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 11-20 of 20 results.

A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 11, 13, 16, 17, 19, 21, 24, 25, 27, 29, 32, 33, 35, 37, 40, 41, 43, 45, 48, 49, 51, 53, 56, 57, 59, 61, 64, 65, 67, 69, 72, 73, 75, 77, 80, 81, 83, 85, 88, 89, 91, 93, 96, 97, 99, 101, 104, 105, 107, 109, 112, 113, 115, 117, 120, 121, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A004526, A016813, A042948, A047470, A140081, A140201, A173562.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 1, 3, 5]]; // Wesley Ivan Hurt, Jun 01 2016
  • Maple
    A047624:=n->(8*n-11-I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047624(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Table[(8n-11-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,3,5,8},100] (* G. C. Greubel, Jun 01 2016 *)

Formula

From Reinhard Zumkeller, Feb 21 2010: (Start)
a(n+1) = A173562(n) - A173562(n-1);
a(n+1) - a(n) = A140081(n-1) + 1;
a(n) = A140201(n-1) + A042948(A004526(n-1)). (End)
G.f.: x^2*(1+2*x+2*x^2+3*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-11-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).
a(2k) = A016813(k-1) for k>0, a(2k-1) = A047470(k). (End)
E.g.f.: (6 + sin(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016
Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(2))*Pi/16 + (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 20 2021

A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8.

Original entry on oeis.org

0, 1, 3, 7, 8, 9, 11, 15, 16, 17, 19, 23, 24, 25, 27, 31, 32, 33, 35, 39, 40, 41, 43, 47, 48, 49, 51, 55, 56, 57, 59, 63, 64, 65, 67, 71, 72, 73, 75, 79, 80, 81, 83, 87, 88, 89, 91, 95, 96, 97, 99, 103, 104, 105, 107, 111, 112, 113, 115, 119, 120, 121, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers n such that the n-th homotopy group of the topological group O(oo) does not vanish [see Baez]. Cf. A195679.
The a(n+1) determine the maximal number of linearly independent smooth nowhere zero vector fields on a (2n+1)-sphere, see A053381. - Johannes W. Meijer, Jun 07 2011

Links

Crossrefs

Cf. A008621. - Johannes W. Meijer, Jun 07 2011
Cf. A047470, A047522, A053381, A195679.

Programs

Formula

From Johannes W. Meijer, Jun 07 2011: (Start)
a(n) = ceiling(n/4) + 2*ceiling((n-1)/4) + 4*ceiling((n-2)/4) + ceiling((n-3)/4).
a(n+1) = A053381(2^p). (End)
G.f.: x^2*(1+2*x+4*x^2+x^3) / ((1+x)*(x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (8n-9+i^(2n)+(2+i)*i^(-n)+(2-i)*i^n)/4, where i=sqrt(-1).
a(2n) = A047522(n), a(2n-1) = A047470(n). (End)
E.g.f.: (2 + sin(x) + 2*cos(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
Sum_{n>=2} (-1)^n/a(n) = (8-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 20 2021

Extensions

More terms from Wesley Ivan Hurt, May 21 2016

A047401 Numbers that are congruent to {0, 1, 3, 6} mod 8.

Original entry on oeis.org

0, 1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 22, 24, 25, 27, 30, 32, 33, 35, 38, 40, 41, 43, 46, 48, 49, 51, 54, 56, 57, 59, 62, 64, 65, 67, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 94, 96, 97, 99, 102, 104, 105, 107, 110, 112, 113, 115, 118, 120, 121, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Partial sums of A068073. - Jeremy Gardiner, Jul 20 2013.

Links

Crossrefs

Cf. A047452, A047470, A068073.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 1, 3, 6]]; // Wesley Ivan Hurt, Jun 01 2016
  • Maple
    A047401:=n->2*(n-1)+(I^(n*(n-1))-1)/2: seq(A047401(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,107], MemberQ[{0, 1, 3, 6}, Mod[#, 8]]&] (* Bruno Berselli, Dec 05 2011 *)
  • Maxima
    makelist(2*(n-1)+(%i^(n*(n-1))-1)/2,n,1,55); /* Bruno Berselli, Dec 05 2011 */
  • PARI
    my(x='x+O('x^100)); concat(0, Vec(x^2*(1+x+2*x^2)/((x^2+1)*(x-1)^2))) \\ Altug Alkan, Jun 02 2016

Formula

G.f.: x^2*(1+x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
a(n) = 2*(n-1)+(i^(n*(n-1))-1)/2, where i=sqrt(-1). - Bruno Berselli, Dec 05 2011
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(2k) = A047452(k), a(2k-1) = A047470(k). (End)
Sum_{n>=2} (-1)^n/a(n) = Pi/16 + (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

A047456 Numbers that are congruent to {0, 2, 3, 4} mod 8.

Original entry on oeis.org

0, 2, 3, 4, 8, 10, 11, 12, 16, 18, 19, 20, 24, 26, 27, 28, 32, 34, 35, 36, 40, 42, 43, 44, 48, 50, 51, 52, 56, 58, 59, 60, 64, 66, 67, 68, 72, 74, 75, 76, 80, 82, 83, 84, 88, 90, 91, 92, 96, 98, 99, 100, 104, 106, 107, 108, 112, 114, 115, 116, 120, 122, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047463, A047470.

Programs

  • Magma
    I:=[0, 2, 3, 4, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 16 2012
  • Maple
    A047456:=n->(-11-(-1)^n-(2-I)*(-I)^n-(2+I)*I^n+8*n)/4: seq(A047456(n), n=1..100); # Wesley Ivan Hurt, May 31 2016
  • Mathematica
    Select[Range[0,300], MemberQ[{0,2,3,4}, Mod[#,8]]&] (* Vincenzo Librandi, May 16 2012 *)

Formula

G.f.: x^2*(2+x+x^2+4*x^3)/((1-x)^2*(1+x)*(1+x^2)). - Colin Barker, May 13 2012
a(n) = (-11-(-1)^n-(2-i)*(-i)^n-(2+i)*i^n+8*n)/4 where i=sqrt(-1). - Colin Barker, May 14 2012
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 16 2012
a(2k) = A047463(k), a(2k-1) = A047470(k). - Wesley Ivan Hurt, May 31 2016
E.g.f.: (8 + sin(x) - 2*cos(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - Ilya Gutkovskiy, May 31 2016
Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 21 2021

A047403 Numbers that are congruent to {0, 2, 3, 6} mod 8.

Original entry on oeis.org

0, 2, 3, 6, 8, 10, 11, 14, 16, 18, 19, 22, 24, 26, 27, 30, 32, 34, 35, 38, 40, 42, 43, 46, 48, 50, 51, 54, 56, 58, 59, 62, 64, 66, 67, 70, 72, 74, 75, 78, 80, 82, 83, 86, 88, 90, 91, 94, 96, 98, 99, 102, 104, 106, 107, 110, 112, 114, 115, 118, 120, 122, 123, 126, 128
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016825, A047470.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 2, 3, 6]]; // Wesley Ivan Hurt, May 24 2016
  • Maple
    A047403:=n->(8*n-9+I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047403(n), n=1..100); # Wesley Ivan Hurt, May 24 2016
  • Mathematica
    Table[(8n-9+I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 24 2016 *)
#+{0,2,3,6}&/@(8*Range[0,20])//Flatten (* or *) LinearRecurrence[{1,0,0,1,-1},{0,2,3,6,8},80] (* Harvey P. Dale, Mar 02 2023 *)

Formula

a(n) = 2*n - ((n mod 4) == 2).
G.f.: x^2*(2+x+3*x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, May 24 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-9+i^(2*n)+i^(1-n)-i^(1+n))/4, where i=sqrt(-1).
a(2k) = A016825(k-1) for k>0, a(2k-1) = A047470. (End)
E.g.f.: (4 + sin(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016
Sum_{n>=2} (-1)^n/a(n) = (4-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 21 2021

A047460 Numbers that are congruent to {0, 1, 3, 4} mod 8.

Original entry on oeis.org

0, 1, 3, 4, 8, 9, 11, 12, 16, 17, 19, 20, 24, 25, 27, 28, 32, 33, 35, 36, 40, 41, 43, 44, 48, 49, 51, 52, 56, 57, 59, 60, 64, 65, 67, 68, 72, 73, 75, 76, 80, 81, 83, 84, 88, 89, 91, 92, 96, 97, 99, 100, 104, 105, 107, 108, 112, 113, 115, 116, 120, 121, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047461, A047470.

Programs

  • Magma
    I:=[0, 1, 3, 4, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 16 2012
  • Maple
    A047460:=n->(-1/4+I/4)*((6+6*I)+(1+I)*I^(2*n)+(-I)^n+I*I^n)+2*n: seq(A047460(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,3000], MemberQ[{0,1,3,4}, Mod[#,8]]&] (* Vincenzo Librandi, May 16 2012 *)
  • PARI
    my(x='x+O('x^100)); concat(0, Vec(x^2*(1+2*x+x^2+4*x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

Formula

From Colin Barker, May 14 2012: (Start)
a(n) = (-1/4+i/4)*((6+6*i)+(1+i)*(-1)^n+(-i)^n+i*i^n)+2*n where i=sqrt(-1).
G.f.: x^2*(1+2*x+x^2+4*x^3)/((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 16 2012
a(2k) = A047461(k), a(2k-1) = A047470(k). - Wesley Ivan Hurt, Jun 01 2016
Sum_{n>=2} (-1)^n/a(n) = Pi/8 + (2-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 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

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047470, A047617.

Programs

Formula

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

A306278 Numbers congruent to 2 or 11 mod 14.

Original entry on oeis.org

2, 11, 16, 25, 30, 39, 44, 53, 58, 67, 72, 81, 86, 95, 100, 109, 114, 123, 128, 137, 142, 151, 156, 165, 170, 179, 184, 193, 198, 207, 212, 221, 226, 235, 240, 249, 254, 263, 268, 277, 282, 291, 296, 305, 310, 319, 324, 333, 338, 347, 352, 361, 366, 375, 380, 389, 394
Offset: 1

Views

Author

Davis Smith, Feb 02 2019

Keywords

Links

Crossrefs

Cf. A007310, A010703, A020639, A047470, A091999, A273669, A306277, A306289.
Primes greater than 2 in this sequence: A045471.

Programs

  • Maple
    seq(seq(14*i+j, j=[2, 11]), i=0..28);
  • Mathematica
    Flatten[Table[{14n + 2, 14n + 11}, {n, 0, 28}]]
LinearRecurrence[{1,1,-1},{2,11,16},60] (* Harvey P. Dale, Jan 16 2023 *)
  • PARI
    for(n=2, 394, if((n%14==2) || (n%14==11), print1(n, ", ")))
  • PARI
    for(n=1,57,print1(7*n-4+(-1)^n,", "))
  • PARI
    for(n=1,500,if(n%14==2,print1(n,", "));if(n%14==11,print1(n,", "))) \\ Jinyuan Wang, Feb 03 2019
  • PARI
    Vec(x*(2 + 9*x + 3*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Mar 14 2019
  • PARI
    upto(n) = forstep(i = 2, n, [9, 5], print1(i", ")) \\ David A. Corneth, Mar 27 2019

Formula

a(n) = 7*n - A010703(n).
a(n) = 7*n - 4 + (-1)^n.
a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3.
A007310(a(n) + 1) = 7*A007310(n)
From Jinyuan Wang, Feb 03 2019: (Start)
For odd number k, a(k) = 7*k - 5.
For even number k, a(k) = 7*k - 3.
(End)
G.f.: x*(2 + 9*x + 3*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Mar 14 2019
E.g.f.: 3 + (7*x - 4)*exp(x) + exp(-x). - David Lovler, Sep 07 2022

A047473 Numbers that are congruent to {2, 3} mod 8.

Original entry on oeis.org

2, 3, 10, 11, 18, 19, 26, 27, 34, 35, 42, 43, 50, 51, 58, 59, 66, 67, 74, 75, 82, 83, 90, 91, 98, 99, 106, 107, 114, 115, 122, 123, 130, 131, 138, 139, 146, 147, 154, 155, 162, 163, 170, 171, 178, 179, 186, 187, 194, 195, 202, 203, 210, 211, 218, 219, 226, 227, 234
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers k such that k and k+2 have the same digital binary sum. - Benoit Cloitre, Dec 01 2002
Also, numbers k such that k*(3*k + 1)/8 + 1/4 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

Links

Crossrefs

Union of A017089 and A017101.
Cf. A047467, A047468, A047470, A047471.

Programs

  • Mathematica
    Flatten[# + {2,3} &/@ (8 Range[0, 30])] (* or *) LinearRecurrence[{1, 1, -1}, {2, 3, 10}, 60] (* Harvey P. Dale, Sep 28 2012 *)

Formula

a(n) = 8*n - a(n-1) - 11 for n>1, a(1)=2. - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 7/2 - 3*(-1)^n/2.
G.f.: x*(2 + x + 5*x^2)/((1 + x)*(1 - x)^2). (End)
a(1)=2, a(2)=3, a(3)=10; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Sep 28 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/16 + sqrt(2)*log(sqrt(2)+1)/8 - log(2)/8. - Amiram Eldar, Dec 18 2021

Extensions

More terms from Vincenzo Librandi, Aug 06 2010
Previous Showing 11-20 of 20 results.

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

Content is available under The OEIS End-User License Agreement.