A052944 -id:A052944 - cp's OEIS frontend

A160692 Iteration of (k terms=k, followed by 2^k-k terms=0), beginning with k=0.

0, 1, 0, 2, 2, 0, 0, 3, 3, 3, 0, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, May 24 2009

For n with 2^k - 1 <= n < 2^k + k-1: a(n) = k;

for n with 2^k + k-1 <= n < 2^(k+1) - 1: a(n) = 0.

A000225, A083706, A052944, A000918.

a(n) = f(n,0) with f(n,k) = if n<2^k then (if n

A210792 Triangle of coefficients of polynomials v(n,x) jointly generated with A210791; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 10, 11, 5, 1, 19, 28, 25, 8, 1, 36, 62, 81, 50, 13, 1, 69, 129, 218, 193, 98, 21, 1, 134, 261, 533, 597, 442, 185, 34, 1, 263, 522, 1235, 1631, 1559, 952, 343, 55, 1, 520, 1040, 2773, 4129, 4763, 3758, 1985, 625, 89, 1, 1033, 2071

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 2: A052944.
Row sums: A000244 (powers of 3).
Alternating row sums: A001333.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 1/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012

			First five rows:
  1;
  1,  2;
  1,  5,  3;
  1, 10, 11,  5;
  1, 19, 28, 25,  8;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 5x + 3x^2
From _Philippe Deléham_, Mar 29 2012: (Start)
(1, 0, 1/2, 3/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   5,   3,   0;
  1,  10,  11,   5,   0;
  1,  19,  28,  25,   8,   0;
  1,  36,  62,  81,  50,  13,   0;
  1,  69, 129, 218, 193,  98,  21,   0; (End)

Cf. A210791, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 2; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210791 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210792 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A007051 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A001129 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A001333 *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 29 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*x - y*x + 3*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

A132736 Row sums of triangle A132735.

Original entry on oeis.org

1, 2, 5, 10, 19, 36, 69, 134, 263, 520, 1033, 2058, 4107, 8204, 16397, 32782, 65551, 131088, 262161, 524306, 1048595, 2097172, 4194325, 8388630, 16777239, 33554456, 67108889, 134217754, 268435483, 536870940, 1073741853, 2147483678

Offset: 0

Gary W. Adamson, Aug 26 2007

nonn

Apart from first term, the same as A052944. - R. J. Mathar, Jun 12 2008

			a(4) = 19 = sum of row 4 terms of triangle A132735: (1 + 5 + 7 + 5 + 1).
a(3) = 10 = (1, 3, 3, 1) dot (1, 1, 2, 0) = (1 + 3 + 6 + 0).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A132735.

[1] cat [2^n + n-1: n in [1..30]]; // G. C. Greubel, Feb 14 2021

a:= proc(n) option remember; if n=0 then 1 else add((binomial(n,j)+1), j=0..n-1) fi end: seq(a(n), n=0..31);# Zerinvary Lajos, Mar 29 2009

Table[2^n + n-1 + Boole[n==0], {n,0,30}] (* G. C. Greubel, Feb 14 2021 *)

[1]+[2^n + n-1 for n in (1..30)] # G. C. Greubel, Feb 14 2021

Binomial transform of [1, 1, 2, 0, 2, 0, 2, 0, 2, ...].
From Colin Barker, Aug 12 2012: (Start)
a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3) for n>3.
G.f.: (1 -2*x +2*x^2 -2*x^3)/((1-x)^2*(1-2*x)). (End)
From G. C. Greubel, Feb 14 2021: (Start)
a(n) = 2^n + n - 1 + [n=0].
E.g.f.: 1 - (1-x)*exp(x) + exp(2*x). (End)

More terms from R. J. Mathar, Jun 12 2008

A247241 In base 2, smallest positive integer k such that k contains all possible strings of n binary digits.

Original entry on oeis.org

2, 19, 558, 272060, 34949682936, 297564712194085870576, 10932581082490851526951746612437309648864, 7425338450010038712278392695279246718801455644988538421150211866664645481447360

Offset: 1

Anthony Sand, Nov 28 2014

In base 2, there are 2^n distinct strings of n digits and the smallest positive integer to contain all of them will have 2^n+(n-1) digits (see A052944). For example, there are 2^2 = 4 distinct strings of two digits: 00, 01, 10, 11. All of them are represented in the 5-digit binary number 10011 = 19 in base 10.

			n = 1: a(1) = 2 = 10 in base 2, which contains 2 distinct strings of one digit: 1, 0.
n = 2: a(2) = 19 = 10011, which contains 4 distinct strings of two digits: 10, 00, 01, 11.
n = 3: a(3) = 558 = 1000101110, which contains 8 distinct strings of three digits: 100, 000, 001, 010, 101, 011, 111, 110.
n = 4: a(4) = 272060 = 1000010011010111100, which contains 16 distinct strings of four digits: 1000, 0000, 0001, 0010, 0100, 1001, 0011, 0110, 1101, 1010, 0101, 1011, 0111, 1111, 1110, 1100.

Anthony Sand, Table of n, a(n) for n = 1..12

Cf. A249907, A033008.

A342298 a(n) is the greatest k such that A342263(k) = n.

Original entry on oeis.org

2, 25, 931, 504455, 67539587599, 585462196329239562271, 21690980800898420269408456526391711768639, 14792097944732868603877386771665610972834204784426907551800717772696470224928895

Offset: 0

Rémy Sigrist, Mar 07 2021

Rémy Sigrist, PARI program for A342298

Cf. A052944, A070939, A247241, A342263.

PARI
```
See Links section.
```

A070939(a(n)) = A052944(n+1).

A309832 Maximum length of a binary sequence S such that every pair of subwords of length n in S is of Hamming distance at least 2.

Original entry on oeis.org

3, 5, 7, 14, 23, 41, 74

Offset: 2

Yeow Meng Chee, Aug 19 2019

Cf. A052944, A309833.

A309833 Maximum length of a binary sequence S such that any pair of distinct subwords of length n in S is of Hamming distance at least 3.

Original entry on oeis.org

4, 5, 7, 12, 20, 25, 39, 71

Offset: 3

Yeow Meng Chee, Aug 19 2019

Cf. A052944, A309832.

A373895 a(n) = ceiling((2^n+n-1)/n).

Original entry on oeis.org

2, 3, 4, 5, 8, 12, 20, 33, 58, 104, 188, 343, 632, 1172, 2186, 4097, 7712, 14565, 27596, 52430, 99866, 190652, 364724, 699052, 1342179, 2581112, 4971028, 9586982, 18512792, 35791396, 69273668, 134217729, 260301050, 505290272, 981706812, 1908874355, 3714566312, 7233629132, 14096302922, 27487790696

Offset: 1

Eric W. Weisstein, Jun 21 2024

a(n) is the irregularity strength of the hypercube graph Q_n for n >= 2.

Michael De Vlieger, Table of n, a(n) for n = 1..3333
G. Ebert, J. Hammenter, F. Lazebnik, and A. Woldar, Irregularity Strengths for Certain Graphs, Congressus Numerantium 71 (1990), 39-52.
Eric Weisstein's World of Mathematics, Hypercube Graph.
Eric Weisstein's World of Mathematics, Irregularity Strength.

Cf. A052944, A053638, A082482.

Array[Ceiling[(2^# + # - 1)/#] &, 120] (* Michael De Vlieger, Jun 21 2024 *)

A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 7, 4, 3, 3, 15, 8, 5, 4, 4, 31, 16, 9, 6, 5, 5, 63, 32, 17, 10, 7, 6, 6, 127, 64, 33, 18, 11, 8, 7, 7, 255, 128, 65, 34, 19, 12, 9, 8, 8, 511, 256, 129, 66, 35, 20, 13, 10, 9, 9, 1023, 512, 257, 130, 67, 36, 21, 14, 11, 10, 10, 2047, 1024, 513, 258, 131, 68, 37, 22

Offset: 0

Ross La Haye, Feb 23 2004

			{0};
{1,1};
{3,2,2};
{7,4,3,3};
{15,8,5,4,4};
{31,16,9,6,5,5};
{63,32,17,10,7,6,6};
a(5,3) = 34 because 2^5 + (3-1) = 34.

Rows: a(0, k) = A001477(k), a(1, k) = A000027(k+1) etc. etc. Columns: a(n, 0) = A000225(n). a(n, 1) = A000079(n). a(n, 2) = A000051(n). a(n, 3) = A052548(n). a(n, 4) = A062709(n). Diagonals: a(n, n+3) = A052968(n+1). a(n, n+2) = A005126(n). a(n, n+1) = A006127(n). a(n, n) = A052944(n). a(n, n-1) = A083706(n-1). Also note that the sums of the antidiagonals = the partial sums of the main diagonal, i.e., a(n, n).

Flatten[ Table[ Table[ a[i, n - i], {i, n, 0, -1}], {n, 0, 11}]] (* both from Robert G. Wilson v, Feb 26 2004 *)
Table[a[n, k], {n, 0, 10}, {k, 0, 10}] // TableForm (* to view the table *)

For k > 0, a(n, k)= a(n, k-1) + 1.

a(n, k) = 2^n + (k-1).

More terms from Robert G. Wilson v, Feb 23 2004

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cp's OEIS Frontend

A094456 Triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

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A160692 Iteration of (k terms=k, followed by 2^k-k terms=0), beginning with k=0.

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A210792 Triangle of coefficients of polynomials v(n,x) jointly generated with A210791; see the Formula section.

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A132736 Row sums of triangle A132735.

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A247241 In base 2, smallest positive integer k such that k contains all possible strings of n binary digits.

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A342298 a(n) is the greatest k such that A342263(k) = n.

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A309832 Maximum length of a binary sequence S such that every pair of subwords of length n in S is of Hamming distance at least 2.

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A309833 Maximum length of a binary sequence S such that any pair of distinct subwords of length n in S is of Hamming distance at least 3.

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A373895 a(n) = ceiling((2^n+n-1)/n).

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A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

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