A047550 -id:A047550 - cp's OEIS frontend

A047578 Numbers that are congruent to {2, 5, 6, 7} mod 8.

2, 5, 6, 7, 10, 13, 14, 15, 18, 21, 22, 23, 26, 29, 30, 31, 34, 37, 38, 39, 42, 45, 46, 47, 50, 53, 54, 55, 58, 61, 62, 63, 66, 69, 70, 71, 74, 77, 78, 79, 82, 85, 86, 87, 90, 93, 94, 95, 98, 101, 102, 103, 106, 109, 110, 111, 114, 117, 118, 119, 122, 125

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A016825, A047404, A047431, A047546, A047550, A056594.

A047578:=n->2*n-cos(Pi*n/2): seq(A047578(n), n=1..100); # Wesley Ivan Hurt, Oct 22 2013

Flatten[#+{2,5,6,7}&/@(8Range[0,20])] (* Harvey P. Dale, Jan 26 2011 *)

[lucas_number1(n,0,1)+2*n+2 for n in range(0,56)] # Zerinvary Lajos, Jul 06 2008

G.f.: x*(1+x)*(x^2-x+2) / ((1+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = 2*n - cos(Pi*n/2). - Wesley Ivan Hurt, Oct 22 2013
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n > 4.
a(n) = (4*n - i^(-n) - i^n)/2 where i=sqrt(-1).
a(2n) = A047550(n), a(2n-1) = A016825(n-1). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/8 - log(2)/4. - Amiram Eldar, Dec 26 2021

More terms from Wesley Ivan Hurt, May 20 2016

A214863 Numbers n such that n XOR 11 = n - 11.

Original entry on oeis.org

11, 15, 27, 31, 43, 47, 59, 63, 75, 79, 91, 95, 107, 111, 123, 127, 139, 143, 155, 159, 171, 175, 187, 191, 203, 207, 219, 223, 235, 239, 251, 255, 267, 271, 283, 287, 299, 303, 315, 319, 331, 335, 347, 351, 363

Offset: 1

Brad Clardy, Mar 09 2013

Links to sequences of the form n XOR m = n - m are found below with the value of m specified.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A005408 (m=1), A042964 (m=2), A131098 (m=3), A047566 (m=4), A047550 (m=5), A047589 (m=6), A004771 (m=7), A115419 (m=8), A214865 (m=9), A214864 (m=10), A133894 (m=12), A125169 (m=15).

Cf. also A016825, A168392.

XOR := func;
m:=11;
for n in [1 .. 500] do
      if (XOR(n, m) eq n-m) then n; end if;
end for;

Select[Range[400],BitXor[#,11]==#-11&] (* or *) LinearRecurrence[{1,1,-1},{11,15,27},50] (* Harvey P. Dale, Jun 05 2021 *)

a(n)= 1+8*n-2*(-1)^n.
a(n)=A016825(n) + A168392(n) +  for n>0.
G.f. x*(11+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Mar 10 2013

A047575 Numbers that are congruent to {0, 5, 6, 7} mod 8.

Original entry on oeis.org

0, 5, 6, 7, 8, 13, 14, 15, 16, 21, 22, 23, 24, 29, 30, 31, 32, 37, 38, 39, 40, 45, 46, 47, 48, 53, 54, 55, 56, 61, 62, 63, 64, 69, 70, 71, 72, 77, 78, 79, 80, 85, 86, 87, 88, 93, 94, 95, 96, 101, 102, 103, 104, 109, 110, 111, 112, 117, 118, 119, 120, 125

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047451, A047550.

[n : n in [0..150] | n mod 8 in [0, 5, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

A047575:=n->(4*n-1+I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/2: seq(A047575(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Select[Range[0,120], MemberQ[{0,5,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Jun 30 2011 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x^2*(5+x+x^2+x^3) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (4*n-1+i^(2*n)-(1+i)*i^(-n)-(1-i)*i^n)/2 where i=sqrt(-1).
a(2k) = A047550(k), a(2k-1) = A047451(k). (End)
E.g.f.: 1 - sin(x) - cos(x) - sinh(x) + 2*x*exp(x). - Ilya Gutkovskiy, May 30 2016
Sum_{n>=2} (-1)^n/a(n) = 5*log(2)/8 - (2*sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 23 2021

A047576 Numbers that are congruent to {1, 5, 6, 7} mod 8.

Original entry on oeis.org

1, 5, 6, 7, 9, 13, 14, 15, 17, 21, 22, 23, 25, 29, 30, 31, 33, 37, 38, 39, 41, 45, 46, 47, 49, 53, 54, 55, 57, 61, 62, 63, 65, 69, 70, 71, 73, 77, 78, 79, 81, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 105, 109, 110, 111, 113, 117, 118, 119, 121, 125

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047452, A047550.

[n : n in [0..150] | n mod 8 in [1, 5, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

A047576:=n->(8*n-1+I^(2*n)-(2+I)*I^(-n)-(2-I)*I^n)/4: seq(A047576(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Flatten[#+{1,5,6,7}&/@(8Range[0,20])] (* Harvey P. Dale, Apr 22 2011 *)
Select[Range[100], MemberQ[{1, 5, 6, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, May 30 2016 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+4*x+x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-1+i^(2*n)-(2+i)*i^(-n)-(2-i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047550(k), a(2k-1) = A047452(k). (End)
E.g.f.: (2 - sin(x) - 2*cos(x) - sinh(x) + 4*x*exp(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*sqrt(2)*Pi/16 - (sqrt(2)+2)*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 24 2021

A047582 Numbers that are congruent to {3, 5, 6, 7} mod 8.

Original entry on oeis.org

3, 5, 6, 7, 11, 13, 14, 15, 19, 21, 22, 23, 27, 29, 30, 31, 35, 37, 38, 39, 43, 45, 46, 47, 51, 53, 54, 55, 59, 61, 62, 63, 67, 69, 70, 71, 75, 77, 78, 79, 83, 85, 86, 87, 91, 93, 94, 95, 99, 101, 102, 103, 107, 109, 110, 111, 115, 117, 118, 119, 123, 125

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047398, A047550.

[n : n in [0..150] | n mod 8 in [3, 5, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

A047582:=n->(8*n+1-I^(2*n)-(2-I)*I^(-n)-(2+I)*I^n)/4: seq(A047582(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Table[(8n+1-I^(2n)-(2-I)*I^(-n)-(2+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(3+2*x+x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n+1-i^(2*n)-(2-i)*i^(-n)-(2+i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047550(k), a(2k-1) = A047398(k). (End)
E.g.f.: (2 + sin(x) - 2*cos(x) + sinh(x) + 4*x*exp(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-2)*Pi/16 - log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 26 2021

A343007 Relative position of the average value between two consecutive partial sums of the Leibniz formula for Pi.

Original entry on oeis.org

6, 13, 26, 41, 62, 85, 114, 145, 182, 221, 266, 313, 366, 421, 482, 545, 614, 685, 762, 841, 926, 1013, 1106, 1201, 1302, 1405, 1514, 1625, 1742, 1861, 1986, 2113, 2246, 2381, 2522, 2665, 2814, 2965, 3122, 3281, 3446, 3613, 3786, 3961, 4142, 4325, 4514, 4705

Offset: 1

Raphael Ranna, Apr 02 2021

Define L(n) to be the n-th partial sum of the Leibniz formula Pi = 4 - 4/3 + 4/5 - 4/7 + ..., i.e., L(n) = Sum_{j=1..n} 4*(-1)^(j+1)/(2*j-1). For every positive integer n, L(n+1) is closer to Pi than L(n) is. If we let V be the average of the two consecutive partial sums L(n) and L(n+1), then the partial sums that lie closest to V are L(a(n)-1) and L(a(n)+1) (one of which is above V, the other below).

			The first several partial sums are as follows:
  n      L(n)
  -  ------------
  1  4.0000000000
  2  2.6666666...
  3  3.4666666...
  4  2.8952380...
  5  3.3396825...
  6  2.9760461...
  7  3.2837384...
  8  3.0170718...
.
For n=1, the average of the partial sums L(1) and L(2) is V = (L(1) + L(2))/2 = (4 + 2.6666666...)/2 = 3.3333333...; the two partial sums closest to V are L(5)=3.3396825... and L(7)=3.2837384..., and V lies in the interval between them, so a(1)=6.
The formula as it is written works for all data in the sequence, but it needs to be proven that it works for all possible integer values of n.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A047550, A102083, A254527.

Rest@ CoefficientList[Series[x (6 + x + x^3)/((1 + x) (1 - x)^3), {x, 0, 48}], x] (* Michael De Vlieger, Apr 05 2021 *)

a(1) = 6; a(n) = a(n-1) + r(n), where r(n) = A047550(n) = 4*n - (-1)^n.
G.f.: x*(6 + x + x^3)/((1 + x)*(1 - x)^3). - Jinyuan Wang, Apr 03 2021
From Stefano Spezia, Apr 03 2021: (Start)
a(n) = (3 + (-1)^(n+1) + 4*n + 4*n^2)/2.
a(2*n) = A102083(n).
a(2*n-1) = A254527(n). (End)

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A047578 Numbers that are congruent to {2, 5, 6, 7} mod 8.

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A214863 Numbers n such that n XOR 11 = n - 11.

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A047575 Numbers that are congruent to {0, 5, 6, 7} mod 8.

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A047576 Numbers that are congruent to {1, 5, 6, 7} mod 8.

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A047582 Numbers that are congruent to {3, 5, 6, 7} mod 8.

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A343007 Relative position of the average value between two consecutive partial sums of the Leibniz formula for Pi.

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