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 A198414 Decimal expansion of x>0 satisfying x^2=2*sin(x). 107
 1, 4, 0, 4, 4, 1, 4, 8, 2, 4, 0, 9, 2, 4, 3, 4, 3, 6, 4, 1, 4, 8, 3, 2, 7, 9, 4, 3, 7, 4, 5, 7, 5, 8, 6, 0, 3, 7, 2, 5, 7, 1, 6, 1, 3, 7, 0, 4, 9, 1, 1, 4, 8, 1, 0, 9, 4, 4, 8, 2, 4, 3, 5, 4, 8, 7, 7, 5, 2, 5, 2, 9, 5, 6, 1, 7, 1, 4, 4, 3, 6, 2, 1, 2, 0, 5, 1, 0, 1, 5, 2, 4, 8, 2, 0, 8, 1, 7, 5 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For many choices of a,b,c, there is a unique nonzero number x satisfying a*x^2+b*x=c*sin(x). Specifically, for a>0 and many choices of b and c, the curves y=ax^2+bx and y=c*sin(x) meet in a single point if and only if b=c, in which case the curves have a common tangent line, y=c*x.  If bc, they meet in quadrant 2. Guide to related sequences (with graphs included in Mathematica programs): a.....b.....c.....x 1.....0.....1.....A124597 1.....0.....2.....A198414 1.....0.....3.....A198415 1.....0.....4.....A198416 1.....1.....2.....A198417 1.....1.....3.....A197418 1.....1.....4.....A197419 1.....2.....1.....A197424 1.....2.....3.....A197425 1.....2.....4.....A197426 1....-1.....1.....A197420 1....-1.....1.....A197420 1....-1.....2.....A197421 1....-1.....3.....A197422 1....-2.....1.....A198427 1....-2.....2.....A198428 1....-2.....3.....A198429 1....-2.....4.....A198430 1....-3.....1.....A198431 1....-3.....2.....A198432 1....-3.....3.....A198433 1....-3.....4.....A198488 1....-4.....1.....A198489 1....-4.....2.....A198490 1....-4.....3.....A198491 1....-4.....4.....A198492 2.....0.....1.....A198583 2.....0.....3.....A198605 2.....1.....2.....A198493 2.....1.....3.....A198494 2.....1.....4.....A198495 2.....2.....1.....A198496 2.....2.....3.....A198497 2.....3.....1.....A198608 2.....3.....2.....A198609 2.....3.....4.....A198610 2.....4.....1.....A198611 2.....4.....3.....A198612 2....-1.....1.....A198546 2....-1.....2.....A198547 2....-1.....3.....A198548 2....-1.....4.....A198549 2....-2.....3.....A198559 2....-3.....1.....A198566 2....-3.....2.....A198567 2....-3.....3.....A198568 2....-3.....4.....A198569 2....-4.....1.....A198577 2....-4.....3.....A198578 3.....0.....1.....A198501 3.....0.....2.....A198502 3.....1.....2.....A198498 3.....1.....3.....A198499 3.....1.....4.....A198500 3.....2.....1.....A198613 3.....2.....3.....A198614 3.....2.....4.....A198615 3.....3.....1.....A198616 3.....3.....2.....A198617 3.....3.....4.....A198618 3.....4.....1.....A198606 3.....4.....2.....A198607 3.....4.....3.....A198619 3....-1.....1.....A198550 3....-1.....2.....A198551 3....-1.....3.....A198552 3....-1.....4.....A198553 3....-2.....1.....A198560 3....-2.....2.....A198561 3....-2.....3.....A198562 3....-2.....4.....A198563 3....-3.....1.....A198570 3....-3.....2.....A198571 3....-3.....4.....A198572 3....-4.....1.....A198579 3....-4.....2.....A198580 3....-4.....3.....A198581 3....-4.....4.....A198582 4.....0.....1.....A198503 4.....0.....3.....A198504 4.....1.....2.....A198505 4.....1.....3.....A198506 4.....1.....4.....A198507 4.....2.....1.....A198539 4.....2.....3.....A198540 4.....3.....1.....A198541 4.....3.....2.....A198542 4.....3.....4.....A198543 4.....4.....1.....A198544 4.....4.....3.....A198545 4....-1.....1.....A198554 4....-1.....2.....A198555 4....-1.....3.....A198556 4....-1.....4.....A198557 4....-1.....1.....A198554 4....-2.....1.....A198564 4....-2.....3.....A198565 4....-3.....1.....A198573 4....-3.....2.....A198574 4....-3.....3.....A198575 4....-3.....4.....A198576 Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f. For an example related to A198414, take f(x,u,v)=x^2+u*x-v*sin(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section. LINKS EXAMPLE x=1.4044148240924343641483279437457586037... MATHEMATICA (* Program 1: A198414 *) a = 1; b = 0; c = 2; f[x_] := a*x^2 + b*x; g[x_] := c*Sin[x] Plot[{f[x], g[x]}, {x, -1, 2}] r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.41}, WorkingPrecision -> 110] RealDigits[r] (* A198414 *) (* Program 2: an implicit surface of x^2+u*x=v*sin(x) *) f[{x_, u_, v_}] := x^2 + u*x - v*Sin[x]; t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .01, 6}]}, {u, .1, 100}, {v, u, 100}]; ListPlot3D[Flatten[t, 1]] CROSSREFS Cf. A197737. Sequence in context: A062524 A152856 A031362 * A110854 A278086 A021716 Adjacent sequences:  A198411 A198412 A198413 * A198415 A198416 A198417 KEYWORD nonn,cons AUTHOR Clark Kimberling, Oct 24 2011 STATUS approved

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Last modified August 24 02:32 EDT 2019. Contains 326260 sequences. (Running on oeis4.)