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.

Showing 1-6 of 6 results.

A047382 Numbers that are congruent to {0, 5} mod 7.

Original entry on oeis.org

0, 5, 7, 12, 14, 19, 21, 26, 28, 33, 35, 40, 42, 47, 49, 54, 56, 61, 63, 68, 70, 75, 77, 82, 84, 89, 91, 96, 98, 103, 105, 110, 112, 117, 119, 124, 126, 131, 133, 138, 140, 145, 147, 152, 154, 159, 161, 166, 168
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Except for the first term, numbers m such that 36*m^2 + 72*m + 35 = (6*m+5)*(6*m+7) is not of the form p*(p+2), with p prime. - Vincenzo Librandi, Aug 05 2010
Nonnegative k such that k or 4*k + 1 is divisible by 7. - Bruno Berselli, Feb 13 2018

Links

Crossrefs

Cf. A008589, A017041.

Programs

  • Magma
    &cat[[7*n,7*n+5]: n in [0..23]];  // Bruno Berselli, Oct 17 2011
  • Mathematica
    {#, 5 + #} &/@ (7 Range[0, 30]) // Flatten (* or *) LinearRecurrence[{1, 1, -1}, {0, 5, 7}, 60] (* Harvey P. Dale, Dec 01 2016 *)
  • PARI
    a(n) = (14*n + 3*(-1)^n - 11)/4 \\ David Lovler, Sep 11 2022

Formula

a(n) = 7*n - a(n-1) - 9 for n>1, with a(1)=0. - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=A005009(k-1)=7*2^(k-1) for k>0. - Philippe Deléham, Oct 17 2011
From Bruno Berselli, Oct 17 2011: (Start)
G.f.: x^2*(5 + 2*x)/((1 + x)*(1 - x)^2).
a(n) = (14*n + 3*(-1)^n - 11)/4.
a(-n) = -A047352(n+2). (End)
a(n) = ceiling((7/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
E.g.f.: 2 + ((14*x - 11)*exp(x) + 3*exp(-x))/4. - David Lovler, Sep 11 2022

A047388 Numbers that are congruent to {0, 1, 2, 5} mod 7.

Original entry on oeis.org

0, 1, 2, 5, 7, 8, 9, 12, 14, 15, 16, 19, 21, 22, 23, 26, 28, 29, 30, 33, 35, 36, 37, 40, 42, 43, 44, 47, 49, 50, 51, 54, 56, 57, 58, 61, 63, 64, 65, 68, 70, 71, 72, 75, 77, 78, 79, 82, 84, 85, 86, 89, 91, 92, 93, 96, 98, 99, 100, 103, 105, 106, 107, 110, 112
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047352, A047383.

Programs

  • Magma
    I:=[0, 1, 2, 5, 7]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 15 2012
  • Maple
    A047388:=n->(-19+I^(2*n)+(1+3*I)*(-I)^n+(1-3*I)*I^n+14*n)/8: seq(A047388(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,300], MemberQ[{0,1,2,5}, Mod[#,7]]&] (* Vincenzo Librandi, May 15 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,5,7},80] (* Harvey P. Dale, Jan 10 2023 *)
  • PARI
    x='x+O('x^100); concat(0, Vec(x^2*(1+x+3*x^2+2*x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Jun 02 2016

Formula

G.f.: x^2*(1+x+3*x^2+2*x^3)/((1-x)^2*(1+x)*(1+x^2)). - Colin Barker, May 13 2012
a(n) = (-19+(-1)^n+(1+3*i)*(-i)^n+(1-3*i)*i^n+14*n)/8 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) = A047383(k), a(2k-1) = A047352(k). - Wesley Ivan Hurt, Jun 01 2016

A047351 Numbers that are congruent to {0, 1, 2, 4} mod 7.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 9, 11, 14, 15, 16, 18, 21, 22, 23, 25, 28, 29, 30, 32, 35, 36, 37, 39, 42, 43, 44, 46, 49, 50, 51, 53, 56, 57, 58, 60, 63, 64, 65, 67, 70, 71, 72, 74, 77, 78, 79, 81, 84, 85, 86, 88, 91, 92, 93, 95, 98, 99, 100, 102, 105, 106, 107, 109, 112
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The set of values for m such that 7i+m is a perfect square (the quadratic residues of 7 including the trivial case of k*7). - Gary Detlefs, Mar 07 2010
The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4 etc. - Bruno Berselli, Dec 03 2012

Links

Crossrefs

Cf. A045373 (primes), A047346, A047352.
Complement of A047327.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [0, 1, 2, 4]]; // Wesley Ivan Hurt, Jun 01 2016
  • Maple
    for i from 1 to 56 do if(i mod 4=0) then print(floor(7*i-3)/4)+1) else print(floor(7*i-3)/4)) fi od; # Gary Detlefs, Mar 07 2010
A047351:=n->n-3+(6*n+(2-I^(2*n))*(1-2*I^(n*(n+1)))+1)/8: seq(A047351(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,100], MemberQ[{0,1,2,4}, Mod[#,7]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,1,2,4,7}, 60] (* Harvey P. Dale, Jun 04 2013 *)
  • PARI
    x='x+O('x^100); concat(0, Vec(x^2*(1+x+2*x^2+3*x^3)/((1+x)*(1+x^2)*(x-1)^2))) \\ Altug Alkan, Jun 02 2016

Formula

If n mod 4 = 0 then a(n) = floor((7*n-3)/4)+1, else a(n) = floor((7*n-3)/4). - Gary Detlefs, Mar 07 2010
G.f.: x^2*(1+x+2*x^2+3*x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
a(n) = n-3+(6*n+(2-(-1)^n)(1-2*i^(n(n+1)))+1)/8, where i=sqrt(-1). - Bruno Berselli, Dec 03 2012
a(0)=0, a(1)=1, a(2)=2, a(3)=4, a(4)=7, a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Harvey P. Dale, Jun 04 2013
a(2k) = A047346(k), a(2k-1) = A047352(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (12 + 3*sin(x) - cos(x) + (7*x - 10)*sinh(x) + (7*x - 11)*cosh(x))/4. - Ilya Gutkovskiy, Jun 02 2016

A047361 Numbers that are congruent to {0, 1, 2, 3} mod 7.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 9, 10, 14, 15, 16, 17, 21, 22, 23, 24, 28, 29, 30, 31, 35, 36, 37, 38, 42, 43, 44, 45, 49, 50, 51, 52, 56, 57, 58, 59, 63, 64, 65, 66, 70, 71, 72, 73, 77, 78, 79, 80, 84, 85, 86, 87, 91, 92, 93, 94, 98, 99, 100, 101, 105, 106, 107, 108, 112
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Nonnegative m for which floor(2*m/7) = 2*floor(m/7). [Bruno Berselli, Dec 03 2015]

Links

Crossrefs

Cf. A047352, A047356.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [0..3]]; // Wesley Ivan Hurt, May 23 2016
  • Maple
    A047361:=n->(14*n-23-3*I^(2*n)-(3-3*I)*I^(-n)-(3+3*I)*I^n)/8: seq(A047361(n), n=1..100); # Wesley Ivan Hurt, May 23 2016
  • Mathematica
    Flatten[#+{0,1,2,3}&/@(7*Range[0,20])] (* Harvey P. Dale, Jan 17 2013 *)
  • PARI
    concat(0, Vec(x^2*(1+x+x^2+4*x^3)/((1+x)*(x^2+1)*(x-1)^2) + O(x^100))) \\ Altug Alkan, Dec 09 2015

Formula

a(n) = 7*floor(n/4) + (n mod 4), with offset 0 and a(0) = 0. - Gary Detlefs, Mar 09 2010
G.f.: x^2*(1+x+x^2+4*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, May 23 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-23-3*i^(2*n)-(3-3*i)*i^(-n)-(3+3*i)*i^n)/8, where i=sqrt(-1).
a(2k) = A047356(k), a(2k-1) = A047352(k). (End)
E.g.f.: (16 + 3*(sin(x) - cos(x)) + (7*x - 10)*sinh(x) + (7*x - 13)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

Extensions

More terms from Wesley Ivan Hurt, May 23 2016

A047279 Numbers that are congruent to {0, 1, 2, 6} mod 7.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 9, 13, 14, 15, 16, 20, 21, 22, 23, 27, 28, 29, 30, 34, 35, 36, 37, 41, 42, 43, 44, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 65, 69, 70, 71, 72, 76, 77, 78, 79, 83, 84, 85, 86, 90, 91, 92, 93, 97, 98, 99, 100, 104, 105, 106, 107, 111, 112
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047322, A047336, A047352, A047361.

Programs

  • Magma
    [n : n in [0..100] | n mod 7 in [0, 1, 2, 6]]; // Wesley Ivan Hurt, May 21 2016
  • Maple
    A047279:=n->(14*n-17+3*(I^(2*n)+(1+I)*I^(-n)+(1-I)*I^n))/8: seq(A047279(n), n=1..100); # Wesley Ivan Hurt, May 21 2016
  • Mathematica
    LinearRecurrence[{1,0,0,1,-1},{0,1,2,6,7},80] (* Harvey P. Dale, Jun 15 2015 *)

Formula

G.f.: x^2*(1+x+4*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 25 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) = (14n-17+3*(i^(2n)+(1+i)*i^(-n)+(1-i)*i^n))/8 where i = sqrt(-1).
a(2n) = A047336(n), a(2n-1) = A047352(n).
a(n) = A047361(n+1) - 1. a(2-n) = - A047322(n). (End)

Extensions

More terms from Wesley Ivan Hurt, May 21 2016

A117795 Heptagonal numbers divisible by 7.

Original entry on oeis.org

0, 7, 112, 189, 469, 616, 1071, 1288, 1918, 2205, 3010, 3367, 4347, 4774, 5929, 6426, 7756, 8323, 9828, 10465, 12145, 12852, 14707, 15484, 17514, 18361, 20566, 21483, 23863, 24850, 27405, 28462, 31192, 32319, 35224, 36421, 39501, 40768, 44023
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006

Keywords

Comments

Intersection of A000566 and A008589. Their indices are given by A047352. - Michel Marcus, Feb 27 2014

Links

Crossrefs

Cf. A000566, A008589, A047352.

Programs

  • Mathematica
    Select[PolygonalNumber[7,Range[0,200]],Divisible[#,7]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 02 2019 *)
  • PARI
    isok(n) = ispolygonal(n, 7) && !(n % 7); \\ Michel Marcus, Feb 27 2014
Showing 1-6 of 6 results.

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

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