A032125 -id:A032125 - cp's OEIS frontend

A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

2, 9, 35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135, 140737496743935, 562949970198527

Offset: 1

Reinhard Zumkeller, Mar 23 2004

Smallest numbers having in binary representation n 0's and n 1's: a(n) = Min{m: A023416(m)=A000120(m)=n}.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Binary
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007088, A001700.
Subsequence of A031443.
Cf. A000120, A023416.
Cf. A007582, A056326, A005367, A063376, A032125, A056309, A028403, A001576, A028400, A049775.

LinearRecurrence[{7, -14, 8}, {2, 9, 35}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
Table[FromDigits[Join[PadRight[{1},n,0],PadRight[{},n-2,1]],2],{n,2,30}]//Sort (* or *) Rest[CoefficientList[Series[x (-2+5x)/((x-1)(2x-1)(4x-1)),{x,0,30}],x]] (* Harvey P. Dale, Jul 30 2021 *)

a(n+1) = 2*a(n) + 4^n + 1.
a(n) = 2^(2*n-1) + 2^(n-1) - 1.
a(n) = A007582(n)-1 = A056326(2n+1) = A005367(n-1)/2 = A063376(n)/2-1 = A032125(n+1)/3-1 = A056309(2n+1)/2 = A028403(n+1)/4-1 = (A001576(n)-3)/2 = (A028400(n+1)-9)/8 = Sum_{k=2..n+1} A049775(k). - Ralf Stephan, Mar 24 2004
G.f.: x*(-2+5*x) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
E.g.f.: exp(x)*(exp(3*x) + exp(x) - 2)/2. - Stefano Spezia, Sep 27 2023

A048240 Number of new colors that can be mixed with n units of yellow, blue, red.

Original entry on oeis.org

1, 3, 3, 7, 9, 18, 15, 33, 30, 45, 42, 75, 54, 102, 81, 108, 108, 168, 117, 207, 156, 210, 195, 297, 204, 330, 270, 351, 306, 462, 300, 525, 408, 510, 456, 612, 450, 738, 567, 708, 600, 900, 594, 987, 750, 900, 825, 1173, 792, 1239, 930, 1200

Offset: 0

Jurjen N.E. Bos, N. J. A. Sloane, Robin Trew (trew(AT)hcs.harvard.edu)

T. D. Noe, Table of n, a(n) for n = 0..1000
Vadym Kurylenko, Thin Simplices via Modular Arithmetic, arXiv:2404.03975 [math.CO], 2024. See p. 33.

Cf. A048134, A048241.
A032125(n) = a(2^n).
Cf. A000217, A008683, A000741, A023023.

A048240 := proc(n) local ans, i, j, k; ans := 0; for i from n by -1 to 0 do for j from n by -1 to 0 do k := n - i - j; if 0 <= k and k <= n and gcd(gcd(i, j), k) = 1 then ans := ans + 1; fi; od; od; RETURN(ans); end;

a[n_] := Sum[ MoebiusMu[n/d]*(d+1)*(d+2)/2, {d, Divisors[n]}]; a[0] = 1; Table[a[n], {n, 0, 51}] (* Jean-François Alcover, Jun 14 2012, after Vladeta Jovovic *)

a(n) = number of triples (i, j, k) with i+j+k = n and gcd(i, j, k) = 1.

a(n) = Sum_{d|n} mu(n/d)*(d+1)*(d+2)/2. G.f.: Sum_{k>0} mu(k)/(1-x^k)^3. - Vladeta Jovovic, Dec 22 2002

cp's OEIS Frontend

A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

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