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.

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A272975 Numbers that are congruent to {0,7} mod 12.

Original entry on oeis.org

0, 7, 12, 19, 24, 31, 36, 43, 48, 55, 60, 67, 72, 79, 84, 91, 96, 103, 108, 115, 120, 127, 132, 139, 144, 151, 156, 163, 168, 175, 180, 187, 192, 199, 204, 211, 216, 223, 228, 235, 240, 247, 252, 259, 264, 271, 276, 283, 288, 295, 300, 307, 312, 319, 324
Offset: 1

Views

Author

Wesley Ivan Hurt, May 30 2016

Keywords

Comments

Numbers that are not congruent to {1, 2, 3, 4, 5, 6, 8, 9, 10, 11} mod 12.
Bisection of A083032.

Links

Crossrefs

Cf. A008594, A017605, A083032, A049453, A049598.

Programs

  • Magma
    [n : n in [0..400] | n mod 12 in [0, 7]];
  • Maple
    A272975:=n->(12*n-11+(-1)^n)/2: seq(A272975(n), n=1..100);
  • Mathematica
    Table[(12n - 11 + (-1)^n)/2, {n, 80}]
  • PARI
    concat(0, Vec(x^2*(7+5*x)/((x-1)^2*(x+1)) + O(x^99))) \\ Altug Alkan, May 31 2016

Formula

G.f.: x^2*(7+5*x) / ((x-1)^2*(x+1)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = (12*n - 11 + (-1)^n)/2.
a(2k) = A017605(k-1) k>0, a(2k-1) = A008594(k-1) k>0, a(2k)-a(2k-1) = 7.
a(n)-a(-n) = A008594(n) for n>0.
Sum_{i=1..n} a(2*i) = A049453(n) for n>0.
Sum_{i=1..n} a(2*i-1) = A049598(n-1) for n>0.
E.g.f.: 5 + ((12*x - 11)*exp(x) + exp(-x))/2. - David Lovler, Sep 04 2022
Sum_{n>=2} (-1)^n/a(n) = log(2)/4 + log(3)/8 - ((sqrt(3)-1)*Pi + 2*(sqrt(3)+3)*log(sqrt(3)+2))/(24*(sqrt(3)+1)). - Amiram Eldar, Sep 17 2023

A308400 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(6*k + 1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 1, 1, 3, 0, 6, 1, 3, 3, 1, 8, 1, 8, 3, 3, 9, 2, 14, 3, 9, 9, 4, 19, 4, 19, 9, 10, 21, 6, 32, 10, 22, 22, 12, 42, 12, 43, 23, 25, 48, 18, 67, 25, 51, 51, 31, 88, 31, 90, 54, 59, 101, 44, 137, 60, 108, 109, 73, 177, 73
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Comments

Number of partitions of n into parts congruent to {0, 5, 7} mod 12.
Convolution inverse of A247223.

Crossrefs

Cf. A006950, A036498, A049453, A210964, A247223, A263536, A308399.

Programs

  • Mathematica
    nmax = 78; CoefficientList[Series[1/Sum[(-x)^(k (6 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]
nmax = 78; CoefficientList[Series[Product[1/((1 - x^(12 k - 7)) * (1 - x^(12 k - 5)) * (1 - x^(12 k))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 / Sum_{k>=1} (-x)^A036498(k).
G.f.: Product_{k>=1} 1 / ((1 - x^(12*k - 7)) * (1 - x^(12*k - 5)) * (1 - x^(12*k))).
a(n) ~ (sqrt(3) - 1) * exp(sqrt(n/6)*Pi) / (2^(5/2)*n). - Vaclav Kotesovec, May 25 2019

A257144 Numbers n not of the form x+y*x^2 for x>1 and y>0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 40, 41, 43, 44, 45, 47, 49, 51, 53, 59, 60, 61, 63, 64, 65, 67, 69, 71, 73, 76, 77, 79, 81, 83, 85, 87, 88, 89, 91, 92, 95, 96, 97, 99, 101, 103, 104, 107, 108, 109, 112, 113, 115, 117, 119, 121
Offset: 1

Views

Author

Gionata Neri, Apr 16 2015

Keywords

Comments

Number n such that (d*k+1) /= (n/d), for k>0 and each value of d, where d is a divisor >1 of n.

Crossrefs

For numbers of the form x+y*x^2 with 0A002378 (y=1), A014105 (y=2), A049451 (y=3), A007742 (y=4), A202803 (y=5), A049453 (y=6), A092277 (y=7), A139275 (y=8), A154517 (y=9), A055437 (y=10). - Danny Rorabaugh, Apr 20 2015

Programs

  • Mathematica
    n = 71; Take[Complement[Range[n^2], DeleteDuplicates@ Sort@ Flatten@ Table[x + y x^2, {x, 2, n}, {y, 1, n}]], n] (* Michael De Vlieger, Apr 17 2015 *)
Previous Showing 21-23 of 23 results.

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