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 A372274 #26 Jun 04 2025 00:25:32 %S A372274 4,0,5,8,4,5,1,5,1,3,7,7,3,9,7,1,6,6,9,0,6,6,0,6,4,1,2,0,7,6,9,6,1,4, %T A372274 6,3,3,4,7,3,8,2,0,1,4,0,9,9,3,7,0,1,2,6,3,8,7,0,4,3,2,5,1,7,9,4,6,6, %U A372274 3,8,1,3,2,2,6,1,2,5,6,5,5,3,2,8,3,1,2 %N A372274 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7. %H A372274 Paolo Xausa, <a href="/A372274/b372274.txt">Table of n, a(n) for n = 0..10000</a> %H A372274 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 A372274 A.H.M. Smeets, <a href="/A382103/a382103.py.txt">Python program for Legendre-Gauss quadrature constants.</a> %H A372274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>. %H A372274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss Quadrature</a>. %H A372274 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>. %H A372274 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>. %F A372274 Smallest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0. %e A372274 0.405845151377397166906606412076961463347382014099370126387043... %t A372274 First[RealDigits[Root[LegendreP[7, #] &, 5], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *) %o A372274 (PARI) solve (x = 0.1, 0.5, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ _A.H.M. Smeets_, May 31 2025 %Y A372274 Cf. A008316, A100258. %Y A372274 There are floor(k/2) positive zeros of the Legendre polynomial of degree k: %Y A372274 k | zeros | corresponding weights for Legendre-Gauss quadrature %Y A372274 ---+---------------------------------+---------------------------------------------------- %Y A372274 2 | A020760 | A000007*10 %Y A372274 3 | A010513/10 | A010716 %Y A372274 4 | A372267, A372268 | A382103, A382104 %Y A372274 5 | A372269, A372270 | A382105, A382106 %Y A372274 6 | A372271, A372272, A372273 | A382107, A382686, A382687 %Y A372274 7 | this sequence, A372275, A372276 | A382688, A382689, A382690 %K A372274 nonn,cons %O A372274 0,1 %A A372274 _Pontus von Brömssen_, Apr 25 2024