A042964 -id:A042964 - cp's OEIS frontend

A352090 Numbers k such that k and k+1 are both tribonacci-Niven numbers (A352089).

1, 6, 7, 12, 13, 20, 26, 27, 39, 68, 75, 80, 81, 87, 115, 128, 135, 149, 176, 184, 185, 195, 204, 215, 224, 230, 236, 243, 264, 278, 284, 291, 344, 364, 399, 447, 506, 507, 519, 548, 555, 560, 575, 595, 615, 635, 656, 664, 665, 684, 704, 725, 744, 777, 804, 824

Offset: 1

Amiram Eldar, Mar 04 2022

Numbers k such that A278043(k) | k and A278043(k+1) | k+1.

The odd tribonacci numbers, A000073(A042964(m)), are all terms.

			6 is a term since 6 and 7 are both tribonacci-Niven numbers: the minimal tribonacci representation of 6, A278038(6) = 110, has 2 1's and 6 is divisible by 2, and the minimal tribonacci representation of 7, A278038(7) = 1000, has one 1 and 7 is divisible by 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000073, A042964, A278038, A278043.
Subsequence of A352089.
Subsequences: A352091, A352092.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; Divisible[n, DigitCount[Total[2^(s - 1)], 2, 1]]]; Select[Range[1000], q[#] && q[# + 1] &]

A047566 Numbers that are congruent to {4, 5, 6, 7} mod 8.

Original entry on oeis.org

4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 68, 69, 70, 71, 76, 77, 78, 79, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109

Offset: 1

N. J. A. Sloane

Numbers having a 1 in position 2 of their binary expansion. One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner, Jan 22 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Maths Magic, Mystery Calculator.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A003628 (primes).

Mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421.

a047566 n = a047566_list !! (n-1)
a047566_list = [n | n <- [1..], mod n 8 > 3]
-- Reinhard Zumkeller, Dec 29 2012

A047566:= n-> n+3 + 4*iquo(n-1, 4):
seq(A047566(n), n=1..100);  # Alois P. Heinz, Aug 22 2011

Flatten[# + {4, 5, 6, 7}&/@(8Range[0, 14])] (* Harvey P. Dale, Feb 02 2011 *)

G.f.: x*(4+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 19 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (4*n+1-(-1)^n-(-1)^((n+1)/2)-(-1)^(n/2)-(-1)^(-(n+1)/2)-(-1)^(-n/2))/2. (End)
E.g.f.: 1 + sin(x) - cos(x) + sinh(x) + 2*x*exp(x). - Ilya Gutkovskiy, May 20 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 - 3*log(2)/8. - Amiram Eldar, Dec 26 2021

A115419 Numbers having a 1 in position 3 of their binary expansion.

Original entry on oeis.org

8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 72, 73, 74, 75, 76, 77, 78, 79, 88, 89, 90, 91, 92, 93, 94, 95, 104, 105, 106, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124, 125, 126, 127

Offset: 1

Jeremy Gardiner, Jan 22 2006

One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421.

Numbers congruent to {8,9,10,11,12,13,14,15} mod 16. Numbers n such that n xor 8 = n - 8. [Brad Clardy, May 06 2013]

			a(1) = 8 = 1000 in binary.

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

Cf. A005408, A042964, A047566, A115420, A115421.

a:= n-> n+7 + 8*iquo(n-1, 8):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2011

a(n)=(n-1)\8*16+(n-1)%8+8 \\ Charles R Greathouse IV, May 06 2013

def A115419(n): return ((n-1<<1)&-7|8)+(n-1&7) # Chai Wah Wu, Mar 28 2024

G.f.: -(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+8)*x / (-x^9+x^8+x-1). - Alois P. Heinz, Aug 22 2011

A047406 Numbers that are congruent to {4, 6} mod 8.

Original entry on oeis.org

4, 6, 12, 14, 20, 22, 28, 30, 36, 38, 44, 46, 52, 54, 60, 62, 68, 70, 76, 78, 84, 86, 92, 94, 100, 102, 108, 110, 116, 118, 124, 126, 132, 134, 140, 142, 148, 150, 156, 158, 164, 166, 172, 174, 180, 182, 188, 190, 196, 198, 204, 206, 212, 214, 220, 222, 228

Offset: 1

N. J. A. Sloane

In groups of four, add the odd and even numbers (4=1+3, 6=2+4; 12=5+7, 14=6+8; etc.). - George E. Antoniou, Dec 12 2001

The first 250 terms (4 through 998) are the 250 non-occurring Fibonacci number residues modulo 1000; i.e., if leading zeros are supplied as necessary for the terms having fewer than three digits, these are the 250 sets of three digits that never appear as the last three digits of a Fibonacci number. - Jon E. Schoenfield, Jul 05 2010

			a(2) = 8*2 - 4 - 6 = 6;
a(3) = 8*3 - 6 - 6 = 12;
a(4) = 8*4 - 12 - 6 = 14.

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

Union of A017113 and A017137.

Cf. A042964.

Select[Range[230], MemberQ[{4, 6}, Mod[#, 8]] &] (* Amiram Eldar, Dec 19 2021 *)

a(n)=4*n-1-(-1)^n \\ Charles R Greathouse IV, May 20 2013

a(n) = A042964(n)*2.
a(n) = (4*n - 1 - (-1)^n). - Jon E. Schoenfield, Jul 05 2010
a(n) = 8*n - a(n-1) - 6 (with a(1)=4). - Vincenzo Librandi, Aug 05 2010
G.f.: 2*x*(2+x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 18 2013: (Start)
a(n) = (8 * ceiling(n/2) - 4) * (n mod 2) + (8 * ceiling(n/2) - 2) * (n+1 mod 2).
a(n) = 8 * ceiling(n/2) - 3 + (-1)^n. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - log(2)/8. - Amiram Eldar, Dec 19 2021
E.g.f.: 2*(1 + 2*x*exp(x) - cosh(x)). - David Lovler, Sep 02 2022

A115420 Numbers having a 1 in position 4 of their binary expansion.

Original entry on oeis.org

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127

Offset: 1

Jeremy Gardiner, Jan 22 2006

One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421.

			a(1) = 16 = 10000 in binary.

Edd Mann, Mystery Calculator.

Cf. A005408, A042964, A047566, A115419, A115421.

a:= n-> n+15 + 16*iquo(n-1, 16):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2011

def A115420(n): return ((n-1<<1)&-15|16)+(n-1&15) # Chai Wah Wu, Mar 28 2024

A115421 Numbers having a 1 in position 5 of their binary expansion.

Original entry on oeis.org

32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Offset: 1

Jeremy Gardiner, Jan 22 2006

One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421.

			a(1) = 32 = 100000 in binary.

Robert Israel, Table of n, a(n) for n = 1..10000
Edd Mann, Mystery Calculator.

Cf. A005408, A042964, A047566, A115419, A115420.

a:= n-> n+31 + 32*iquo(n-1, 32):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2011

Select[Range[256], BitAnd[#, 32] == 32 &] (* Alonso del Arte, Nov 14 2016 *)

def A115421(n): return ((n-1<<1)&-31|32)+(n-1&31) # Chai Wah Wu, Mar 28 2024

From Robert Israel, Nov 14 2016: (Start)
a(n + 32) = a(n) + 64.
O.g.f.: (31 + (1 - x^33)/(1 - x))*x/((1 - x)*(1 - x^32)). (End)

A090964 Permutation of natural numbers generated by 2-rowed array shown below.

Original entry on oeis.org

1, 4, 2, 5, 3, 8, 6, 9, 7, 12, 10, 13, 11, 16, 14, 17, 15, 20, 18, 21, 19, 24, 22, 25, 23, 28, 26, 29, 27, 32, 30, 33, 31, 36, 34, 37, 35, 40, 38, 41, 39, 44, 42, 45, 43, 48, 46, 49, 47, 52, 50, 53, 51, 56, 54, 57, 55, 60, 58, 61, 59, 64, 62, 65, 63, 68, 66, 69, 67, 72, 70, 73

Offset: 0

Giovanni Teofilatto, Feb 29 2004

nonn

1 4 5 8 9 12 13 16 17 20...(A042948)

2 3 6 7 10 11 14 15 18 19...(A042964)

For n > 1, a(n+8) = a(n)+8. - David Wasserman, Feb 23 2006

More terms from David Wasserman, Feb 23 2006

A098180 Odd numbers with twice the odd numbers repeated in order between them.

Original entry on oeis.org

1, 2, 2, 3, 5, 6, 6, 7, 9, 10, 10, 11, 13, 14, 14, 15, 17, 18, 18, 19, 21, 22, 22, 23, 25, 26, 26, 27, 29, 30, 30, 31, 33, 34, 34, 35, 37, 38, 38, 39, 41, 42, 42, 43, 45, 46, 46, 47, 49, 50, 50, 51, 53, 54, 54, 55, 57, 58, 58, 59, 61, 62, 62, 63, 65, 66, 66, 67, 69, 70, 70, 71

Offset: 0

Paul Barry, Aug 30 2004

Partial sums of A098178.
Also A042968 with the even terms repeated. - Michel Marcus, Apr 14 2015
Fixed points are [2,3,6,7,10,11,..] = A042964. - Wesley Ivan Hurt, Oct 13 2015

Iain Fox, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A002265, A042964, A042968, A098178.

[Floor((2*n+1-(-1)^((n+1)*(n+2)/2))/2): n in [0..80]]; // Vincenzo Librandi, Apr 13 2015

A098180:=n->(2*n+1-(-1)^((n+1)*(n+2)/2))/2: seq(A098180(n), n=0..100); # Wesley Ivan Hurt, Apr 12 2015

Table[(2 n + 1 - (-1)^((n + 1) (n + 2)/2))/2, {n, 0, 40}] (* Wesley Ivan Hurt, Apr 12 2015 *)

first(n) = Vec((1+x)*(1-x+x^2)/((1-x)^2*(1+x^2)) + O(x^n)) \\ Iain Fox, Oct 17 2018

a(n) = (2*n+1-(-1)^((n+1)*(n+2)/2))/2 \\ Iain Fox, Oct 17 2018

G.f.: (1+x)(1-x+x^2)/((1-x)^2(1+x^2)).
a(n) = sqrt(2)*sin(Pi*n/2+Pi/4)/2+n+1/2.
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4), n>3.
From Wesley Ivan Hurt, Apr 12 2015, Oct 13 2015: (Start)
a(n) = (2n+1-(-1)^((n+1)*(n+2)/2))/2.
a(n) = n + A002265(n) - A002265(n-2). (End)
E.g.f: (exp(-i*x)*((1+i) + (1-i)*exp(2*i*x) + exp((1+i)*x)*(2+4*x)))/4, where i = sqrt(-1). - Iain Fox, Oct 17 2018

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  *)

cp's OEIS Frontend

A352090 Numbers k such that k and k+1 are both tribonacci-Niven numbers (A352089).

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A047566 Numbers that are congruent to {4, 5, 6, 7} mod 8.

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A115419 Numbers having a 1 in position 3 of their binary expansion.

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A047406 Numbers that are congruent to {4, 6} mod 8.

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A115420 Numbers having a 1 in position 4 of their binary expansion.

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A115421 Numbers having a 1 in position 5 of their binary expansion.

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A090964 Permutation of natural numbers generated by 2-rowed array shown below.

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A098180 Odd numbers with twice the odd numbers repeated in order between them.