A047345 -id:A047345 - cp's OEIS frontend

A047365 Numbers that are congruent to {0, 3, 4, 5} mod 7.

0, 3, 4, 5, 7, 10, 11, 12, 14, 17, 18, 19, 21, 24, 25, 26, 28, 31, 32, 33, 35, 38, 39, 40, 42, 45, 46, 47, 49, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 73, 74, 75, 77, 80, 81, 82, 84, 87, 88, 89, 91, 94, 95, 96, 98, 101, 102, 103, 105, 108, 109, 110

Offset: 1

N. J. A. Sloane

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

Cf. A047345, A047389.

[n : n in [0..150] | n mod 7 in [0, 3, 4, 5]]; // Wesley Ivan Hurt, Jun 04 2016

A047365:=n->(14*n-11+I^(2*n)-(3+I)*I^(-n)-(3-I)*I^n)/8: seq(A047365(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016

Select[Range[0,100], MemberQ[{0,3,4,5}, Mod[#,7]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,3,4,5,7}, 60] (* Harvey P. Dale, May 26 2012 *)

G.f.: x^2*(3+x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
a(1)=0, a(2)=3, a(3)=4, a(4)=5, a(5)=7, a(n)=a(n-1)+a(n-4)-a(n-5) for n>5. - Harvey P. Dale, May 26 2012
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = (14*n-11+i^(2*n)-(3+i)*i^(-n)-(3-i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047389(k), a(2k-1) = A047345(k). (End)

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

A198269 Ceiling(n*sqrt(12)).

Original entry on oeis.org

0, 4, 7, 11, 14, 18, 21, 25, 28, 32, 35, 39, 42, 46, 49, 52, 56, 59, 63, 66, 70, 73, 77, 80, 84, 87, 91, 94, 97, 101, 104, 108, 111, 115, 118, 122, 125, 129, 132, 136, 139, 143, 146, 149, 153, 156, 160, 163, 167, 170, 174, 177, 181, 184, 188

Offset: 0

Vincenzo Librandi, Oct 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A047345, A047355, A194028.

Magma
```
[Ceiling(n*Sqrt(12)): n in [0..60]]
```

Ceiling[Sqrt[12]Range[0,60]] (* Harvey P. Dale, Aug 27 2013 *)

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A047365 Numbers that are congruent to {0, 3, 4, 5} mod 7.

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

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A198269 Ceiling(n*sqrt(12)).

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Mathematica