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 1-2 of 2 results.

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.

A265227 Nonnegative m for which k*floor(m^2/9) = floor(k*m^2/9), with 2 < k < 9.

Original entry on oeis.org

0, 1, 3, 6, 8, 9, 10, 12, 15, 17, 18, 19, 21, 24, 26, 27, 28, 30, 33, 35, 36, 37, 39, 42, 44, 45, 46, 48, 51, 53, 54, 55, 57, 60, 62, 63, 64, 66, 69, 71, 72, 73, 75, 78, 80, 81, 82, 84, 87, 89, 90, 91, 93, 96, 98, 99, 100, 102, 105, 107, 108, 109, 111, 114
Offset: 1

Views

Author

Bruno Berselli, Dec 06 2015

Keywords

Comments

Also, nonnegative m congruent to 0, 1, 3, 6 or 8 (mod 9). The product of any two terms belongs to the sequence and so also a(n)^2, a(n)^3, a(n)^4, etc.
Integers x >= 0 satisfying k*floor(x^2/9) = floor(k*x^2/9) 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, 3, 6, 7, 8, 9, 10, 11, 12, 15, ... (A060464);
k = 3..8: x = 0, 1, 3, 6, 8, 9, 10, 12, 15, 17, 18, ... (this sequence);
k > 8: x = 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... (A008585).
Primes in sequence: 3, 17, 19, 37, 53, 71, 73, 89, 107, 109, 127, ...

Links

Crossrefs

Cf. A001477, A008585, A060464.
Cf. similar sequences listed in A265188.

Programs

  • Magma
    [n: n in [0..120] | 3*Floor(n^2/9) eq Floor(3*n^2/9)]; /* or, by the definition: */ K:=[3..8]; [: k in K];
  • Mathematica
    Select[Range[0, 120], 3 Floor[#^2/9] == Floor[3 #^2/9] &]
Select[Range[0, 120], MemberQ[{0, 1, 3, 6, 8}, Mod[#, 9]] &]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 3, 6, 8, 9}, 70]
  • Sage
    [n for n in (0..120) if 3*floor(n^2/9) == floor(3*n^2/9)]

Formula

G.f.: x^2*(1 + 2*x + 3*x^2 + 2*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) for n>6.
Showing 1-2 of 2 results.

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