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 1-4 of 4 results.

A152728 a(n) + a(n+1) + a(n+2) = n^3.

Original entry on oeis.org

0, 0, 0, 1, 7, 19, 38, 68, 110, 165, 237, 327, 436, 568, 724, 905, 1115, 1355, 1626, 1932, 2274, 2653, 3073, 3535, 4040, 4592, 5192, 5841, 6543, 7299, 8110, 8980, 9910, 10901, 11957, 13079, 14268, 15528, 16860, 18265, 19747, 21307, 22946, 24668, 26474
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Comments

The differences between the terms are (1) a(3*k) - a(3*k-1) = 9*k*(k-1)+1; (2) otherwise, a(n) - a(n-1) = (n-2)*(n-1). - J. M. Bergot, Jul 10 2013
Second differences give A047266. - J. M. Bergot, Dec 01 2014

Links

Crossrefs

Cf. A000578, A047266, A057078.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); [0,0,0] cat Coefficients(R!(x^3*(1+4*x+x^2)/((1+x+x^2)*(x-1)^4))); // G. C. Greubel, Sep 01 2018
  • Maple
    seq(ceil((n^3 - 3*n^2 + n)/3), n=0..100); # Robert Israel, Dec 01 2014
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^3-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,4!}];lst
LinearRecurrence[{3,-3,2,-3,3,-1}, {0,0,0,1,7,19}, 50] (* G. C. Greubel, Sep 01 2018 *)
  • PARI
    x='x+O('x^50); concat([0,0,0], Vec(x^3*(1+4*x+x^2)/((1+x+x^2)*(x -1)^4 ))) \\ G. C. Greubel, Sep 01 2018

Formula

From R. J. Mathar, Aug 15 2010: (Start)
a(n) = ( (n-1)*(n^2-2*n-1) - A057078(n))/3.
G.f.: x^3*(1+4*x+x^2) / ( (1+x+x^2)*(x-1)^4 ). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-5). - Charles R Greathouse IV, Jul 10 2013
a(3n) = n*(9n^2-9n+1), a(3n+1) = n*(9n^2-2), a(3n+2) = n*(9n^2+9n+1). - Ralf Stephan, Jul 12 2013
a(n) = ceiling((n^3 - 3*n^2 + n)/3). - Robert Israel, Dec 01 2014
E.g.f.: (3*exp(x)*(1 - x + x^3) - exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Mar 04 2023

A265187 Nonnegative m for which 2*floor(m^2/11) = floor(2*m^2/11).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 82, 83, 84
Offset: 1

Views

Author

Bruno Berselli, Dec 04 2015

Keywords

Comments

Also, nonnegative m not congruent to 3 or 8 (mod 11).
Integers x >= 0 satisfying k*floor(x^2/11) = floor(k*x^2/11) with k >= 0:
k = 0, 1: x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... (A001477);
k = 2: x = 0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, ... (this sequence);
k = 3: x = 0, 1, 5, 6, 10, 11, 12, 16, 17, 21, 22, ... (A265188);
k = 4..10: x = 0, 1, 10, 11, 12, 21, 22, 23, 32, 33, ... (A112654);
k > 10: x = 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, ... (A008593).
Primes in sequence: 2, 5, 7, 11, 13, 17, 23, 29, 31, 37, 43, 53, 59, ...

Links

Crossrefs

Cf. A008593, A112654, A265188.
Cf. similar sequences provided by 2*floor(m^2/h) = floor(2*m^2/h): A005843 (h=2), A001477 (h=3,4), A008854 (h=5), A047266 (h=6), A047299 (h=7), A042965 (h=8), A060464 (h=9), A237415 (h=10), this sequence (h=11), A047263 (h=12).

Programs

  • Magma
    [n: n in [0..100] | 2*Floor(n^2/11) eq Floor(2*n^2/11)];
  • Mathematica
    Select[Range[0, 100], 2 Floor[#^2/11] == Floor[2 #^2/11] &]
Select[Range[0, 100], ! MemberQ[{3, 8}, Mod[#, 11]] &]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 4, 5, 6, 7, 9, 10, 11}, 80]
  • PARI
    is(n)=2*(n^2\11) == (2*n^2)\11 \\ Anders Hellström, Dec 05 2015
  • Sage
    [n for n in (0..100) if 2*floor(n^2/11) == floor(2*n^2/11)]

Formula

G.f.: x^2*(1 + x + 2*x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + x^8)/((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)).
a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.

A265188 Nonnegative m for which 3*floor(m^2/11) = floor(3*m^2/11).

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 12, 16, 17, 21, 22, 23, 27, 28, 32, 33, 34, 38, 39, 43, 44, 45, 49, 50, 54, 55, 56, 60, 61, 65, 66, 67, 71, 72, 76, 77, 78, 82, 83, 87, 88, 89, 93, 94, 98, 99, 100, 104, 105, 109, 110, 111, 115, 116, 120, 121, 122, 126, 127, 131, 132, 133, 137, 138, 142
Offset: 1

Views

Author

Bruno Berselli, Dec 04 2015

Keywords

Comments

See the second comment in A265187.
Also, nonnegative m congruent to 0, 1, 5, 6 or 10 (mod 11).
Primes in sequence: 5, 11, 17, 23, 43, 61, 67, 71, 83, 89, 109, 127, ...

Links

Crossrefs

Cf. A265187.
Cf. similar sequences provided by 3*floor(n^2/h) = floor(3*n^2/h): A005843 (h=2), A008585 (h=3), A001477 (h=4), A008854 (h=5), A047266 (h=6), A047299 (h=7), A042965 (h=8), A265227 (h=9), A054967 (h=10), this sequence (h=11), A047266 (h=12).

Programs

  • Magma
    [n: n in [0..150] | 3*Floor(n^2/11) eq Floor(3*n^2/11)];
  • Mathematica
    Select[Range[0, 150], 3 Floor[#^2/11] == Floor[3 #^2/11] &]
Select[Range[0, 150], MemberQ[{0, 1, 5, 6, 10}, Mod[#, 11]] &]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 5, 6, 10, 11}, 70]
  • PARI
    is(n) = 3*(n^2\11) == (3*n^2)\11 \\ Anders Hellström, Dec 05 2015
  • PARI
    concat(0, Vec(x^2*(1 + 4*x + x^2 + 4*x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^100))) \\ Michel Marcus, Dec 05 2015
  • Sage
    [n for n in (0..150) if 3*floor(n^2/11) == floor(3*n^2/11)]

Formula

G.f.: x^2*(1 + 4*x + x^2 + 4*x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6), n>6.

A305859 Numbers that are congruent to {1, 3, 11} mod 12.

Original entry on oeis.org

1, 3, 11, 13, 15, 23, 25, 27, 35, 37, 39, 47, 49, 51, 59, 61, 63, 71, 73, 75, 83, 85, 87, 95, 97, 99, 107, 109, 111, 119, 121, 123, 131, 133, 135, 143, 145, 147, 155, 157, 159, 167, 169, 171, 179, 181, 183, 191, 193, 195, 203, 205, 207, 215, 217, 219, 227, 229, 231, 239
Offset: 1

Views

Author

Vincenzo Librandi, Jun 12 2018

Keywords

Links

Crossrefs

Equals 2*A047240 - 1 and 2*A047266 + 1 (after 0).

Programs

  • Magma
    [n: n in [0..300] | n mod 12 in [1,3,11]]; // Bruno Berselli, Jun 13 2018
  • Mathematica
    Table[2 n + 6 Floor[n/3] - 1, {n, 1, 60}] (* Bruno Berselli, Jun 13 2018 *)
LinearRecurrence[{1,0,1,-1},{1,3,11,13},60] (* Harvey P. Dale, Mar 15 2023 *)

Formula

G.f.: x*(1 + 2*x + 8*x^2 + x^3)/((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>12.
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) for n>6.
a(n) = 2*n + 6*floor(n/3) - 1. - Bruno Berselli, Jun 13 2018

Extensions

Edited by Bruno Berselli, Jun 13 2018
Showing 1-4 of 4 results.

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