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.

A279001 Numbers of the form (11*h+j)*11^k-1 for h,k in N and j in {2,6,7,8,10}.

Original entry on oeis.org

1, 5, 6, 7, 9, 12, 16, 17, 18, 20, 21, 23, 27, 28, 29, 31, 34, 38, 39, 40, 42, 45, 49, 50, 51, 53, 56, 60, 61, 62, 64, 65, 67, 71, 72, 73, 75, 76, 78, 82, 83, 84, 86, 87, 89, 93, 94, 95, 97, 100, 104, 105, 106, 108, 109, 111, 115, 116, 117, 119, 122, 126, 127
Offset: 1

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Comments

Created in a failed attempt to explain sequences J and K on page 10 of Fu and Han (2016). See A279194 and A279195. - N. J. A. Sloane, Dec 15 2016

Links

Crossrefs

Complement of A279000.
Cf. A279194, A279195.

Programs

  • Maple
    isA279001 := proc(n)
    not isA279000(n) ;
end proc:
for n from 0 to 200 do
    if isA279001(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 15 2016
  • Mathematica
    okQ[n_] := Not @ MatchQ[IntegerDigits[n+1, 11], {_, 1|3|4|5|9, 0...}];
Select[Range[200], okQ] (* Jean-François Alcover, Feb 25 2018, after R. J. Mathar *)
  • Python
    from sympy import integer_log
def A279001(n):
    def f(x): return n-1+sum(((m:=(x+1)//11**i)-1)//11+(m-3)//11+(m-4)//11+(m-5)//11+(m-9)//11+5 for i in range(integer_log(x+1,11)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 23 2025

Extensions

Corrected by Lars Blomberg (base 5 replaced by base 11. 10 removed, 21 added,...), Dec 15 2016

A279194 Numbers of the form {(11*h+p)*11^2k-1 | h,k in N and p in {1,3,4,5,9} } U {(11*h+q)*11^(2k+1)-1 | h,k in N and q in {2,6,7,8,10} }.

Original entry on oeis.org

0, 2, 3, 4, 8, 11, 13, 14, 15, 19, 21, 22, 24, 25, 26, 30, 33, 35, 36, 37, 41, 44, 46, 47, 48, 52, 55, 57, 58, 59, 63, 65, 66, 68, 69, 70, 74, 76, 77, 79, 80, 81, 85, 87, 88, 90, 91, 92, 96, 99, 101, 102, 103, 107, 109, 110, 112, 113, 114, 118, 120, 121, 123, 124, 125, 129
Offset: 1

Views

Author

N. J. A. Sloane, Dec 15 2016

Keywords

Comments

The sequence J related to the Apwenian power series F_{11}(x).

Links

Crossrefs

Cf. A279000, A279001, A279195.

Programs

  • Mathematica
    isok[n_]:=Module[{ord=IntegerExponent[n+1,11],pq={{1,3,4,5,9},{2,6,7,8,10}}},MemberQ[pq[[Mod[ord,2]+1]],Mod[(n+1)/11^ord,11]]];Select[Range[0,131],isok] (* Ray Chandler, Dec 17 2016 *)

Extensions

Definition (from p. 5, Definition 2.1 of the arXiv reference) provided by Arie Groeneveld, Dec 16 2016
More terms from Ray Chandler, Dec 17 2016

A279195 Numbers of the form {(11*h+p)*11^(2k+1)-1 | h,k in N and p in {1,3,4,5,9} } U {(11*h+q)*11^2k-1 | h,k in N and q in {2,6,7,8,10} }.

Original entry on oeis.org

1, 5, 6, 7, 9, 10, 12, 16, 17, 18, 20, 23, 27, 28, 29, 31, 32, 34, 38, 39, 40, 42, 43, 45, 49, 50, 51, 53, 54, 56, 60, 61, 62, 64, 67, 71, 72, 73, 75, 78, 82, 83, 84, 86, 89, 93, 94, 95, 97, 98, 100, 104, 105, 106, 108, 111, 115, 116, 117, 119, 122, 126, 127, 128, 130, 131
Offset: 1

Views

Author

N. J. A. Sloane, Dec 15 2016

Keywords

Comments

The sequence K related to the Apwenian power series F_{11}(x).

Links

Crossrefs

Cf. A279000, A279001, A279194.

Programs

  • Mathematica
    isok[n_]:=Module[{ord=IntegerExponent[n+1,11],pq={{2,6,7,8,10},{1,3,4,5,9}}},MemberQ[pq[[Mod[ord,2]+1]],Mod[(n+1)/11^ord,11]]];Select[Range[0,131],isok] (* Ray Chandler, Dec 17 2016 *)

Extensions

Definition (from p. 5, Definition 2.1 of the arXiv reference) provided by Arie Groeneveld, Dec 16 2016
More terms from Ray Chandler, Dec 17 2016

A299647 Positive solutions to x^2 == -2 (mod 11).

Original entry on oeis.org

3, 8, 14, 19, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 80, 85, 91, 96, 102, 107, 113, 118, 124, 129, 135, 140, 146, 151, 157, 162, 168, 173, 179, 184, 190, 195, 201, 206, 212, 217, 223, 228, 234, 239, 245, 250, 256, 261, 267, 272, 278, 283, 289, 294, 300, 305, 311, 316
Offset: 1

Views

Author

Bruno Berselli, Mar 06 2018

Keywords

Comments

Positive numbers congruent to {3, 8} mod 11.
Equivalently, interleaving of A017425 and A017485.

Links

Crossrefs

Subsequence of A106252, A279000.
Cf. A017425, A017485.
Cf. A017497: positive solutions to x == -2 (mod 11).
Cf. A017437: positive solutions to x^3 == -2 (mod 11).
Nonnegative solutions to x^2 == -2 (mod j): A005843 (j=2), A001651 (j=3), A047235 (j=6), A156638 (j=9), this sequence (j=11).

Programs

  • GAP
    List([1..60], n -> 5*n-2+(2*n-(-1)^n-3)/4);
  • Julia
    [(11(2n-1)-(-1)^n)>>2 for n in 1:60] # Peter Luschny, Mar 07 2018
  • Magma
    [5*n-2+(2*n-(-1)^n-3)/4: n in [1..60]];
  • Mathematica
    Table[5 n - 2 + (2 n - (-1)^n - 3)/4, {n, 1, 60}]
CoefficientList[ Series[(3 + 5x + 3x^2)/((x - 1)^2 (x + 1)), {x, 0, 57}], x] (* or *)
LinearRecurrence[{1, 1, -1}, {3, 8, 14}, 58] (* Robert G. Wilson v, Mar 08 2018 *)
  • Maxima
    makelist(5*n-2+(2*n-(-1)^n-3)/4, n, 1, 60);
  • PARI
    vector(60, n, nn; 5*n-2+(2*n-(-1)^n-3)/4)
  • Python
    [5*n-2+(2*n-(-1)**n-3)/4 for n in range(1, 60)]
  • Sage
    [5*n-2+(2*n-(-1)^n-3)/4 for n in (1..60)]

Formula

O.g.f.: x*(3 + 5*x + 3*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (-1 + 12*exp(x) - 11*exp(2*x) + 22*x*exp(2*x))*exp(-x)/4.
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 5*n - 2 + (2*n - (-1)^n - 3)/4.
a(n) = 4*n - 1 + floor((n - 1)/2) + floor((3*n - 1)/3).
a(n+k) - a(n) = 11*k/2 + (1 - (-1)^k)*(-1)^n/4.
a(n+k) + a(n) = 11*(2*n + k - 1)/2 - (1 + (-1)^k)*(-1)^n/4.
E.g.f.: 3 + ((22*x - 11)*exp(x) - exp(-x))/4. - David Lovler, Aug 08 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/22)*Pi/11. - Amiram Eldar, Feb 27 2023
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(3*Pi/22)/2.
Product_{n>=1} (1 + (-1)^n/a(n)) = sec(5*Pi/22)*sin(2*Pi/11). (End)
Showing 1-4 of 4 results.

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

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