A198680 -id:A198680 - cp's OEIS frontend

A198679 Decimal expansion of the absolute minimum of sin(x)+sin(2x)+sin(3x).

2, 4, 9, 9, 6, 0, 7, 6, 0, 4, 3, 2, 0, 0, 5, 6, 6, 0, 4, 3, 6, 8, 1, 2, 6, 2, 8, 5, 6, 1, 8, 0, 4, 3, 2, 1, 8, 2, 4, 3, 7, 2, 3, 8, 5, 6, 7, 4, 0, 0, 3, 1, 7, 2, 2, 7, 3, 3, 7, 1, 4, 1, 8, 1, 3, 5, 8, 0, 1, 7, 5, 3, 0, 5, 1, 8, 0, 6, 0, 8, 8, 8, 8, 1, 9, 0, 5, 6, 6, 5, 4, 4, 7, 1, 4, 7, 5, 3, 8

Offset: 1

Clark Kimberling, Oct 29 2011

See A198677 for a guide to related sequences.

			x=5.615894235655113730430510613029770525768423...
min=-2.499607604320056604368126285618043218243...

Cf. A198677.

n = 3; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198679 *)
RealDigits[v] (* A198680 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
Root[-108 + 2124x^2 - 4892x^4 + 729x^6, 1] // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)

A198681 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.

Original entry on oeis.org

3, 9, 24, 27, 42, 48, 60, 66, 72, 81, 96, 102, 114, 120, 126, 138, 144, 159, 168, 174, 180, 192, 198, 213, 216, 231, 237, 243, 258, 264, 276, 282, 288, 300, 306, 321, 330, 336, 342, 354, 360, 375, 378, 393, 399, 408, 414, 429, 432, 447, 453, 465, 471, 477, 492, 498

Offset: 1

John W. Layman, Oct 28 2011

It appears that Sum[k^j, 0<=k<=2^n-1, k in A198680] = Sum[k^j, 0<=k<=2^n-1, k in A198681] = Sum[k^j, 0<=k<=2^n-1, k in A180682], for 0<=j<=n-1, which has been verified numerically in a number of cases. This is a generalization of Prouhet's Theorem (see the reference). To illustrate for j=3, we have Sum[k^3, 0<=k<=2^n-1, k in A198680] = {0,0,12636,1108809,94478400,7780827681,633724260624,51425722195929,4168024588857600,...}, Sum[k^3, 0<=k<=2^n-1, k in A198681] = {0,27,14580,1095687,94478400,7780827681,633724260624,51425722195929,4168024588857600,..., Sum[k^3, 0<=k<=2^n-1, k in A198682] = {0,216,7776,1121931,94478400,7780827681,633724260624,51425722195929,4168024588857600,...}, and it is seen that all three sums agree for n>=4=j+1.

Michel Marcus, Table of n, a(n) for n = 1..1001 (after Harvey P. Dale).
Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag. 82 (2009), pp. 57-62.
Eric Weisstein's World of Mathematics, Prouhet-Tarry-Escott Problem

Cf. A000069, A001969, A157971, A158704, A198680, A198682.

Select[3Range[200],IntegerQ[(Total[IntegerDigits[#,3]]-1)/3]&] (* Harvey P. Dale, Feb 05 2012 *)

Offset corrected by Michel Marcus, Mar 02 2016

A198682 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3*k+2.

Original entry on oeis.org

6, 12, 18, 30, 36, 51, 54, 69, 75, 84, 90, 105, 108, 123, 129, 141, 147, 153, 162, 177, 183, 195, 201, 207, 219, 225, 240, 246, 252, 267, 270, 285, 291, 303, 309, 315, 324, 339, 345, 357, 363, 369, 381, 387, 402, 411, 417, 423, 435, 441, 456, 459, 474, 480, 486

Offset: 1

John W. Layman, Oct 28 2011

It appears that Sum[k^j, 0<=k<=2^n-1, k in A198680] = Sum[k^j, 0<=k<=2^n-1, k in A198681] = Sum[k^j, 0<=k<=2^n-1, k in A180682], for 0<=j<=n-1, which has been verified numerically in a number of cases. This is a generalization of Prouhet's Theorem (see the reference). To illustrate for j=3, we have Sum[k^3, 0<=k<=2^n-1, k in A198680] = {0, 0, 12636, 1108809, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,...}, Sum[k^3, 0<=k<=2^n-1, k in A198681] = {0, 27, 14580, 1095687, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,..., Sum[k^3, 0<=k<=2^n-1, k in A198682] = {0, 216, 7776, 1121931, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,...}, and it is seen that all three sums agree for n>=4=j+1.

For each m, the sequence contains exactly one of 9*m, 9*m+3, 9*m+6. - Robert Israel, Mar 04 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag. 82 (2009), pp. 57-62.
Eric Weisstein's World of Mathematics, Prouhet-Tarry-Escott Problem

Cf. A000069, A001969, A157971, A158704, A198680, A198681.

select(t -> convert(convert(t,base,3),`+`) mod 3 = 2, [seq(3*i,i=1..1000)]); # Robert Israel, Mar 04 2016

Select[Range[3, 498, 3], IntegerQ[(-2 + Plus@@IntegerDigits[#, 3])/3] &] (* Alonso del Arte, Nov 02 2011 *)

isok(n) = !(n % 3) && !((vecsum(digits(n, 3)) - 2) % 3); \\ Michel Marcus, Mar 02 2016

Offset corrected by Michel Marcus, Mar 02 2016

A198728 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x)+sin(3x).

Original entry on oeis.org

5, 6, 1, 5, 8, 9, 4, 2, 3, 5, 6, 5, 5, 1, 1, 3, 7, 3, 0, 4, 3, 0, 5, 1, 0, 6, 1, 3, 0, 2, 9, 7, 7, 0, 5, 2, 5, 7, 6, 8, 4, 2, 3, 2, 5, 4, 2, 6, 3, 1, 8, 2, 4, 8, 9, 8, 0, 3, 8, 4, 3, 5, 2, 7, 4, 2, 3, 0, 7, 3, 8, 9, 7, 4, 8, 5, 0, 8, 7, 6, 9, 5, 9, 2, 4, 5, 1, 2, 5, 7, 6, 5, 1, 3, 7, 6, 3, 8, 5

Offset: 1

Clark Kimberling, Oct 29 2011

See A198677 for a guide to related sequences.

			x=5.615894235655113730430510613029770525768423...
min=-2.499607604320056604368126285618043218243...

Cf. A198677.

n = 3; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198679 *)
RealDigits[v] (* A198680 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
RealDigits[3*Pi/2 + ArcSin[Root[-1 - 4x + 2x^2 + 6x^3, 3]], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)

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A198679 Decimal expansion of the absolute minimum of sin(x)+sin(2x)+sin(3x).

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A198681 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.

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Mathematica

Extensions

A198682 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3*k+2.

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Maple

Mathematica

PARI

Extensions

A198728 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x)+sin(3x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica