A266977 -id:A266977 - cp's OEIS frontend

A004116 a(n) = floor((n^2 + 6n - 3)/4).

1, 3, 6, 9, 13, 17, 22, 27, 33, 39, 46, 53, 61, 69, 78, 87, 97, 107, 118, 129, 141, 153, 166, 179, 193, 207, 222, 237, 253, 269, 286, 303, 321, 339, 358, 377, 397, 417, 438, 459

Offset: 1

N. J. A. Sloane

a(n)-3 is the maximal size of a regular triangulation of a prism over a regular n-gon.
Solution to a postage stamp problem with 2 denominations.
This sequence is half the degree of the denominator of a certain sequence of rational polynomials defined in the referenced paper by G. Alkauskas. Although this fact is not documented in the paper it can be verified by running the author's code and evaluating degree(denom(...)). - Stephen Crowley, Sep 18 2011
From Griffin N. Macris, Jul 19 2016: (Start)
Consider quadratic functions x^2+ax+b. Then a(n) is the number of these functions with 0 <= a+b < n, modulo changing x to x+c for a constant c.
For a(6)=17, four functions are excluded, because:
x^2 + 2x + 1 = (x+1)^2 + 0(x+1) + 0
x^2 + 2x + 2 = (x+1)^2 + 0(x+1) + 1
x^2 + 2x + 3 = (x+1)^2 + 0(x+1) + 2
x^2 + 3x + 2 = (x+1)^2 + 1(x+1) + 0 (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
G. Alkauskas, Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator, arXiv:1004.1783 [math.NT], 2010-2012. See also code.
R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. Develin, Maximal triangulations of a regular prism, arXiv:math/0309220 [math.CO], 2003.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 420.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
David Singmaster, David Fielker, and N. J. A. Sloane, Correspondence, August 1979.
Wikipedia, Gauss-Kuzmin-Wirsing operator.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

[Floor( (n^2 + 6*n - 3)/4 ) : n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

A004116:=(-1-z+z**3)/(z+1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

Table[Floor[(n^2 + 6 n - 3)/4], {n, 40}] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {1, 3, 6, 9}, 40] (* Michael De Vlieger, Jul 19 2016 *)

PARI
```
a(n)=(n^2+6*n-3)>>2
```

a(n) = floor((1/4)*n^2 + (3/2)*n + 1/4) - 1.
a(n) = (1/8)*(-1)^(n+1) - 7/8 + (3/2)*n + (1/4)*n^2.
From Ilya Gutkovskiy, Jul 20 2016: (Start)
O.g.f.: x*(1 + x - x^3)/((1 - x)^3*(1 + x)).
E.g.f.: (8 + sinh(x) - cosh(x) + (2*x^2 + 14*x - 7)*exp(x))/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = Sum_{k=0..n-1} A266977(k). (End)
Sum_{n>=1} 1/a(n) = 2 + tan(sqrt(13)*Pi/2)*Pi/sqrt(13) - cot(sqrt(3)*Pi)*Pi/(2*sqrt(3)). - Amiram Eldar, Aug 13 2022

A333416 Irregular triangle T read by rows: each row represents a finite (increasing) oscillation contained in the infinite (increasing) oscillation O.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 4, 1, 3, 3, 1, 5, 2, 4, 2, 4, 1, 5, 3, 3, 1, 5, 2, 6, 4, 2, 4, 1, 6, 3, 5

Offset: 1

Stefano Spezia, Mar 24 2020

The oscillations are bounded affine permutations. For the definition of a bounded affine permutation, see Definitions 1 and 2 in Madras and Troyka.
The infinite (increasing) oscillation O is described by the function f defined as f(s) = s - 4*(-1)^s - 2 with s in the set of integers, while the finite (increasing) oscillations are indecomposable permutations, i.e., that are not the sum of two permutations of nonzero size, and that are contained in O.
For each m >= 3, there are exactly two oscillations of size m: 312 and 231, 3142 and 2413, and so on (see p. 22 of Madras and Troyka).

			1
2  1
3  1  2
2  3  1
3  1  4  2
2  4  1  3
3  1  5  2  4
2  4  1  5  3
3  1  5  2  6  4
2  4  1  6  3  5

Michael H. Albert, Robert Brignall, Vincent Vatter, Large infinite antichains of permutations, arXiv:1212.3346 [math.CO], 2012; Pure Mathematics and Applications, 24(2) pp. 47-57 (2013).
Neal Madras, Justin M. Troyka, Bounded affine permutations I. Pattern avoidance and enumeration, arXiv:2003.00267 [math.CO], 2020.
Vincent Vatter, Small permutation classes, arXiv:0712.4006 [math.CO], 2007; Proc. Lond. Math. Soc. 103 (2011), 879-921.

Cf. A000142, A266977, A158478, A333616 (row sums).

T(n, 1) = A158478(n).

A270059 Number of distinct digits needed to write n in all bases >= 2.

Original entry on oeis.org

1, 1, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34

Offset: 0

Gionata Neri, Mar 09 2016

			4 written in all bases 2,3,4,5,6,7,... is 100,11,10,4,4,4,... and the distinct digits needed are 0, 1 and 4, so a(4) = 3.

Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A008619, A266977.

Table[Length[Union@ Flatten@ Map[IntegerDigits[n, #] &, Range[2, n + 1]] /. {} -> {0}], {n, 0, 120}] (* or *)
Table[Length[Union@ Flatten@ Map[IntegerDigits[n, #] &, Range[2, n + 1]] /. {} -> {0}], {n, 0, 2}]~Join~Table[Floor[(n + 3)/2], {n, 3, 120}] (* or *)
CoefficientList[Series[1 + (x (1 + 2 x - x^2 - 2 x^3 + x^4))/((-1 + x)^2 (1 + x)), {x, 0, 120}], x] (* Michael De Vlieger, Mar 09 2016 *)

a(n) = {if (n <= 1, 1, v = []; for (b=2, n+1, v = vecsort(concat(v, digits(n, b)),,8)); #v;);} \\ Michel Marcus, Mar 09 2016

For n >= 3, a(n) = floor((n + 3)/2).

G.f.: 1 + (x*(1 + 2*x - x^2 - 2*x^3 + x^4))/((-1 + x)^2*(1 + x)). - Michael De Vlieger, Mar 09 2016

cp's OEIS Frontend

A004116 a(n) = floor((n^2 + 6n - 3)/4).

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A333416 Irregular triangle T read by rows: each row represents a finite (increasing) oscillation contained in the infinite (increasing) oscillation O.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A270059 Number of distinct digits needed to write n in all bases >= 2.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula