A047383 -id:A047383 - cp's OEIS frontend

A047388 Numbers that are congruent to {0, 1, 2, 5} mod 7.

0, 1, 2, 5, 7, 8, 9, 12, 14, 15, 16, 19, 21, 22, 23, 26, 28, 29, 30, 33, 35, 36, 37, 40, 42, 43, 44, 47, 49, 50, 51, 54, 56, 57, 58, 61, 63, 64, 65, 68, 70, 71, 72, 75, 77, 78, 79, 82, 84, 85, 86, 89, 91, 92, 93, 96, 98, 99, 100, 103, 105, 106, 107, 110, 112

Offset: 1

N. J. A. Sloane

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).

Cf. A047352, A047383.

I:=[0, 1, 2, 5, 7]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 15 2012

A047388:=n->(-19+I^(2*n)+(1+3*I)*(-I)^n+(1-3*I)*I^n+14*n)/8: seq(A047388(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,300], MemberQ[{0,1,2,5}, Mod[#,7]]&] (* Vincenzo Librandi, May 15 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,5,7},80] (* Harvey P. Dale, Jan 10 2023 *)

x='x+O('x^100); concat(0, Vec(x^2*(1+x+3*x^2+2*x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Jun 02 2016

G.f.: x^2*(1+x+3*x^2+2*x^3)/((1-x)^2*(1+x)*(1+x^2)). - Colin Barker, May 13 2012
a(n) = (-19+(-1)^n+(1+3*i)*(-i)^n+(1-3*i)*i^n+14*n)/8 where i=sqrt(-1). - Colin Barker, May 14 2012
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 16 2012
a(2k) = A047383(k), a(2k-1) = A047352(k). - Wesley Ivan Hurt, Jun 01 2016

A047392 Numbers that are congruent to {0, 1, 3, 5} mod 7.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 22, 24, 26, 28, 29, 31, 33, 35, 36, 38, 40, 42, 43, 45, 47, 49, 50, 52, 54, 56, 57, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 91, 92, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110

Offset: 1

N. J. A. Sloane

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

Cf. A047355, A047383.

Cf. A047371: n + floor(3*n/4-1/2) - 1; A047379: n + floor(3*n/4-1/4) - 1.

[n: n in [0..100] | n mod 7 in [0, 1, 3, 5]]; // Wesley Ivan Hurt, May 21 2016

A047392:=n->(14*n-17-I^(2*n)+(1+I)*I^(-n)+(1-I)*I^n)/8: seq(A047392(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(14n-17-I^(2n)+(1+I)*I^(-n)+(1-I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
Table[n + Floor[3 n/4 - 3/4] - 1, {n, 1, 70}] (* Bruno Berselli, Jun 15 2016 *)

G.f.: x^2*(1+2*x+2*x^2+2*x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n - 17 - i^(2*n) + (1 + i)*i^(-n) + (1 - i)*i^n)/8.
a(2k) = A047383(k), a(2k-1) = A047355(k). (End)
a(n) = n + floor(3*n/4-3/4) - 1. - Bruno Berselli, Jun 15 2016

More terms from Wesley Ivan Hurt, May 21 2016

A047329 Numbers that are congruent to {1, 3, 5, 6} mod 7.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 110, 111

Offset: 1

N. J. A. Sloane

Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.

Robert Fludd, Page 158 of "Utriusque Cosmi" in Beinecke Rare Book and Manuscript Library Photonegatives Collection.
Robert Fludd, Larger version of the same image
Robert Fludd, Utriusque Cosmi, Maioris scilicet et Minoris, metaphysica, physica, atque technica Historia, available as ZIP or PDF download.
Wikipedia, Robert Fludd
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047280, A047383, A082977.

a047329 n = a047329_list !! (n-1)
a047329_list = [1, 3, 5, 6] ++ map (+ 7) a047329_list
-- Reinhard Zumkeller, Jan 07 2014

[Floor((7*n-1)/4) : n in [1..100]]; // Wesley Ivan Hurt, May 21 2016

A047329:=n->floor((7*n-1)/4): seq(A047329(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[Floor[(7*n-1)/4], {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
#+{1,3,5,6}&/@(7*Range[0,20])//Flatten (* Harvey P. Dale, Jan 07 2021 *)

a(n) = floor((7n-1)/4). - Gary Detlefs, Mar 07 2010
G.f.: (x*(1+2*x+2*x^2+x^3+x^4)) / ((1+x)*(x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (14n-5-i^(2n)-(1+i)*i^(-n)-(1-i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047280(n), a(2n-1) = A047383(n). (End)
E.g.f.: (4 - sin(x) - cos(x) + (7*x - 2)*sinh(x) + (7*x - 3)*cosh(x))/4. - Ilya Gutkovskiy, May 21 2016

Fludd reference from Brendan McKay, May 27 2003

More terms from Wesley Ivan Hurt, May 21 2016

A257772 Numbers n>=0 such that (n+1)^3 - n^3 = 3n^2+3n+1 is not prime.

Original entry on oeis.org

0, 5, 7, 8, 12, 15, 16, 18, 19, 20, 21, 22, 26, 29, 31, 33, 35, 36, 39, 40, 43, 44, 46, 47, 50, 51, 53, 54, 56, 57, 59, 60, 61, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 82, 83, 84, 85, 87, 89, 92, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106

Offset: 1

Miguel Angel Velilla Mula, May 08 2015

Complement of A111251.

Includes all members of A047383 except 1. - Robert Israel, May 12 2015

			5 is a term since (5+1)^3 - 5^3 = 91 = 13*7 is not prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A003215, A047383, A111251.

[n: n in [0..120] | not IsPrime(3*n^2+3*n+1)]; // Vincenzo Librandi, May 13 2015

remove(t -> isprime((t+1)^3-t^3), [$0..300]); # Robert Israel, May 12 2015

Select[Range[0, 200], ! PrimeQ[(#+1)^3 - #^3] &] (* Giovanni Resta, May 08 2015 *)

for(n=0,100,if(!isprime(3*n^2+3*n+1),print1(n,", "))) \\ Derek Orr, May 19 2015

10 print 0
20 for n=1 to 200
30   s = (n+1)^3 - n^3
40   if prmdiv(s)<>s then print n
50 next n

a(n) ~ n. - Charles R Greathouse IV, May 22 2015

A047313 Numbers that are congruent to {1, 4, 5, 6} mod 7.

Original entry on oeis.org

1, 4, 5, 6, 8, 11, 12, 13, 15, 18, 19, 20, 22, 25, 26, 27, 29, 32, 33, 34, 36, 39, 40, 41, 43, 46, 47, 48, 50, 53, 54, 55, 57, 60, 61, 62, 64, 67, 68, 69, 71, 74, 75, 76, 78, 81, 82, 83, 85, 88, 89, 90, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111

Offset: 1

N. J. A. Sloane

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).

Cf. A047288, A047383.

[n : n in [0..150] | n mod 7 in [1, 4, 5, 6]]; // Wesley Ivan Hurt, May 23 2016

A047313:= n-> iquo(n-1, 4, 'r')*7 +[1, 4, 5, 6][r+1]: seq(A047313(n), n=1..80);  # Alois P. Heinz, Dec 04 2011

Select[Range[100],MemberQ[{1,4,5,6},Mod[#,7]]&]  (* Harvey P. Dale, Apr 17 2011 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 4, 5, 6, 8}, 80] (* Jean-François Alcover, Feb 18 2016 *)

G.f.: x*(1+3*x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 03 2011
From Wesley Ivan Hurt, May 23 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14n-3+i^(2n)-(3+i)*i^(-n)-(3-i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047288(n), a(2n-1) = A047383(n). (End)
E.g.f.: (4 - sin(x) - 3*cos(x) + (7*x - 2)*sinh(x) + (7*x - 1)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

A047325 Numbers that are congruent to {1, 2, 5, 6} mod 7.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 12, 13, 15, 16, 19, 20, 22, 23, 26, 27, 29, 30, 33, 34, 36, 37, 40, 41, 43, 44, 47, 48, 50, 51, 54, 55, 57, 58, 61, 62, 64, 65, 68, 69, 71, 72, 75, 76, 78, 79, 82, 83, 85, 86, 89, 90, 92, 93, 96, 97, 99, 100, 103, 104, 106, 107, 110, 111

Offset: 1

N. J. A. Sloane

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).

Cf. A047276, A047383.

[n : n in [0..150] | n mod 7 in [1, 2, 5, 6]]; // Wesley Ivan Hurt, May 23 2016

A047325:=n->(14*n-7-3*I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/8: seq(A047325(n), n=1..100); # Wesley Ivan Hurt, May 23 2016

Table[(14n-7-3*I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 23 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 5, 6, 8}, 80] (* Vincenzo Librandi, May 24 2016 *)
#+{1,2,5,6}&/@(7*Range[0,20])//Flatten (* Harvey P. Dale, Aug 16 2018 *)

G.f.: x*(1+x+3*x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, May 23 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14n-7-3*i^(2n)+(1-i)*i^(-n)+(1+i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047276(n), a(2n-1) = A047383(n). (End)
E.g.f.: (4 - sin(x) + cos(x) + (7*x - 2)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

More terms from Wesley Ivan Hurt, May 23 2016

A047377 Numbers that are congruent to {0, 1, 4, 5} mod 7.

Original entry on oeis.org

0, 1, 4, 5, 7, 8, 11, 12, 14, 15, 18, 19, 21, 22, 25, 26, 28, 29, 32, 33, 35, 36, 39, 40, 42, 43, 46, 47, 49, 50, 53, 54, 56, 57, 60, 61, 63, 64, 67, 68, 70, 71, 74, 75, 77, 78, 81, 82, 84, 85, 88, 89, 91, 92, 95, 96, 98, 99, 102, 103, 105, 106, 109, 110

Offset: 1

N. J. A. Sloane

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

Cf. A030308, A047345, A047383.

[n : n in [0..150] | n mod 7 in [0, 1, 4, 5]]; // Wesley Ivan Hurt, May 24 2016

A047377:=n->(14*n-15-3*I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/8: seq(A047377(n), n=1..100); # Wesley Ivan Hurt, May 24 2016

Table[(14n-15-3*I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 24 2016 *)
Select[Range@ 120, MemberQ[{0, 1, 4, 5}, Mod[#, 7]] &] (* Michael De Vlieger, May 24 2016 *)
a[n_] :=  n + Floor[(n - 1)/2] +  Floor[(n - 3)/4];
Table[a[n], {n, 1, 64}] (* Peter Luschny, Dec 23 2021 *)

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=4 and b(k)=7*2^(k-2) for k>1. - Philippe Deléham, Oct 25 2011
G.f.: x^2*(1+3*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, May 24 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-15-3*i^(2*n)+(1-i)*i^(-n)+(1+i)*i^n)/8, where i=sqrt(-1).
a(2k) = A047383(k), a(2k-1) = A047345(k). (End)
E.g.f.: (8 - sin(x) + cos(x) + (7*x - 6)*sinh(x) + (7*x - 9)*cosh(x))/4. - Ilya Gutkovskiy, May 25 2016

More terms from Wesley Ivan Hurt, May 24 2016

A131227 2*A051340 - A128174.

Original entry on oeis.org

1, 2, 3, 1, 2, 5, 2, 1, 2, 7, 1, 2, 1, 2, 9, 2, 1, 2, 1, 2, 11, 1, 2, 1, 2, 1, 2, 13, 2, 1, 2, 1, 2, 1, 2, 15, 1, 2, 1, 2, 1, 2, 1, 2, 17, 2, 1, 2, 1, 2, 1, 2, 1, 2, 19

Offset: 0

Gary W. Adamson, Jun 20 2007

Row sums = A047383, numbers congruent to {1,5} mod 7: (1, 5, 8, 12, 15, 19, ...)

			First few rows of the triangle:
  1;
  2, 3;
  1, 2, 5;
  2, 1, 2, 7;
  1, 2, 1, 2, 9;
  2, 1, 2, 1, 2, 11;
  1, 2, 1, 2, 1,  2, 13;
  ...

Cf. A051340, A128174, A047383, A131228, A131229.

2*A051340 - A128174 as infinite lower triangular matrices.

A144652 Triangle, read by rows, where T(m,n) = floor((2mn+m+n)/2) with m >= n >= 1.

Original entry on oeis.org

2, 3, 6, 5, 8, 12, 6, 11, 15, 20, 8, 13, 19, 24, 30, 9, 16, 22, 29, 35, 42, 11, 18, 26, 33, 41, 48, 56, 12, 21, 29, 38, 46, 55, 63, 72, 14, 23, 33, 42, 52, 61, 71, 80, 90, 15, 26, 36, 47, 57, 68, 78, 89, 99, 110, 17, 28, 40, 51, 63, 74, 86, 97, 109, 120, 132, 18, 31, 43, 56, 68

Offset: 1

Vincenzo Librandi, Jan 27 2009

From Vincenzo Librandi, Nov 16 2012: (Start)
First column:  A007494(n+1);
second column: A047219(n+2);
third column:  A047383(n+3);
fourth column: A193910(n+4).
Conjecture: If h does not belong to the sequence, then 4*h+1 is prime. (End)

			Triangle begins:
2;
3,  6;
5,  8,  12;
6,  11, 15, 20;
8,  13, 19, 24, 30;
9,  16, 22, 29, 35, 42;
11, 18, 26, 33, 41, 48, 56; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A007494, A047219, A047383, A193910.

[Floor((2*n*k+n+k)/2): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 16 2012

Flatten[Table[Floor[(2*n*m + m + n)/2], {n, 1, 20}, {m, n}]] (* Vincenzo Librandi, Nov 16 2012 *)

Definition edited (specifying m >= n >= 1), and terms recomputed to match definition, as was done with the similar sequence A140869, by Jon E. Schoenfield, Jun 24 2010

A131230 Triangle read by rows: 2*A130296 - A128174.

Original entry on oeis.org

1, 4, 1, 5, 2, 1, 8, 1, 2, 1, 9, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 13, 2, 1, 2, 1, 2, 1, 16, 1, 2, 1, 2, 1, 2, 1, 17, 2, 1, 2, 1, 2, 1, 2, 1, 20, 1, 2, 1, 2, 1, 2, 1, 2, 1, 21, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 25, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Gary W. Adamson, Jun 20 2007

Left column = A042948, numbers congruent to {1,0} mod 4: (1, 4, 5, 8, 9, 12, ...).

Row sums = A047383, numbers congruent to {1,5} mod 7: (1, 5, 8, 12, 15, ...).

			First few rows of the triangle:
   1;
   4, 1;
   5, 2, 1;
   8, 1, 2, 1;
   9, 2, 1, 2, 1;
  12, 1, 2, 1, 2, 1;
  ...

Cf. A047383, A042948, A128174, A131231, A130296.

Incorrect formula removed and more terms from Georg Fischer, Jun 08 2023

cp's OEIS Frontend

A047388 Numbers that are congruent to {0, 1, 2, 5} mod 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A047392 Numbers that are congruent to {0, 1, 3, 5} mod 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A047329 Numbers that are congruent to {1, 3, 5, 6} mod 7.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

Extensions

A257772 Numbers n>=0 such that (n+1)^3 - n^3 = 3*n^2+3*n+1 is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

UBASIC

Formula

A047313 Numbers that are congruent to {1, 4, 5, 6} mod 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A047325 Numbers that are congruent to {1, 2, 5, 6} mod 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A047377 Numbers that are congruent to {0, 1, 4, 5} mod 7.

Views

Author

A257772 Numbers n>=0 such that (n+1)^3 - n^3 = 3n^2+3n+1 is not prime.