A042948 -id:A042948 - cp's OEIS frontend

A151972 Numbers that are congruent to {0, 1, 6, 10} mod 15.

0, 1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225

Offset: 1

N. J. A. Sloane, Aug 23 2009

Also, numbers n such that n^2 - n is divisible by 15.

Also, numbers n such that n^2 - n is divisible by 30.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

For m^2 == m (mod n), see: n=2: A001477, n=3: A032766, n=4: A042948, n=5: A008851, n=6: A032766, n=7: A047274, n=8: A047393, n=9: A090570, n=10: A008851, n=11: A112651, n=12: A112652, n=13: A112653, n=14: A047274, n=15: A151972, n=16: A151977, n=17: A151978, n=18: A090570, n=19: A151979, n=20: A151980, n=21: A151971, n=22, A112651, n=24: A151973, n=26: A112653, n=30: A151972, n=32: A151983, n=34: A151978, n=38: A151979, n=42: A151971, n=48: A151981, n=64: A151984.

Cf. A215202.

[ n : n in [0..1000] | n mod 15 in [0, 1, 6, 10]]; // Vincenzo Librandi, Apr 02 2011, simplified by Eric M. Schmidt, Aug 05 2012

[ n: n in [0..1000] | (n^2-n) mod (15) eq 0 ]; // Vincenzo Librandi, Apr 03 2011, altered by Eric M. Schmidt, Aug 05 2012

A151972:=n->(30*n-41-5*I^(2*n)+(3+3*I)*I^(-n)+(3-3*I)*I^n)/8: seq(A151972(n), n=1..100); # Wesley Ivan Hurt, Jun 07 2016

Select[Range[0, 300], Divisible[#^2-#, 15]&]  (* Harvey P. Dale, Apr 01 2011, altered by Eric M. Schmidt, Aug 05 2012 *)

G.f.: x^2*(1+5*x+4*x^2+5*x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Jun 07 2016: (Start)
a(n) = (30*n-41-5*i^(2*n)+(3+3*i)*i^(-n)+(3-3*i)*i^n)/8 where i=sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. (End)
E.g.f.: (20 + (15*x - 23)*cosh(x) + 3*(sin(x) + cos(x) + (5*x - 6)*sinh(x)))/4. - Ilya Gutkovskiy, Jun 07 2016

This is a merge of two identical sequences, A151972 and A151975.

A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 15, 8, 11, 6, 31, 16, 23, 12, 9, 63, 32, 47, 24, 19, 10, 127, 64, 95, 48, 39, 20, 13, 255, 128, 191, 96, 79, 40, 27, 14, 511, 256, 383, 192, 159, 80, 55, 28, 17, 1023, 512, 767, 384, 319, 160, 111, 56, 35, 18, 2047, 1024, 1535, 768, 639

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A000225 (-1+2^n)
Row 2:  A000079 (2^n)
Row 3:  A055010
Row 4:  3*A000079
Row 5:  A153894
Row 6:  5*A000079
Row 7:  A086224
Row 8:  A005009
Row 9:  A052996
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
This sequence is a permutation of the natural numbers.  - L. Edson Jeffery, Aug 13 2014

			Northwest corner:
1...3...7....15...31
2...4...8....16...32
5...11..23...47...95
6...12..24...48...96
9...19..39...79...159

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191665, A191426.

Cf. A241957.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A014601(n+2): (4+4k,5+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191664 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191664  *)
(* Clark Kimberling, Jun 11 2011 *)
Grid[Table[2^k*(2*Floor[(n + 1)/2] - 1) - Mod[n, 2], {n, 12}, {k, 12}]] (* L. Edson Jeffery, Aug 13 2014 *)

A191665 Dispersion of A042963 (numbers >1, congruent to 1 or 2 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 10, 13, 9, 7, 21, 26, 18, 14, 8, 42, 53, 37, 29, 17, 11, 85, 106, 74, 58, 34, 22, 12, 170, 213, 149, 117, 69, 45, 25, 15, 341, 426, 298, 234, 138, 90, 50, 30, 16, 682, 853, 597, 469, 277, 181, 101, 61, 33, 19, 1365, 1706, 1194, 938, 554

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A000975
Row 2:  A081254
Row 3:  A081253
Row 4:  A052997
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

			Northwest corner:
1...2...5....10...21
3...6...13...26...53
4...9...18...37...74
7...14..29...58...117
8...17..34...69...138

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191426, A191664.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 5; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042963: (2+4k,5+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191665 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191665  *)

A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 6, 14, 10, 8, 11, 27, 19, 15, 9, 22, 54, 38, 30, 18, 12, 43, 107, 75, 59, 35, 23, 13, 86, 214, 150, 118, 70, 46, 26, 16, 171, 427, 299, 235, 139, 91, 51, 31, 17, 342, 854, 598, 470, 278, 182, 102, 62, 34, 20, 683, 1707, 1195, 939, 555, 363

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A005578
Row 2:  A160113
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

			Northwest corner:
1...2...3....6...11
4...7...14....27...54
5...10...19...38...75
8...15..30...59...118
8...18..35...70...139

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191426, A191663.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 3; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042964: (2+4k,3+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191666 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191666  *)

A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Mar 24 2017

Partition the natural number sequence into quadruples starting with (1, 2, 3, 4); swap the third and fourth element, then swap the second and third element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699.
Subsequences:
  elements with odd index: A042963(n), n > 0
  elements with even index: A014601(A103889(n)), n > 0
  odd elements: A005408(n-1), n > 0
  indices of odd elements: A042948(n), n > 0
  even elements: 2*A103889(n), n > 0
  indices of even elements: A042964(n), n > 0
Sequence of fixed points: A016813(n-1), n > 0
Every fourth element starting at:
  n=1: a(4n-3) = 4n-3 = A016813(n-1), n > 0
  n=2: a(4n-2) = 4n = A008586(n), n > 0
  n=3: a(4n-1) = 4n-2 = A016825(n-1), n > 0
  n=4: a(4n) = 4n-1 = A004767(n-1), n > 0
Difference between pairs of elements:
  a(2n+1)-a(2n-1) = A010684(n-1), n > 0
Compositions:
  a(n) = A133256(A116966(n-1)), n > 0
  a(A042948(n)) = A005408(n-1), n > 0
  A067060(a(n)) = A092486(n), n > 0

a = [1 4 2 3];
max = (specify);
for n = 5:max
   a(n) = a(n-4) + 4;
end;

Table[n + ((-1)^n - (-1)^(n (n - 1)/2) (1 + 2 (-1)^n))/2, {n, 68}] (* Michael De Vlieger, Mar 28 2017 *)
LinearRecurrence[{1,0,0,1,-1},{1,4,2,3,5},70] (* or *) {#[[1]],#[[4]], #[[2]],#[[3]]}&/@Partition[Range[70],4]//Flatten(* Harvey P. Dale, Sep 27 2017 *)

for(n=1, 68, print1(n + ((-1)^n - (-1)^(n*(n - 1)/2)*(1 + 2*(-1)^n))/2,", ")) \\ Indranil Ghosh, Mar 29 2017

a(1)=1, a(2)=4, a(3)=2, a(4)=3, a(n) = a(n-4) + 4, n > 4.
O.g.f.: (x^4 + x^3 - 2*x^2 + 3x - 1)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^n - (-1)^(n*(n-1)/2)*(1 + 2*(-1)^n))/2.
a(n) = n + (-1)^n*(1 - (-1)^(n*(n-1)/2) - (i^n - (-i)^n))/2.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5), n > 5.
First differences, periodic: (3, -2, 1, 2), repeat.
a(n) = (2*n - 3*cos(n*Pi/2) + cos(n*Pi) + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Apr 01 2017

A292680 Rule 6: 000, ..., 111 -> 0, 1, 1, 0, 0, 0, 0, 0.

Original entry on oeis.org

0, 6, 12, 8, 24, 26, 16, 16, 48, 54, 52, 48, 32, 34, 32, 32, 96, 102, 108, 104, 104, 106, 96, 96, 64, 70, 68, 64, 64, 66, 64, 64, 192, 198, 204, 200, 216, 218, 208, 208, 208, 214, 212, 208, 192, 194, 192, 192, 128, 134, 140, 136, 136, 138, 128, 128, 128, 134, 132, 128

Offset: 0

M. F. Hasler, Oct 09 2017

The orbit of 1 under this rule is A266180.
Rule 6 is the smallest rule which is even (otherwise infinitely many bits would be switched on at step 1, for any finite starting value) and nontrivial (i.e., does not lead to extinction nor simple reproduction, possibly shifted left or right, of a single-bit initial state).
As is customary in the context of elementary cellular automata, the result is the bitmap obtained from the argument extended by one bit to the right (as to consider the cell which has bit 0 of the input as left neighbor), cf. example. Since the rule has a value < 16, no cell having its left neighbor 'on' will be on. Therefore all values a(n) are even. See A292681 for the variant without this extension beyond bit 0, i.e., a(n)/2.

			     n        |          a(n)
   0 =   0[2] |       0[2] =  0
   1 =   1[2] |     110[2] =  6 (bits below 001 and 010 are on, below 100 is off)
   2 =  10[2] |    1100[2] = 12 (as above, plus an additional bit 0 below 000)
   3 =  11[2] |    1000[2] =  8 (1 below 001, 0 below 011, 110 and 100.)
   4 = 100[2] |   11000[2] = 24 (as n = 1 and n = 2, shifted right once more)
   5 = 101[2] |   11010[2] = 26 (1 below 001 and 010 (twice), 0 below 101 and 100)
   6 = 110[2] |   10000[2] = 16 (as n = 3, shifted right once)
   7 = 111[2] |   10000[2] = 16 (1 below 001, 0 below 011, 111, 110 and 100).

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A266178, A266179, A266180, A019590, A003953, A003945, A000034, A032766, A042948, A035608.

apply( A292680(n,r=6)=sum(i=0,logint(!n+n<<=2,2)+1,bittest(r,(n>>i)%8)<

A301504 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-3)).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 2, 0, 2, 5, 4, 1, 2, 7, 7, 2, 3, 10, 11, 4, 4, 14, 17, 8, 6, 19, 25, 13, 8, 25, 36, 21, 12, 33, 50, 33, 18, 43, 69, 49, 26, 56, 93, 71, 38, 72, 124, 102, 55, 92, 163, 142, 79, 118, 212, 195, 112, 151, 273, 265, 157, 193, 350, 354, 217, 246, 444

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 1 mod 4.

			a(9) = 3 because we have [9], [8, 1] and [5, 4].

Index entries for sequences related to partitions

Cf. A035362, A035451, A035457, A042948, A108932, A132463, A261734, A301505, A301507, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k)) (1 + x^(4 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[x^3 QPochhammer[-1, x^4] QPochhammer[-x^(-3), x^4]/(2 (1 + x) (1 - x + x^2)), {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 1}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042948(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A110185 Coefficients of x in the partial quotients of the continued fraction expansion exp(1/x) = [1, x - 1/2, 12x, 5x, 28x, 9x, 44x, 13x, ...]. The partial quotients all have the form a(n)*x except the constant term of 1 and the initial partial quotient which equals (x - 1/2).

Original entry on oeis.org

0, 1, 12, 5, 28, 9, 44, 13, 60, 17, 76, 21, 92, 25, 108, 29, 124, 33, 140, 37, 156, 41, 172, 45, 188, 49, 204, 53, 220, 57, 236, 61, 252, 65, 268, 69, 284, 73, 300, 77, 316, 81, 332, 85, 348, 89, 364, 93, 380, 97, 396, 101, 412, 105, 428, 109, 444, 113, 460, 117, 476

Offset: 0

Paul D. Hanna, Jul 14 2005

Simple continued fraction expansion of 2*(e - 1)/(e + 1) = 2*tanh(1/2) = 1/(1 + 1/(12 + 1/(5 + 1/(28 + ...)))). - Peter Bala, Oct 01 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. continued fraction expansions: A004273 ( tanh(1) ), A204877 ( 3*tanh(1/3) ), A130824 ( tanh(1/2) ).

a(n)=polcoeff(x*(1+12*x+3*x^2+4*x^3)/(1-x^2)^2+x*O(x^n),n)

G.f.: x*((1+3*x^2) + 4*x*(3+x^2))/(1-x^2)^2 = sum_{n>=0} a(n)*x^n.
From Carl R. White, Feb 11 2010: (Start)
a(n) = sign(n) * (2*n+1) * (3*cos(Pi*n)+5)/2.
a(2n+1) = a(2n-1) + 4, a(2n+2) = a(2n) + 16, with a(0)=0, a(1)=1, a(2)=12. (End)
a(n) = (5+3*(-1)^n)*(2*n-1)/2, with a(0)=0. Sum_{i=0..n} a(i) = A085787(A042948(n)). - Bruno Berselli, Jan 20 2012

A123223 Triangle read by rows: T(n,k) = number of ternary Lyndon words of length n with exactly k 1's.

Original entry on oeis.org

1, 2, 1, 1, 2, 0, 2, 4, 2, 0, 3, 8, 5, 2, 0, 6, 16, 16, 8, 2, 0, 9, 32, 38, 26, 9, 2, 0, 18, 64, 96, 80, 40, 12, 2, 0, 30, 128, 220, 224, 137, 56, 13, 2, 0, 56, 256, 512, 596, 448, 224, 74, 16, 2, 0, 99, 512, 1144, 1536, 1336, 806, 332, 96, 17, 2, 0, 186, 1024, 2560, 3840, 3840

Offset: 0

Mike Zabrocki, Nov 05 2006

Sum of rows equal to number of ternary Lyndon words A027376 first column (k=0) is equal to the number of binary Lyndon words A001037 third through sixth column (k=2,3,4,5) equal to A124720, A124721, A124722, A124723 T(n+1,n-1) entry equal to A042948.

			Triangle begins:
   1;
   2,  1;
   1,  2,  0;
   2,  4,  2,  0;
   3,  8,  5,  2,  0;
   6, 16, 16,  8,  2,  0;
   9, 32, 38, 26,  9,  2, 0;
  18, 64, 96, 80, 40, 12, 2, 0;
T(n,1) = 2^(n-1) because all words beginning with a 1 and consisting of the rest 2's or 3's are ternary Lyndon words with exactly one 1.

Alois P. Heinz, Rows n = 0..140, flattened.

Cf. A027376, A001037, A124720, A124721, A124722, A124723, A051168, A042948.

G.f. for columns (except for k=0) given by 1/k*Sum_{d|k} mu(d) x^k/(1-2*x^d)^(k/d) T(0,0) = 1 and T(n,0) = 1/n*Sum_{d|n} mu(d)*2^(n/d) T(n,n) = 0 if n>1, T(n,n-1) = 2.

A151971 Numbers n such that n^2 - n is divisible by 21.

Original entry on oeis.org

0, 1, 7, 15, 21, 22, 28, 36, 42, 43, 49, 57, 63, 64, 70, 78, 84, 85, 91, 99, 105, 106, 112, 120, 126, 127, 133, 141, 147, 148, 154, 162, 168, 169, 175, 183, 189, 190, 196, 204, 210, 211, 217, 225, 231, 232, 238, 246, 252, 253, 259, 267, 273, 274, 280, 288, 294, 295, 301, 309

Offset: 1

N. J. A. Sloane, Aug 23 2009

Equivalently, numbers that are congruent to {0, 1, 7, 15} mod 21. - Bruno Berselli, Aug 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

For m^2 == m (mod n), see: n=2: A001477; n=3: A032766; n=4: A042948; n=5: A008851; n=6: A032766; n=7: A047274; n=8: A047393; n=9: A090570; n=10: A008851; n=11: A112651; n=12: A112652; n=13:A112653; n=14: A047274; n=15: A151972; n=16: A151977; n=17: A151978; n=18: A090570; n=19: A151979; n=20: A151980; n=21: A151971; n=22: A112651; n=24: A151973; n=26: A112653; n=30: A151972; n=32: A151983; n=34: A151978; n=38: A151979; n=42: A151971; n=48: A151981; n=64: A151984.

Cf. A215202.

[n: n in [0..309] | IsZero((n^2-n) mod 21)]; // Bruno Berselli, Aug 06 2012

A151971:=n->(42*n+14*I^((n-1)*n)-3*I^(2*n)-3)/8-7: seq(A151971(n), n=1..100); # Wesley Ivan Hurt, Jun 07 2016

Select[Range[0,400], Divisible[#^2-#,21]&] (* Harvey P. Dale, Jun 04 2012 *)

makelist((42*n+14*%i^((n-1)*n)-3*(-1)^n-3)/8-7, n, 1, 60); /* Bruno Berselli, Aug 06 2012 */

From Bruno Berselli, Aug 06 2012: (Start)
G.f.: x^2*(1+6*x+8*x^2+6*x^3)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = (42*n +14*i^((n-1)*n) -3*(-1)^n -3)/8 -7, where i=sqrt(-1). (End)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Wesley Ivan Hurt, Jun 07 2016
E.g.f.: (24 + (21*x - 31)*cosh(x) + 7*(sin(x) + cos(x) + (3*x - 4)*sinh(x)))/4. - Ilya Gutkovskiy, Jun 07 2016

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A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

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A292680 Rule 6: 000, ..., 111 -> 0, 1, 1, 0, 0, 0, 0, 0.

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A301504 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-3)).

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A110185 Coefficients of x in the partial quotients of the continued fraction expansion exp(1/x) = [1, x - 1/2, 12*x, 5*x, 28*x, 9*x, 44*x, 13*x, ...]. The partial quotients all have the form a(n)*x except the constant term of 1 and the initial partial quotient which equals (x - 1/2).

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