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 1-6 of 6 results.

A237274 a(n) = A236283(n) mod 9.

Original entry on oeis.org

2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1, 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1, 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1, 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1
Offset: 0

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Author

Paul Curtz, Feb 05 2014

Keywords

Comments

(Conjecture) This has period 18: repeat 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1.
The first 19 terms and the following 17 are palindromes.
The sorted terms in the conjectured period are 1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 4, 5, 5, 5, 7, 7, 7, 7.
Via the extended differences of A236283(n+1) and A236283(n+18) - A236283(n) which is A008600(n+9)=162, 180,... ,it is easy to see that A236283(0)=2.
A236283(-n) = A236283(n).
A236283(n) difference table:
2, 1, 4, 5, 10, 13, 20, 25, 34, 41,...
-1, 3, 1, 5, 3, 7, 5, 9, 7, 11,... = A097062(n+1)
4, -2, 4, -2, 4, -2, 4, -2, 4, -2,...
-6, 6, -6, 6, -6, 6, -6, 6, -6, 6,... .
A097062(n+1) mod 9 = (a(n+1) -a(n)) mod 9 =
period 18: repeat 8, 3, 1, 5, 3, 7, 5, 0, 7, 2, 0, 4, 2, 6, 4, 8, 6, 1 =b(n). b(n) + b(18-n)= 9, 9, 9, 9, 9, 9, 9, 0, 9.
Ordered b(n)=
period 18: repeat 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8.

Formula

a(n) = A236283(n) mod 9.

A081352 Main diagonal of square maze arrangement of natural numbers A081349.

Original entry on oeis.org

1, 7, 11, 21, 29, 43, 55, 73, 89, 111, 131, 157, 181, 211, 239, 273, 305, 343, 379, 421, 461, 507, 551, 601, 649, 703, 755, 813, 869, 931, 991, 1057, 1121, 1191, 1259, 1333, 1405, 1483, 1559, 1641, 1721, 1807, 1891, 1981, 2069, 2163, 2255, 2353, 2449, 2551
Offset: 0

Views

Author

Paul Barry, Mar 19 2003

Keywords

Comments

Conjecture: let a and b be integers such that 0 < a < b so that 0 < a/b is a proper fraction. Define the map f(a,b,D) = a/b + gcd(a,b)/D. Of course, all such a/b can be partially ordered by value, i.e., 1/2 = 0.5 < 2/3 = 4/6 = 6/9 = 0.6666... < 3/4 = 6/8 = 0.75 < 4/5 = 0.8 etc. The map f appears to specify a total strict order on the co-domain for all a/b that is consistent with the given partial order of the domain, i.e., the partial order remains intact, while equivalent fractions are given a total strict order themselves. Moreover, equivalent fractions are strictly ordered by numerator (or denominator), e.g., 1/2 < 2/4 < 3/6 etc. The conditions are that for n >= 4 all of the fractions with denominator b <= n are listed and the minimum integer value of D to achieve the total strict order of the co-domain is 2*C(n-1,2) - (-1)^(n-1). So, a(n-3) = D for n >= 4. Example: given n = 4, we have D = 2*(4-1,2) - (-1)^(4-1) = 2*3 + 1 = 7 = a(4-3) = a(1). Partial order of domain. 1/4 < 1/3 < 1/2 = 2/4 < 2/3 < 3/4. Total order of co-domain. f(1,4,7) = 1/4 + 1/7 = 33/84 < f(1,3,7) = 1/3 + 1/7 = 40/84 < f(1,2,7) = 1/2 + 1/7 = 54/84 < f(2,4,7) = 2/4 + 2/7 = 66/84 < f(2,3,7) = 2/3 + 2/7 = 68/84 < f(3,4,7) = 3/4 + 1/7 = 75/84. Observe that if D = 6, then f(2,4,6) = 2/4 + 2/6 = 10/12 = f(2,3,6) = 2/3 + 1/6. Computation shows the same failure to achieve total strict order of the co-domain for D = 2..5. (As a >= 1, then b >=2, from the above). Computation also shows that the conjecture holds for n = 4..17. - Ross La Haye, Oct 02 2016

Links

Crossrefs

Cf. A081350, A081351, A054569.

Programs

  • Magma
    I:=[1,7,11,21]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 08 2013
  • Maple
    A081352:=n->(n + 1)*(n + 2) - (-1)^n; seq(A081352(n), n=0..50); # Wesley Ivan Hurt, Feb 26 2014
  • Mathematica
    CoefficientList[Series[(1 + 5 x - 3 x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)
  • PARI
    x='x+O('x^99); Vec((1+5*x-3*x^2+x^3)/((1+x)*(1-x)^3)) \\ Altug Alkan, Mar 26 2016

Formula

a(n) = (n + 1)*(n + 2) - (-1)^n = 2*C(n+2, 2) - (-1)^n.
G.f.: (1 +5*x -3*x^2 +x^3) / ((1+x)*(1-x)^3). [Bruno Berselli, Aug 01 2010]
a(n) -2*a(n-1) +2*a(n-3) -a(n-4) = 0 with n>3. [Bruno Berselli, Aug 01 2010]
a(n) = 3*A000982(n + 2) - A000982(n + 3). - Miko Labalan, Mar 26 2016
a(n) = A116940(n) + A236283(n + 1). - Miko Labalan, Dec 04 2016
a(n) = (2*n^2 + 6*n - 2*(-1)^n + (-1)^(2*n) + 3)/2. - Kritsada Moomuang, Oct 24 2019

A116940 Greatest m such that A116939(m) = n.

Original entry on oeis.org

0, 3, 6, 11, 16, 23, 30, 39, 48, 59, 70, 83, 96, 111, 126, 143, 160, 179, 198, 219, 240, 263, 286, 311, 336, 363, 390, 419, 448, 479, 510, 543, 576, 611, 646, 683, 720, 759, 798, 839, 880, 923, 966, 1011, 1056, 1103, 1150, 1199, 1248, 1299, 1350, 1403, 1456
Offset: 0

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Author

Reinhard Zumkeller, Feb 27 2006

Keywords

Comments

From Andrew Rupinski, Nov 30 2009: (Start)
For n > 0, a(n) appears to be the set such that binomial(2*a(n),r) - binomial(2*a(n),r-2) = binomial(2*a(n),s) - binomial(2*a(n),s-2) for some r != s.
As a consequence of the Weyl Dimension Formula and the above comment, a(n) also appears to be the indices k such that the Symplectic Group Sp(k) has two fundamental irreducible representations of the same dimension. (End)

Examples

			a(n) = A000982(n) + A005843(n).
From _Andrew Rupinski_, Nov 30 2009: (Start)
a(1) = 3 and binomial(6,3)-binomial(6,1) = binomial(6,2)-binomial(6,0).
a(1) = 3 and the fundamental representations of Sp(3) are of dimensions 6, 14 and 14. a(2) = 6 and the fundamental representations of Sp(6) are of dimensions 12, 65, 208, 429, 572, and 429. (End)

Links

Crossrefs

Cf. A116939.
Cf. A081352, A236283.
Cf. A028884, A099392.
Cf. A000217, A001859.

Programs

  • Haskell
    import Data.List (elemIndices)
a116940 n = last $ elemIndices n $ takeWhile (<= n + 1) a116939_list
-- Reinhard Zumkeller, Jun 28 2013
  • Magma
    [(2*n*(n+4) -(-1)^n +1)/4: n in [0..55]]; // G. C. Greubel, Jan 26 2020
  • Maple
    seq( (2*(n+2)^2 -(-1)^n -7)/4, n=0..55); # G. C. Greubel, Jan 26 2020
  • Mathematica
    a = {0}; Do[AppendTo[a, If[Count[a, #-1] > #-1, #+1, #-1]] &@ a[[n]], {n, 1500}]; Most@ Values@ Map[Last, PositionIndex@ a] - 1 (* Michael De Vlieger, Dec 07 2016, Version 10 *)
Table[(2*(n+2)^2 -(-1)^n -7)/4, {n,0,55}] (* G. C. Greubel, Jan 26 2020 *)
  • PARI
    vector(56, n, (2*(n+1)^2 +(-1)^n -7)/4) \\ G. C. Greubel, Jan 26 2020
  • Sage
    [(2*n*(n+4) -(-1)^n +1)/4 for n in (0..55)] # G. C. Greubel, Jan 26 2020

Formula

a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 3.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Joerg Arndt, Apr 02 2011
G.f.: x*(3 - x^2)/((1 + x)*(1 - x)^3). - Arkadiusz Wesolowski, Jan 01 2012
a(n) = 2n + ceiling(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
a(n) = (2*n*(n + 4) - (-1)^n + 1)/4. - Bruno Berselli, Jun 14 2013
a(n) = A081352(n) - A236283(n + 1). - Miko Labalan, Dec 04 2016
From Klaus Purath, Jan 26 2020: (Start)
a(n) = binomial(n+2, 2) + floor((n-1)/2).
a(n) = floor(A028884(n)/2) - n.
a(n) = (n+1)^2 - A099392(n+1).
a(2*n)^2 - a(2*n-1)*a(2*n+1) = 3, n > 0.
a(2*n+1)^2 - a(2*n)*a(2*n+2) = (2*n+3)^2. (End)
E.g.f.: (1/2)*(x*(5 + x)*cosh(x) + (1 + 5*x + x^2)*sinh(x)). - Stefano Spezia, Jan 26 2020
a(n) = A000217(2*n) - 2*A001859(n-1) for n>0. - John Tyler Rascoe, Jul 31 2022
Sum_{n>=1} 1/a(n) = 11/8 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022

A236332 The number of orbits of 4-tuples of the dihedral group of order 2n acting on {1,2,...,n}.

Original entry on oeis.org

1, 8, 14, 36, 63, 112, 172, 260, 365, 504, 666, 868, 1099, 1376, 1688, 2052, 2457, 2920, 3430, 4004, 4631, 5328, 6084, 6916, 7813, 8792, 9842, 10980, 12195, 13504, 14896, 16388, 17969, 19656, 21438, 23332, 25327, 27440, 29660, 32004, 34461, 37048, 39754, 42596, 45563
Offset: 1

Views

Author

W. Edwin Clark, Jan 22 2014

Keywords

Comments

In other words, a(n) is the number of equivalence classes of length 4 words with an alphabet of size n where equivalence is up to rotation or reflection of the alphabet. - Andrew Howroyd, Jan 17 2020

Examples

			For n = 3 there are the following 14 orbits of 4-tuples for the group D6 = S3:
1) [[1,1,1,1], [2,2,2,2], [3,3,3,3]],
2) [[1,1,1,2], [2,2,2,3], [1,1,1,3], [3,3,3,1], [3,3,3,2], [2,2,2,1]],
3) [[1,1,2,1], [2,2,3,2], [1,1,3,1], [3,3,1,3], [3,3,2,3], [2,2,1,2]],
4) [[1,1,2,2], [2,2,3,3], [1,1,3,3], [3,3,1,1], [3,3,2,2], [2,2,1,1]],
5) [[1,1,2,3], [2,2,3,1], [1,1,3,2], [3,3,1,2], [3,3,2,1], [2,2,1,3]],
6) [[1,2,1,1], [2,3,2,2], [1,3,1,1], [3,1,3,3], [3,2,3,3], [2,1,2,2]],
7) [[1,2,1,2], [2,3,2,3], [1,3,1,3], [3,1,3,1], [3,2,3,2], [2,1,2,1]],
8) [[1,2,1,3], [2,3,2,1], [1,3,1,2], [3,1,3,2], [3,2,3,1], [2,1,2,3]],
9) [[1,2,2,1], [2,3,3,2], [1,3,3,1], [3,1,1,3], [3,2,2,3], [2,1,1,2]],
10) [[1,2,2,2], [2,3,3,3], [1,3,3,3], [3,1,1,1], [3,2,2,2], [2,1,1,1]],
11) [[1,2,2,3], [2,3,3,1], [1,3,3,2], [3,1,1,2], [3,2,2,1], [2,1,1,3]],
12) [[1,2,3,1], [2,3,1,2], [1,3,2,1], [3,1,2,3], [3,2,1,3], [2,1,3,2]],
13) [[1,2,3,2], [2,3,1,3], [1,3,2,3], [3,1,2,1], [3,2,1,2], [2,1,3,1]],
14) [[1,2,3,3], [2,3,1,1], [1,3,2,2], [3,1,2,2], [3,2,1,1], [2,1,3,3]].

Links

Crossrefs

Cf. A236283 (3-tuples).

Programs

  • GAP
    a:=function(n)
local g,orbs;
g:=DihedralGroup(IsPermGroup,2*n);
orbs := OrbitsDomain(g, Tuples( [ 1 .. n ], 4), OnTuples );
return Size(orbs);
end;;
  • PARI
    a(n) = {(9 + 7*(-1)^n + 2*n^3)/4} \\ Andrew Howroyd, Jan 17 2020

Formula

Conjectures from Colin Barker, Jan 22 2014: (Start)
a(n) = (9 + 7*(-1)^n + 2*n^3)/4.
G.f.: -x*(4*x^4-12*x^3+8*x^2-5*x-1) / ((x-1)^4*(x+1)).
(End)
From Andrew Howroyd, Jan 17 2020: (Start)
The above conjectures are true and can be derived from the following formulas for even and odd n.
a(n) = (n-2)*(n^2 + 2*n + 4)/2 + 8 for even n.
a(n) = (n-1)*(n^2 + n + 1)/2 + 1 for odd n.
(End)

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 17 2020

A340528 Radio number of the path graph P_n.

Original entry on oeis.org

0, 1, 3, 5, 10, 13, 20, 25, 34, 41, 52, 61, 74, 85, 100, 113, 130, 145, 164, 181, 202, 221, 244, 265, 290, 313, 340, 365, 394, 421, 452, 481, 514, 545, 580, 613, 650, 685, 724, 761, 802, 841, 884, 925, 970, 1013, 1060, 1105, 1154, 1201, 1252, 1301, 1354, 1405, 1460, 1513
Offset: 1

Views

Author

Eric W. Weisstein, Jan 10 2021

Keywords

Comments

a(n) is also the largest possible radio number for a graph on n nodes.

Links

Crossrefs

Cf. A236283.

Formula

a(n) = A236283(n-1) for n > 1, n != 3.
a(n) = 0 for n = 1,
= 3 for n = 3,
= (n - 2)*n/2 + 1 for n even,
= (n - 3)*(n + 1)/2 + 4 for n odd.
G.f.: x^2*(1 + x^2)*(1 + x - 2*x^2 + x^3)/((1 - x)^3*(1 + x)).

A376133 Triangle T read by rows: T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 and T(n, k) = T(n, k-1) + (-1)^k * 2 * (n+1-k) for k >= 2.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 6, 12, 8, 10, 9, 17, 11, 15, 13, 14, 24, 16, 22, 18, 20, 19, 31, 21, 29, 23, 27, 25, 26, 40, 28, 38, 30, 36, 32, 34, 33, 49, 35, 47, 37, 45, 39, 43, 41, 42, 60, 44, 58, 46, 56, 48, 54, 50, 52, 51, 71, 53, 69, 55, 67, 57, 65, 59, 63, 61, 62, 84, 64, 82, 66, 80, 68, 78, 70, 76, 72, 74
Offset: 1

Views

Author

Werner Schulte, Sep 11 2024

Keywords

Comments

Row n consists of the next n odd/even natural numbers if n is odd/even. So the sequence yields a permutation of the natural numbers.

Examples

			Row n=5: Next (1,3,5,7 see rows 1 and 3) five odd numbers are 9,11,13,15 and 17; with "9+8-6+4-2" we get 9,17,11,15,13 for row 5.
Row n=8: Next (2,4,..,24 see rows 2, 4 and 6) eight even numbers are 26,28,..,40; with "26+14-12+10-8+6-4+2" we get 26,40,28,38,30,36,32,34 for row 8.
Triangle T(n, k) for 1 <= k <= n starts:
n\ k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   2   4
   3 :   3   7   5
   4 :   6  12   8  10
   5 :   9  17  11  15  13
   6 :  14  24  16  22  18  20
   7 :  19  31  21  29  23  27  25
   8 :  26  40  28  38  30  36  32  34
   9 :  33  49  35  47  37  45  39  43  41
  10 :  42  60  44  58  46  56  48  54  50  52
  11 :  51  71  53  69  55  67  57  65  59  63  61
  12 :  62  84  64  82  66  80  68  78  70  76  72  74
  etc.

Crossrefs

Cf. A061925 (column 1), A074148 (column 2), A074149 (row sums), A236283 (main diagonal).

Programs

  • Maple
    T := (n, k) -> ((-1)^k*(2 + 4*(n - k)) + 2*n^2 + (-1)^n + 5)/4:
seq(seq(T(n, k), k = 1..n), n = 1..12);  # Peter Luschny, Sep 13 2024
  • PARI
    T(n,k)=(2*n*n+(-1)^k*4*(n-k)+5+2*(-1)^k+(-1)^n)/4

Formula

T(n, k) = (2*n*n + (-1)^k * 4 * (n - k) + 5 + 2 * (-1)^k + (-1)^n) / 4.
T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 = A061925(n-1).
T(n, 2) = (2*n*n + 4*n - 1 + (-1)^n) / 4 = A074148(n) for n > 1.
T(n, k) = T(n, k-2) - (-1)^k * 2 for 3 <= k <= n.
G.f.: x*y*(1 + 2*x*y + 2*x^5*y^2 + x^6*y^3 - x^4*y*(3 + y + y^2) - x^2*(1 + y + 3*y^2) + 2*x^3*(1 + y^3))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Sep 12 2024
Showing 1-6 of 6 results.

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