cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A047482 Numbers that are congruent to {1, 2, 5, 7} mod 8.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 13, 15, 17, 18, 21, 23, 25, 26, 29, 31, 33, 34, 37, 39, 41, 42, 45, 47, 49, 50, 53, 55, 57, 58, 61, 63, 65, 66, 69, 71, 73, 74, 77, 79, 81, 82, 85, 87, 89, 90, 93, 95, 97, 98, 101, 103, 105, 106, 109, 111, 113, 114, 117, 119, 121, 122, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016813, A047524.

Programs

  • Magma
    I:=[1, 2, 5, 7, 9]; [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
  • Maple
    A047482:=n->(-5-(-1)^n+(-I)^n+I^n+8*n)/4: seq(A047482(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,300], MemberQ[{1,2,5,7}, Mod[#,8]]&] (* Vincenzo Librandi, May 17 2012 *)
LinearRecurrence[{1,0,0,1,-1},{1, 2, 5, 7, 9},100] (* G. C. Greubel, Jun 01 2016 *)
  • PARI
    my(x='x+O('x^100)); Vec(x*(1+x+3*x^2+2*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

Formula

From Colin Barker, May 14 2012: (Start)
a(n) = (-5 - (-1)^n + (-i)^n + i^n + 8*n)/4 where i=sqrt(-1).
G.f.: x*(1 + x + 3*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) = A047524(k), a(2k-1) = A016813(k-1) for k>0. - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (2 + cos(x) + (4*x - 2)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+2)*Pi/16 + log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 24 2021

A047541 Numbers that are congruent to {1, 2, 4, 7} mod 8.

Original entry on oeis.org

1, 2, 4, 7, 9, 10, 12, 15, 17, 18, 20, 23, 25, 26, 28, 31, 33, 34, 36, 39, 41, 42, 44, 47, 49, 50, 52, 55, 57, 58, 60, 63, 65, 66, 68, 71, 73, 74, 76, 79, 81, 82, 84, 87, 89, 90, 92, 95, 97, 98, 100, 103, 105, 106, 108, 111, 113, 114, 116, 119, 121, 122, 124
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047461, A047524.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [1, 2, 4, 7]]; // Wesley Ivan Hurt, Jun 04 2016
  • Maple
    A047541:=n->(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4: seq(A047541(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016
  • Mathematica
    Table[(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
Select[Range[200],MemberQ[{1,2,4,7},Mod[#,8]]&] (* or  *) LinearRecurrence[ {2,-2,2,-1},{1,2,4,7},70] (* Harvey P. Dale, Jul 09 2020 *)
  • PARI
    a(n)=n\4*8+[-1,1,2,4][n%4+1] \\ Charles R Greathouse IV, Nov 04 2011

Formula

From Wesley Ivan Hurt, Jun 04 2016: (Start)
G.f.: x*(1+2*x^2+x^3)/(x-1)^2*(1+x^2).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (1+i)*(n*(4-4*i)+3*i-3+i^(-n)-i^(1+n))/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047461(k). (End)
E.g.f.: (2 + sin(x) + cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, Jun 04 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)+1)*Pi/16 - log(2)/8. - Amiram Eldar, Dec 24 2021

A047554 Numbers that are congruent to {1, 2, 6, 7} mod 8.

Original entry on oeis.org

1, 2, 6, 7, 9, 10, 14, 15, 17, 18, 22, 23, 25, 26, 30, 31, 33, 34, 38, 39, 41, 42, 46, 47, 49, 50, 54, 55, 57, 58, 62, 63, 65, 66, 70, 71, 73, 74, 78, 79, 81, 82, 86, 87, 89, 90, 94, 95, 97, 98, 102, 103, 105, 106, 110, 111, 113, 114, 118, 119, 121, 122, 126
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047452, A047524, A193887.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [1, 2, 6, 7]]; // Wesley Ivan Hurt, May 29 2016
  • Maple
    A047554:=n->2*n+(1+I)*(2*I-2-(1-I)*I^(2*n)-I^(1-n)+I^n)/4: seq(A047554(n), n=1..100); # Wesley Ivan Hurt, May 29 2016
  • Mathematica
    Select[Range[120],MemberQ[{1,2,6,7},Mod[#,8]]&] (* Harvey P. Dale, Nov 29 2011 *)

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+x+4*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) = 2*n+(1+i)*(2*i-2-(1-i)*i^(2*n)-i^(1-n)+i^n)/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047452(k). (End)
E.g.f.: (2 - sin(x) + cos(x) + (4*x - 1)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/8 (A193887). - Amiram Eldar, Dec 24 2021

A047532 Numbers that are congruent to {0, 2, 3, 7} mod 8.

Original entry on oeis.org

0, 2, 3, 7, 8, 10, 11, 15, 16, 18, 19, 23, 24, 26, 27, 31, 32, 34, 35, 39, 40, 42, 43, 47, 48, 50, 51, 55, 56, 58, 59, 63, 64, 66, 67, 71, 72, 74, 75, 79, 80, 82, 83, 87, 88, 90, 91, 95, 96, 98, 99, 103, 104, 106, 107, 111, 112, 114, 115, 119, 120, 122, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047470, A047524.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 2, 3, 7]]; // Wesley Ivan Hurt, May 29 2016
  • Maple
    A047532:=n->2*n+(1+I)*(4*I-4+(1-I)*I^(2*n)+I^(-n)-I^(1+n))/4: seq(A047532(n), n=1..100); # Wesley Ivan Hurt, May 29 2016
  • Mathematica
    Table[2n+(1+I)*(4*I-4+(1-I)*I^(2n)+I^(-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x^2*(2+x+4*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) = 2*n+(1+i)*(4*i-4+(1-i)*i^(2n)+i^(-n)-i^(1+n))/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047470(k). (End)
E.g.f.: (2 + sin(x) + cos(x) + (4*x - 5)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 29 2016
Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4 - (2*sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 21 2021

A047533 Numbers that are congruent to {1, 2, 3, 7} mod 8.

Original entry on oeis.org

1, 2, 3, 7, 9, 10, 11, 15, 17, 18, 19, 23, 25, 26, 27, 31, 33, 34, 35, 39, 41, 42, 43, 47, 49, 50, 51, 55, 57, 58, 59, 63, 65, 66, 67, 71, 73, 74, 75, 79, 81, 82, 83, 87, 89, 90, 91, 95, 97, 98, 99, 103, 105, 106, 107, 111, 113, 114, 115, 119, 121, 122, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047471, A047524.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [1, 2, 3, 7]]; // Wesley Ivan Hurt, May 29 2016
  • Maple
    A047533:=n->(8*n-7+I^(2*n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4: seq(A047533(n), n=1..100); # Wesley Ivan Hurt, May 29 2016
  • Mathematica
    Table[(8n-7+I^(2n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 7, 9}, 50] (* G. C. Greubel, May 29 2016 *)

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+x+x^2+4*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-7+i^(2*n)+(1+2*i)*i^(-n)+(1-2*i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047471(k). (End)
E.g.f.: (2 + 2*sin(x) + cos(x) + 4*(x - 1)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 29 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*sqrt(2)*Pi/16 + log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 23 2021

A047540 Numbers that are congruent to {0, 2, 4, 7} mod 8.

Original entry on oeis.org

0, 2, 4, 7, 8, 10, 12, 15, 16, 18, 20, 23, 24, 26, 28, 31, 32, 34, 36, 39, 40, 42, 44, 47, 48, 50, 52, 55, 56, 58, 60, 63, 64, 66, 68, 71, 72, 74, 76, 79, 80, 82, 84, 87, 88, 90, 92, 95, 96, 98, 100, 103, 104, 106, 108, 111, 112, 114, 116, 119, 120, 122, 124
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The products of an odd number of terms as well as products of one term each of this sequence and one term of A047409 are members. The products of an even number of terms belong to A047409. The union of this sequence and A047409 is closed under multiplication. - Klaus Purath, Apr 23 2023

Links

Crossrefs

Cf. A008586, A047524.

Programs

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x^2*(2+2*x+3*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) = (8*n-7+i^(2*n)+i^(-n)+i^n)/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A008586(k-1) for k>0. (End)
Sum_{n>=2} (-1)^n/a(n) = (10-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - sqrt(2)*Pi/16. - Amiram Eldar, Dec 21 2021

A047553 Numbers that are congruent to {0, 2, 6, 7} mod 8.

Original entry on oeis.org

0, 2, 6, 7, 8, 10, 14, 15, 16, 18, 22, 23, 24, 26, 30, 31, 32, 34, 38, 39, 40, 42, 46, 47, 48, 50, 54, 55, 56, 58, 62, 63, 64, 66, 70, 71, 72, 74, 78, 79, 80, 82, 86, 87, 88, 90, 94, 95, 96, 98, 102, 103, 104, 106, 110, 111, 112, 114, 118, 119, 120, 122, 126
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047451, A047524.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 2, 6, 7]]; // Wesley Ivan Hurt, May 29 2016
  • Maple
    A047553:=n->(8*n-5-I^(2*n)+(1-2*I)*I^(-n)+(1+2*I)*I^n)/4: seq(A047553(n), n=1..100); # Wesley Ivan Hurt, May 29 2016
  • Mathematica
    Select[Range[0,200], MemberQ[{0,2,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Aug 09 2013 *)

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x^2*(2+4*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) = (8*n-5-i^(2*n)+(1-2*i)*i^(-n)+(1+2*i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047451(k). (End)
E.g.f.: (2 - 2*sin(x) + cos(x) + (4*x - 2)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 29 2016
Sum_{n>=2} (-1)^n/a(n) = (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 21 2021
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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