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 1-5 of 5 results.

A244151 0-additive sequence: start with a(1) = 2; thereafter, a(n) = smallest number not already in sequence which is not the sum of any previous two terms.

Original entry on oeis.org

2, 3, 4, 8, 9, 14, 15, 20, 21, 26, 27, 32, 33, 38, 39, 44, 45, 50, 51, 56, 57, 62, 63, 68, 69, 74, 75, 80, 81, 86, 87, 92, 93, 98, 99, 104, 105, 110, 111, 116, 117, 122, 123, 128, 129, 134, 135, 140, 141, 146, 147, 152, 153, 158, 159, 164, 165, 170, 171, 176, 177, 182, 183, 188, 189, 194, 195, 200, 201, 206, 207, 212, 213
Offset: 1

Views

Author

Leonardo Sznajder, Jun 21 2014

Keywords

Comments

As A033627, but first term is 2.
4 and the numbers in A047243. - Joerg Arndt, Jun 22 2014

Links

Crossrefs

Cf. A033627, A047243.

Programs

  • Mathematica
    f[s_List] := Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s, {2}]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[f, {2}, 70] (* Robert G. Wilson v, Jun 23 2014 *)
  • PARI
    Vec(x*(x^5+3*x^3-x^2+x+2)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Jun 26 2014

Formula

a(2n) = 6(n-1)+2 & a(2n+1) = 6(n-1)+3 for n>1. - Robert G. Wilson v, Jun 23 2014
From Colin Barker, Jun 26 2014: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 6.
G.f.: x*(x^5 + 3*x^3 - x^2 + x + 2)/((x - 1)^2*(x + 1)). (End)
E.g.f.: (x^3 + 3*x^2 + 30*x + 24)/6 + (3*x - 4)*cosh(x) + 3*(x - 2)*sinh(x). - Stefano Spezia, Apr 15 2023

Extensions

Added terms >= 20, Joerg Arndt, Jun 22 2014

A047260 Numbers that are congruent to {0, 1, 4, 5} mod 6.

Original entry on oeis.org

0, 1, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 25, 28, 29, 30, 31, 34, 35, 36, 37, 40, 41, 42, 43, 46, 47, 48, 49, 52, 53, 54, 55, 58, 59, 60, 61, 64, 65, 66, 67, 70, 71, 72, 73, 76, 77, 78, 79, 82, 83, 84, 85, 88, 89, 90, 91, 94, 95, 96, 97
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers x which are not a solution to 3^x - 2^x == 5 mod 7. - Cino Hilliard, May 14 2003
The sequence is the interleaving of A047233 with A007310. - Guenther Schrack, Feb 13 2019

Links

Crossrefs

Cf. A007310, A047233, A047269.
Complement: A047243.

Programs

  • GAP
    Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 4 or n mod 6 = 5); # Muniru A Asiru, Feb 19 2019
  • Magma
    [n : n in [0..100] | n mod 6 in [0, 1, 4, 5]]; // Wesley Ivan Hurt, May 21 2016
  • Maple
    A047260:=n->(6*n-5-I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/4: seq(A047260(n), n=1..100); # Wesley Ivan Hurt, May 21 2016
  • Mathematica
    Table[(6n-5-I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,4,5,6},70] (* Harvey P. Dale, Sep 20 2023 *)
  • PARI
    my(x='x+O('x^70)); concat([0], Vec(x^2*(1+3*x+x^2+x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Feb 16 2019
  • Sage
    a=(x^2*(1+3*x+x^2+x^3)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

Formula

G.f.: x^2*(1+3*x+x^2+x^3) / ((1+x)*(1+x^2)*(1-x)^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) = (6*n - 5 - i^(2*n) + (1-i)*i^(-n) + (1+i)*i^n)/4 where i=sqrt(-1).
a(2*n) = A007310(n), a(2*n-1) = A047233(n). (End)
From Guenther Schrack, Feb 13 2019: (Start)
a(n) = (6*n - 5 - (-1)^n + 2*(-1)^(n*(n + 1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=4, a(4)=5, for n > 4.
a(-n) = -A047269(n+2). (End)
Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(3)/4 + 2*log(2)/3. - Amiram Eldar, Dec 16 2021

Extensions

More terms from Wesley Ivan Hurt, May 21 2016

A030531 Value of 3^x - 2^x - 5 for the solutions of 3^x - 2^x == 5 (mod 7).

Original entry on oeis.org

0, 14, 6300, 19166, 4766580, 14316134, 3485735820, 10458256046, 2541798719460, 7625463267254, 1853015893884540, 5559051976620926, 1350851442795085140, 4052554603263162374, 984770884591425188460, 2954312671366461609806
Offset: 1

Views

Author

Cino Hilliard, May 09 2003

Keywords

References

  • Emil Grosswald, Topics From the Theory of Numbers. 1966 p 65 problem 23

Crossrefs

Cf. A047243.

Programs

  • PARI
    f(n) = for(x=1,n,y=3^x-2^x-5; if(y%7==0,print1(y" ")))

A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.

Original entry on oeis.org

2, 3, 0, 1, 4, 5, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 18, 19, 16, 17, 20, 21, 22, 23, 26, 27, 24, 25, 28, 29, 30, 31, 34, 35, 32, 33, 36, 37, 38, 39, 42, 43, 40, 41, 44, 45, 46, 47, 50, 51, 48, 49, 52, 53, 54, 55, 58, 59, 56, 57, 60, 61, 62, 63, 66, 67, 64
Offset: 0

Views

Author

Franck Maminirina Ramaharo, Aug 01 2018

Keywords

Comments

Write n in base 8, then apply the following substitution to the rightmost digit: '0'->'2, '1'->'3', and vice versa. Convert back to decimal.
A self-inverse permutation: a(a(n)) = n.
Array whose columns are, in this order, A047463, A047621, A047451 and A047522, read by rows.

Examples

			a(25) = a('3'1') = '3'3' = 27.
a(26) = a('3'2') = '3'0' = 24.
a(27) = a('3'3') = '3'1' = 25.
a(28) = a('3'4') = '3'4' = 28.
a(29) = a('3'5') = '3'5' = 29.
The sequence as array read by rows:
  A047463, A047621, A047451, A047522;
        2,       3,       0,       1;
        4,       5,       6,       7;
       10,      11,       8,       9;
       12,      13,      14,      15;
       18,      19,      16,      17;
       20,      21,      22,      23;
       26,      27,      24,      25;
       28,      29,      30,      31;
  ...

Links

Crossrefs

Cf. A047225, A047243, A047257, A064429, A080412, A159959, A026185.

Programs

  • Magma
    m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1)))); // G. C. Greubel, Sep 25 2018
  • Mathematica
    Table[(4*(Floor[1/4 Mod[2*n + 4, 8]] - Floor[1/4 Mod[n + 2, 8]]) + 2*n)/2, {n, 0, 100}]
f[n_] := Block[{id = IntegerDigits[n, 8]}, FromDigits[ Join[Most@ id /. {{} -> {0}}, {id[[-1]] /. {0 -> 2, 1 -> 3, 2 -> 0, 3 -> 1}}], 8]]; Array[f, 67, 0] (* or *)
CoefficientList[ Series[(x^7 + x^5 + 3x^3 - 2x^2 - x + 2)/((x - 1)^2 (x^6 + x^4 + x^2 + 1)), {x, 0, 70}], x] (* or *)
LinearRecurrence[{2, -2, 2, -2, 2, -2, 2, -1}, {2, 3, 0, 1, 4, 5, 6, 7}, 70] (* Robert G. Wilson v, Aug 01 2018 *)
  • Maxima
    makelist((4*(floor(mod(2*n + 4, 8)/4) - floor(mod(n + 2, 8)/4)) + 2*n)/2, n, 0, 100);
  • PARI
    my(x='x+O('x^100)); Vec((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1))) \\ G. C. Greubel, Sep 25 2018

Formula

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - a(n-8), n > 7.
a(n) = (4*(floor(((2*n + 4) mod 8)/4) - floor(((n + 2) mod 8)/4)) + 2*n)/2.
a(n) = lod_4(A047247(n+1)).
a(4*n) = A047463(n+1).
a(4*n+1) = A047621(n+1).
a(4*n+2) = A047451(n+1).
a(4*n+3) = A047522(n+1).
a(A042948(n)) = A047596(n+1).
a(A042964(n+1)) = A047551(n+1).
G.f.: (x^7 + x^5 + 3*x^3 - 2*x^2 - x + 2)/((x-1)^2 * (x^2+1) * (x^4+1)).
E.g.f.: x*exp(x) + cos(x) + sin(x) + cos(x/sqrt(2))*cosh(x/sqrt(2)) + (sqrt(2)*cos(x/sqrt(2)) - sin(x/sqrt(2)))*sinh(x/sqrt(2)).
a(n+8) = a(n) + 8 . - Philippe Deléham, Mar 09 2023
Sum_{n>=3} (-1)^(n+1)/a(n) = 1/6 + log(2). - Amiram Eldar, Mar 12 2023

A174041 Composites of the form 6n+2 or 6n+3.

Original entry on oeis.org

8, 9, 14, 15, 20, 21, 26, 27, 32, 33, 38, 39, 44, 45, 50, 51, 56, 57, 62, 63, 68, 69, 74, 75, 80, 81, 86, 87, 92, 93, 98, 99, 104, 105, 110, 111, 116, 117, 122, 123, 128, 129, 134, 135, 140, 141, 146, 147, 152, 153, 158, 159, 164, 165, 170, 171, 176, 177, 182, 183
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 06 2010

Keywords

Examples

			a(1)=8 because 6*1+2=8; a(2)=9 because 6*1+3=9.

Crossrefs

Cf. A047243.

Formula

a(n)=A047243(n+2).
a(2n-1) = 6n+2, a(2n) = 6n+3.

Extensions

Entries checked by R. J. Mathar, Apr 26 2010
Showing 1-5 of 5 results.

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

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