y A372275 5 | A372269, A372270 | A382105, A382106 %Y A372275 6 | A372271, A372272, A372273 | A382107, A382686, A382687 %Y A372275 7 | A372274, this sequence, A372276 | A382688, A382689, A382690 %K A372275 nonn,cons %O A372275 0,1 %A A372275 _Pontus von Brömssen_, Apr 25 2024