A047617 -id:A047617 - cp's OEIS frontend

A047613 Numbers that are congruent to {1, 2, 4, 5} mod 8.

1, 2, 4, 5, 9, 10, 12, 13, 17, 18, 20, 21, 25, 26, 28, 29, 33, 34, 36, 37, 41, 42, 44, 45, 49, 50, 52, 53, 57, 58, 60, 61, 65, 66, 68, 69, 73, 74, 76, 77, 81, 82, 84, 85, 89, 90, 92, 93, 97, 98, 100, 101, 105, 106, 108, 109, 113, 114, 116, 117, 121, 122, 124

Offset: 1

N. J. A. Sloane

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

Cf. A047461, A047617.

[n: n in [1..120] | n mod 8 in [1,2,4,5]];

A047613:=n->2*n-2-(I^(2*n)+I^(n*(n+1)))/2: seq(A047613(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[120], MemberQ[{1, 2, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 4, 5, 9}, 60] (* Bruno Berselli, Jul 17 2012 *)

makelist(2*n-2-((-1)^n+%i^(n*(n+1)))/2,n,1,60);

Vec((1+x+2*x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

From Bruno Berselli, Jul 17 2012: (Start)
G.f.: x*(1+x+2*x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-2-((-1)^n+i^(n*(n+1)))/2, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047617(k), a(2k-1) = A047461(k). (End)
E.g.f.: (6 + sin(x) - cos(x) + (4*x - 3)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 + sqrt(2)*log(sqrt(2)+2)/4 - (sqrt(2)+1)*log(2)/8. - Amiram Eldar, Dec 23 2021

A047433 Numbers that are congruent to {2, 4, 5, 6} mod 8.

Original entry on oeis.org

2, 4, 5, 6, 10, 12, 13, 14, 18, 20, 21, 22, 26, 28, 29, 30, 34, 36, 37, 38, 42, 44, 45, 46, 50, 52, 53, 54, 58, 60, 61, 62, 66, 68, 69, 70, 74, 76, 77, 78, 82, 84, 85, 86, 90, 92, 93, 94, 98, 100, 101, 102, 106, 108, 109, 110, 114, 116, 117, 118, 122, 124

Offset: 1

N. J. A. Sloane

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

Cf. A047406, A047617.

[n : n in [0..150] | n mod 8 in [2, 4, 5, 6]]; // Wesley Ivan Hurt, May 26 2016

A047433:=n->(8*n-3-I^(2*n)-(2-I)*I^(-n)-(2+I)*I^n)/4: seq(A047433(n), n=1..100); # Wesley Ivan Hurt, May 26 2016

Select[Range[120], MemberQ[{2,4,5,6}, Mod[#,8]]&]  (* Harvey P. Dale, Mar 04 2011 *)

G.f.: x*(2+2*x+x^2+x^3+2*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
From Wesley Ivan Hurt, May 26 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-3-i^(2*n)-(2-i)*i^(-n)-(2+i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047406(k), a(2k-1) = A047617(k). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (3-sqrt(2))*Pi/16 + log(2)/4 + sqrt(2)*log(sqrt(2)-1)/8. - Amiram Eldar, Dec 25 2021

A047487 Numbers that are congruent to {2, 3, 5, 7} mod 8.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 15, 18, 19, 21, 23, 26, 27, 29, 31, 34, 35, 37, 39, 42, 43, 45, 47, 50, 51, 53, 55, 58, 59, 61, 63, 66, 67, 69, 71, 74, 75, 77, 79, 82, 83, 85, 87, 90, 91, 93, 95, 98, 99, 101, 103, 106, 107, 109, 111, 114, 115, 117, 119, 122, 123

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. A004767, A047617.

I:=[2, 3, 5, 7, 10]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 17 2012

A047487:=n->(-3-(-1)^n+I*(-I)^n-I*I^n+8*n)/4: seq(A047487(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,300], MemberQ[{2,3,5,7}, Mod[#,8]]&] (* Vincenzo Librandi, May 17 2012 *)

my(x='x+O('x^100)); Vec(x*(2+x+2*x^2+2*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

From Colin Barker, May 14 2012: (Start)
a(n) = (-3-(-1)^n+i*(-i)^n-i*i^n+8*n)/4 where i=sqrt(-1).
G.f.: x*(2+x+2*x^2+2*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 17 2012
a(2k) = A004767(k-1) for n>0, a(2k-1) = A047617(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (2 + sin(x) + (4*x - 1)*sinh(x) + (4*x - 2)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (4-sqrt(2))*Pi/16 - log(2)/8 + sqrt(2)*log(sqrt(2)-1)/8. - Amiram Eldar, Dec 25 2021

A047495 Numbers that are congruent to {2, 4, 5, 7} mod 8.

Original entry on oeis.org

2, 4, 5, 7, 10, 12, 13, 15, 18, 20, 21, 23, 26, 28, 29, 31, 34, 36, 37, 39, 42, 44, 45, 47, 50, 52, 53, 55, 58, 60, 61, 63, 66, 68, 69, 71, 74, 76, 77, 79, 82, 84, 85, 87, 90, 92, 93, 95, 98, 100, 101, 103, 106, 108, 109, 111, 114, 116, 117, 119, 122, 124

Offset: 1

N. J. A. Sloane

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

Cf. A047535, A047617.

[n : n in [0..150] | n mod 8 in [2, 4, 5, 7]]; // Wesley Ivan Hurt, May 27 2016

A047495:=n->(1+I)*(4*n-4*n*I+I-1+I^(1-n)-I^n)/4: seq(A047495(n), n=1..100); # Wesley Ivan Hurt, May 27 2016

Table[(1+I)*(4n-4n*I+I-1+I^(1-n)-I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *)

G.f.: x*(2+x^2+x^3) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Nov 06 2015
From Wesley Ivan Hurt, May 27 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (1+i)*(4*n-4*n*i+i-1+i^(1-n)-i^n)/4 where i=sqrt(-1).
a(2k) = A047535(k), a(2k-1) = A047617(k). (End)
E.g.f.: (2 + sin(x) - cos(x) + (4*x - 1)*exp(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 - (sqrt(2)-1)*log(2)/8 + sqrt(2)*log(2-sqrt(2))/4. - Amiram Eldar, Dec 25 2021

A047605 Numbers that are congruent to {0, 2, 3, 5} mod 8.

Original entry on oeis.org

0, 2, 3, 5, 8, 10, 11, 13, 16, 18, 19, 21, 24, 26, 27, 29, 32, 34, 35, 37, 40, 42, 43, 45, 48, 50, 51, 53, 56, 58, 59, 61, 64, 66, 67, 69, 72, 74, 75, 77, 80, 82, 83, 85, 88, 90, 91, 93, 96, 98, 99, 101, 104, 106, 107, 109, 112, 114, 115, 117, 120, 122, 123

Offset: 1

N. J. A. Sloane

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

Cf. A047470, A047617.

[n : n in [0..150] | n mod 8 in [0, 2, 3, 5]]; // Wesley Ivan Hurt, Jun 04 2016

A047605:=n->2*(n-1)-(I^(n*(n+1))+1)/2: seq(A047605(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016

Table[(1+I)*((4-4*I)*n+5*I-5+I^(1-n)-I^n)/4, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
Flatten[#+{0,2,3,5}&/@(8*Range[0,20])] (* or *) LinearRecurrence[{2,-2,2,-1},{0,2,3,5},100] (* Harvey P. Dale, Sep 30 2018 *)

a(n)=n\4*8+[-3,0,2,3][n%4+1] \\ Charles R Greathouse IV, Dec 05 2011

From Bruno Berselli, Dec 05 2011: (Start)
G.f.: x^2*(2-x+3*x^2)/((1-x)^2*(1+x^2)).
a(n) = 2*(n-1)-(i^(n*(n+1))+1)/2, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(2k) = A047617(k), a(2k-1) = A047470(k). (End)
E.g.f.: (6 + sin(x) - cos(x) + (4*x - 5)*exp(x))/2. - Ilya Gutkovskiy, Jun 05 2016
Sum_{n>=2} (-1)^n/a(n) = (3-2*sqrt(2))*Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 21 2021

A047606 Numbers that are congruent to {1, 2, 3, 5} mod 8.

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 11, 13, 17, 18, 19, 21, 25, 26, 27, 29, 33, 34, 35, 37, 41, 42, 43, 45, 49, 50, 51, 53, 57, 58, 59, 61, 65, 66, 67, 69, 73, 74, 75, 77, 81, 82, 83, 85, 89, 90, 91, 93, 97, 98, 99, 101, 105, 106, 107, 109, 113, 114, 115, 117, 121, 122, 123

Offset: 1

N. J. A. Sloane

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

Cf. A047471, A047617.

[n: n in [1..120] | n mod 8 in [1,2,3,5]];

A047606:=n->2*n-3+(3-I^(2*n))*(1-I^(n*(n+1)))/4: seq(A047606(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[120], MemberQ[{1, 2, 3, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 5, 9}, 60] (* Bruno Berselli, Jul 17 2012 *)

makelist(2*n-3+(3-(-1)^n)*(1-%i^(n*(n+1)))/4,n,1,60);

Vec((1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

From Bruno Berselli, Jul 17 2012: (Start)
G.f.: x*(1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-3+(3-(-1)^n)*(1-i^(n*(n+1)))/4, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047617(k), a(2k-1) = A047471(k). (End)
E.g.f.: (6 + 2*sin(x) - cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-2)*Pi/16 + (2-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 23 2021

A047612 Numbers that are congruent to {0, 2, 4, 5} mod 8.

Original entry on oeis.org

0, 2, 4, 5, 8, 10, 12, 13, 16, 18, 20, 21, 24, 26, 28, 29, 32, 34, 36, 37, 40, 42, 44, 45, 48, 50, 52, 53, 56, 58, 60, 61, 64, 66, 68, 69, 72, 74, 76, 77, 80, 82, 84, 85, 88, 90, 92, 93, 96, 98, 100, 101, 104, 106, 108, 109, 112, 114, 116, 117, 120, 122, 124

Offset: 1

N. J. A. Sloane

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

Cf. A008586, A047617.

[n: n in [0..120] | n mod 8 in [0,2,4,5]];

A047612:=n->2*n-2-(1+I^(2*n))*(1+I^n)/4: seq(A047612(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[0,120], MemberQ[{0, 2, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {0, 2, 4, 5, 8}, 60] (* Bruno Berselli, Jul 18 2012 *)

makelist(2*n-2-(1+(-1)^n)*(1+%i^n)/4,n,1,60);

concat(0, Vec((2+2*x+x^2+3*x^3)/((1+x)*(1-x)^2*(1+x^2))+O(x^60))) (End)

From Bruno Berselli, Jul 18 2012: (Start)
G.f.: x^2*(2+2*x+x^2+3*x^3)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-2-(1+(-1)^n)*(1+i^n)/4, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047617(k), a(2k-1) = A008586(k-1) for k>0. (End)
E.g.f.: (6 - cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016
Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + 5*log(2)/8 + sqrt(2)*log(sqrt(2)-1)/8. - Amiram Eldar, Dec 21 2021

A322261 Square array T(n, k) (n >= 0, k >= 0) read by antidiagonals upwards: the lengths of runs in binary expansion of T(n, k) correspond to the lengths of runs in binary expansion of n followed by the lengths of runs in binary expansion of k.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 5, 5, 3, 4, 6, 10, 4, 4, 5, 9, 13, 11, 11, 5, 6, 10, 18, 12, 20, 10, 6, 7, 13, 21, 19, 27, 21, 9, 7, 8, 14, 26, 20, 36, 26, 22, 8, 8, 9, 17, 29, 27, 43, 37, 25, 23, 23, 9, 10, 18, 34, 28, 52, 42, 38, 24, 40, 22, 10, 11, 21, 37, 35, 59, 53

Offset: 0

Rémy Sigrist, Dec 01 2018

The array T is associative.

			Array T(n, k) begins (in decimal):
  n\k|  0   1   2   3   4   5   6   7    8    9   10   11   12
  ---+--------------------------------------------------------
    0|  0   1   2   3   4   5   6   7    8    9   10   11   12
    1|  1   2   5   4  11  10   9   8   23   22   21   20   19
    2|  2   5  10  11  20  21  22  23   40   41   42   43   44
    3|  3   6  13  12  27  26  25  24   55   54   53   52   51
    4|  4   9  18  19  36  37  38  39   72   73   74   75   76
    5|  5  10  21  20  43  42  41  40   87   86   85   84   83
    6|  6  13  26  27  52  53  54  55  104  105  106  107  108
    7|  7  14  29  28  59  58  57  56  119  118  117  116  115
    8|  8  17  34  35  68  69  70  71  136  137  138  139  140
Array T(n, k) begins (in binary):
  n\k |     0      1      10      11      100      101      110      111      1000
  ----+---------------------------------------------------------------------------
     0|     0      1      10      11      100      101      110      111      1000
     1|     1     10     101     100     1011     1010     1001     1000     10111
    10|    10    101    1010    1011    10100    10101    10110    10111    101000
    11|    11    110    1101    1100    11011    11010    11001    11000    110111
   100|   100   1001   10010   10011   100100   100101   100110   100111   1001000
   101|   101   1010   10101   10100   101011   101010   101001   101000   1010111
   110|   110   1101   11010   11011   110100   110101   110110   110111   1101000
   111|   111   1110   11101   11100   111011   111010   111001   111000   1110111
  1000|  1000  10001  100010  100011  1000100  1000101  1000110  1000111  10001000

Index entries for sequences related to binary expansion of n

Cf. A004757, A005811, A010078, A020330, A042963, A047457, A047617, A101211, A163621.

torl(n) = my (r=[]); while (n, r = concat(valuation(n+(n%2),2), r); n \= 2^r[1];); r
fromrl(r) = my (v=0); for (i=1, #r, v = (v + (i%2))*2^r[i]-(i%2)); v
T(n,k) = fromrl(concat(torl(n), torl(k)))

T(n, 0) = T(0, n) = n.
T(n, 1) = A042963(n+1).
T(n, 2) = A047617(n+1).
T(n, 3) = A047457(n+1).
T(1, n) = A010078(n+1).
T(2, n) = A004757(n) for any n > 0.
A005811(T(n, k)) = A005811(n) + A005811(k).
T(2*n, k) = A163621(2*n, k) for any n > 0 and k > 0.
T(2*n, 2*n) = A020330(2*n) for any n > 0.

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A047613 Numbers that are congruent to {1, 2, 4, 5} mod 8.

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A047433 Numbers that are congruent to {2, 4, 5, 6} mod 8.

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A047487 Numbers that are congruent to {2, 3, 5, 7} mod 8.

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A047495 Numbers that are congruent to {2, 4, 5, 7} mod 8.

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A047605 Numbers that are congruent to {0, 2, 3, 5} mod 8.

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A047606 Numbers that are congruent to {1, 2, 3, 5} mod 8.

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A047612 Numbers that are congruent to {0, 2, 4, 5} mod 8.

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