A036563 -id:A036563 - cp's OEIS frontend

A261693 Irregular triangle read by rows in which row n lists the positive odd numbers in decreasing order starting with 2^n - 1. T(0, 1) = 0 and T(n, k) for n >= 1, 1 <= k <= 2^(n-1).

0, 1, 3, 1, 7, 5, 3, 1, 15, 13, 11, 9, 7, 5, 3, 1, 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 63, 61, 59, 57, 55, 53, 51, 49, 47, 45, 43, 41, 39, 37, 35, 33, 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 127, 125, 123, 121, 119, 117, 115, 113, 111, 109, 107, 105, 103, 101, 99, 97, 95, 93

Offset: 0

Omar E. Pol, Sep 25 2015

Also the first differences of A261692.
Number of cells turned ON at n-th stage of the cellular automaton of A261692.
This irregular triangle A (instead of T) appears also in the linearization of the following product of Chebyshev T polynomials (A053120): PrT(n) := Product_{j=1..n} T(2^j, x)  = (1/2^(n-1))*Sum_{k=1..2^(n-1)} T(2*A(n, k), x), for n >= 1. Proof via 2*T(n, x)*T(m, x) = T(n+m, x) + T(|n-m|, x). - Wolfdieter Lang, Oct 26 2019

			With the terms written as an irregular triangle T in which row lengths are the terms of A011782 the sequence begins:
0;
1;
3, 1;
7, 5, 3, 1;
15, 13, 11, 9, 7, 5, 3, 1;
31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1;
...
-------------------------------------------------------------------------------
From _Wolfdieter Lang_, Oct 26 2019: (Start)
Chebyshev T(2^j)-products (the argument x is here omitted):
n = 1: T(2) = (2^0)*T(2*1),
n = 2: T(2)*T(4) = (1/2)*(T(2*3) + T(2*1)) = (T(6) + T(2))/2,
n = 3: T(2)*T(4)*T(8) =  (1/2^2)*(T(2*7) + T(2*5) + T(2*3) + T(2*1))
       = (T(14) + T(10) + T(6) + T(2))/4.
... (End)

Cf. A036563, A005408, A011782, A080079, A099375, A147582, A262617, A261692, A262621.

Column 1 is A000225. Row sums give A000302, n >= 1.

A261693 := n -> Bits:-Nor(2*n, 2*n):
seq(A261693(n), n=0..81); # Peter Luschny, Sep 23 2019

Table[Reverse[2 Range[2^(n - 1)] - 1], {n, 0, 7}] /. {} -> 0 // Flatten (* Michael De Vlieger, Oct 05 2015 *)

tabf(nn) = {for (n=0, nn, print1(n, ":"); for (k=1, 2^(n-2), print1(2^(n-1) - 2*k + 1, ", ");); print(););} \\ Michel Marcus, Oct 27 2015

T(n, k) = 2^n + 1 - 2*k, n >= 1, 1 <= k <= 2^(n-1), and T(0, 0) = 0.

As a sequence: a(n) = A262621(n)/4, n >= 1, and a(0) = 0.

Corrections by Wolfdieter Lang, Nov 15 2019

A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Original entry on oeis.org

1, 1, 1, 4, -1, 1, 0, 4, -5, 1, 24, -16, 4, -13, 1, -8, 40, -48, 4, -29, 1, 104, -88, 72, -112, 4, -61, 1, -48, 184, -248, 136, -240, 4, -125, 1, 352, -400, 344, -568, 264, -496, 4, -253, 1, 80, 544, -1104, 664, -1208, 520, -1008, 4, -509, 1, 1424, -784, 928, -2512, 1304, -2488, 1032, -2032, 4, -1021, 1

Offset: 1

Antti Karttunen, Nov 22 2019

			The top left corner of the array:
   n   p_n |k=1,     2, 3,      4,     5,      6,     7,       8,      9,      10
  ---------+----------------------------------------------------------------------
   1 ->  2 |  1,     1, 4,      0,    24,     -8,   104,     -48,    352,      80,
   2 ->  3 |  1,    -1, 4,    -16,    40,    -88,   184,    -400,    544,    -784,
   3 ->  5 |  1,    -5, 4,    -48,    72,   -248,   344,   -1104,    928,   -2512,
   4 ->  7 |  1,   -13, 4,   -112,   136,   -568,   664,   -2512,   1696,   -5968,
   5 -> 11 |  1,   -29, 4,   -240,   264,  -1208,  1304,   -5328,   3232,  -12880,
   6 -> 13 |  1,   -61, 4,   -496,   520,  -2488,  2584,  -10960,   6304,  -26704,
   7 -> 17 |  1,  -125, 4,  -1008,  1032,  -5048,  5144,  -22224,  12448,  -54352,
   8 -> 19 |  1,  -253, 4,  -2032,  2056, -10168, 10264,  -44752,  24736, -109648,
   9 -> 23 |  1,  -509, 4,  -4080,  4104, -20408, 20504,  -89808,  49312, -220240,
  10 -> 29 |  1, -1021, 4,  -8176,  8200, -40888, 40984, -179920,  98464, -441424,
  11 -> 31 |  1, -2045, 4, -16368, 16392, -81848, 81944, -360144, 196768, -883792,

Cf. A000079, A000203, A000225, A075708, A182944, A329610, A329644, A329890.
Rows 1-2: A329891, A329892 (from n>=1).
Column 1: A000012, Column 2: -A036563(n) (from n>=1), Column 3: A010709.

up_to = 105;
A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329637sq(n,k) = ((2^(n+k-1)) - (((2^n)-1) * A329890(k)));
A329637list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A329637sq(col,(a-(col-1))))); (v); };
v329637 = A329637list(up_to);
A329637(n) = v329637[n];

A(n, k) = A329644(A182944(n, k)).

A(n, k) = A000079(n+k-1) - (A000225(n) * A329890(k)).

A131437 (A000012 * A131436) + (A131436 * A000012) - A000012.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 21, 29, 31, 33, 37, 45, 61, 63, 65, 69, 77, 93, 125, 127, 129, 133, 141, 157, 189, 253, 255, 257, 261, 269, 285, 317, 381, 509, 511, 513, 517, 525, 541, 573, 637, 765, 1021, 1023, 1025, 1029, 1037, 1053, 1085, 1149, 1277, 1533, 2045

Offset: 1

Gary W. Adamson, Jul 11 2007

Left column = 2^n - 1; right border = A036563, 2^(n+1) - 3: (1, 5, 13, 29, 61, 125, ...). Row sums = A131438: (1, 8, 29, 82, 207, 492, 1129, ...).

			First few rows of the triangle are:
1;
3, 5;
7, 9, 13;
15, 17, 21, 29;
31, 33, 37, 45, 61;
63, 65, 69, 77, 93, 125;
...

Cf. A036563, A131436, A131438, A131439.

A000012 := proc(n,k)
        1 ;
end proc:
A131436 := proc(n,k)
        if k = n then
                2^n-1 ;
        else
                0;
        end if;
end proc:
A131437 := proc(n,k)
        add( A000012(n,i)*A131436(i,k) + A131436(n,i)*A000012(i,k),i=k..n) -1 ;
end proc:
seq(seq(A131437(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Sep 24 2011

(A000012 * A131436) + (A131436 * A000012) - A000012; as infinite lower triangular matrices.

Corrected by R. J. Mathar, Sep 24 2011

A220087 a(n) = 2^n - 27.

Original entry on oeis.org

-26, -25, -23, -19, -11, 5, 37, 101, 229, 485, 997, 2021, 4069, 8165, 16357, 32741, 65509, 131045, 262117, 524261, 1048549, 2097125, 4194277, 8388581, 16777189, 33554405, 67108837, 134217701, 268435429, 536870885, 1073741797, 2147483621, 4294967269

Offset: 0

Andreas Rieber, Dec 04 2012

sign

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A036563, A185346.

Table[2^n - 27, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)
LinearRecurrence[{3,-2},{-26,-25},40] (* Harvey P. Dale, May 17 2018 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (53*x - 26)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 08 2023: (Start)
a(n) = 2*a(n-1) + 27 with a(0) = -26.
E.g.f.: exp(2*x) - 27*exp(x). (End)

A226618 Irregular array read by rows in which row n lists the positive integers k in ascending order for which 1 is in a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 13, 29, 11, 61, 17, 125, 253, 509, 145, 203, 1021, 43, 2045, 55, 4093, 355, 1169, 8189, 137, 3275, 16381, 1129, 32765, 1007, 5957, 9361, 65533, 131069, 97, 52427, 262141, 643, 74897, 524285, 41, 1048573, 553, 28727, 110375, 2097149, 281, 673, 2075, 9731, 34663

Offset: 1

Geoffrey H. Morley, Jul 02 2013

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd, where k is odd.
For primitive cycles, GCD(k,6)=1.

			The irregular array starts:
1;
5;
13;
29;
11, 61;
17, 125; ...
Row 1 is empty.

Geoffrey H. Morley, Rows 2..26 of array, flattened

The first element in row n is A226616(n), and the last is A036563(n) = 2^n-3.

Cf. A226609, A226619.

A226619 Irregular array read by rows in which row n lists the integers k, in ascending order, for which there is a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

-1, 1, 1, -1, 5, -11, 7, 13, -49, 5, 23, 29, -179, -17, 11, 37, 55, 61, -601, -115, 17, 47, 101, 119, 125, -1931, -473, 13, 25, 35, 175, 229, 247, 253, -6049, -1675, -217, -31, 97, 269, 431, 485, 503, 509, -18659, -5537, -1163, -791, 59, 71, 145, 203, 295, 781, 943, 997, 1015, 1021

Offset: 1

Geoffrey H. Morley, Jul 02 2013

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd, where k is odd.
For primitive cycles, GCD(k,6)=1.
We associate the cycle {0} with k = A226606(2) = 1.
For n>1 the first term of row n is 2^n-3^(n-1), and the last term is A036563(n) = 2^n-3.

			The irregular array starts:
-1, 1;
1;
-1, 5;
-11, 7, 13;
-49, 5, 23, 29; ...

Geoffrey H. Morley, Rows 1..26 of array, flattened

Cf. A226605, A226606, A226618, A226620, A226621.

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A290113 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 16381, 32765, 65533, 131069, 262141, 524285, 1048573, 2097149, 4194301, 8388605, 16777213, 33554429, 67108861, 134217725, 268435453, 536870909, 1073741821, 2147483645, 4294967293, 8589934589

Offset: 0

Robert Price, Jul 19 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Essentially the same as A091270.

Cf. A036563, A290111, A290112, A290114.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 643; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

For n>1, a(n) = 2^(n+1)-3.
a(n) = A036563(n+1) for n > 1. - Georg Fischer, Oct 30 2018
From Chai Wah Wu, Apr 02 2024: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 3.
G.f.: (4*x^3 - 2*x^2 + 1)/(2*x^2 - 3*x + 1). (End)

A331105 T(n,k) = -k*(k+1)/2 mod 2^n; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 2, 0, 7, 5, 2, 6, 1, 3, 4, 0, 15, 13, 10, 6, 1, 11, 4, 12, 3, 9, 14, 2, 5, 7, 8, 0, 31, 29, 26, 22, 17, 11, 4, 28, 19, 9, 30, 18, 5, 23, 8, 24, 7, 21, 2, 14, 25, 3, 12, 20, 27, 1, 6, 10, 13, 15, 16, 0, 63, 61, 58, 54, 49, 43, 36, 28, 19, 9

Offset: 0

Alois P. Heinz, Jan 09 2020

Row n is a permutation of {0, 1, ..., A000225(n)}.

			Triangle T(n,k) begins:
  0;
  0,  1;
  0,  3,  1,  2;
  0,  7,  5,  2, 6, 1,  3, 4;
  0, 15, 13, 10, 6, 1, 11, 4, 12, 3, 9, 14, 2, 5, 7, 8;
  ...

Alois P. Heinz, Rows n = 0..15, flattened

Columns k=0-2 give: A000004, A000225, A036563 (for n>1).
Row sums give A006516.
Row lengths give A000079.
T(n,n) gives A014833 (for n>0).
T(n,2^n-1) gives A131577.
Cf. A329278, A363674.

T:= n-> (p-> seq(modp(-k*(k+1)/2, p), k=0..p-1))(2^n):
seq(T(n), n=0..6);
# second Maple program:
T:= proc(n, k) option remember;
      `if`(k=0, 0, T(n, k-1)-k mod 2^n)
    end:
seq(seq(T(n, k), k=0..2^n-1), n=0..6);

T[n_, k_] := T[n, k] = If[k == 0, 0, Mod[T[n, k - 1] - k, 2^n]];
Table[Table[T[n, k], {k, 0, 2^n - 1}], {n, 0, 6}] // Flatten (* Jean-François Alcover, Mar 28 2022, after Alois P. Heinz *)

A382674 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^4.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 77, 29, 1, 1, 61, 325, 325, 61, 1, 1, 125, 1181, 2357, 1181, 125, 1, 1, 253, 3973, 13621, 13621, 3973, 253, 1, 1, 509, 12797, 69269, 118061, 69269, 12797, 509, 1, 1, 1021, 40165, 326005, 862261, 862261, 326005, 40165, 1021, 1

Offset: 0

Seiichi Manyama, Apr 03 2025

			Square array begins:
  1,   1,    1,     1,      1,       1, ...
  1,   5,   13,    29,     61,     125, ...
  1,  13,   77,   325,   1181,    3973, ...
  1,  29,  325,  2357,  13621,   69269, ...
  1,  61, 1181, 13621, 118061,  862261, ...
  1, 125, 3973, 69269, 862261, 8712245, ...
  ...

Columns k=0..2 give A000012, A036563(n+2), A382677.
Main diagonal gives A382678.
Cf. A099594, A136126, A382673.
Cf. A382736.

a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2));

E.g.f.: exp(x+y) / (exp(x) + exp(y) - exp(x+y))^4.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+3,3) * Stirling2(n+1,j+1) * Stirling2(k+1,j+1).

cp's OEIS Frontend

A261693 Irregular triangle read by rows in which row n lists the positive odd numbers in decreasing order starting with 2^n - 1. T(0, 1) = 0 and T(n, k) for n >= 1, 1 <= k <= 2^(n-1).

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A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

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A131437 (A000012 * A131436) + (A131436 * A000012) - A000012.

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A220087 a(n) = 2^n - 27.

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A226618 Irregular array read by rows in which row n lists the positive integers k in ascending order for which 1 is in a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

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A226619 Irregular array read by rows in which row n lists the integers k, in ascending order, for which there is a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

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A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

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A290113 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

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