A047346 -id:A047346 - cp's OEIS frontend

A047380 Numbers that are congruent to {1, 2, 4, 5} mod 7.

1, 2, 4, 5, 8, 9, 11, 12, 15, 16, 18, 19, 22, 23, 25, 26, 29, 30, 32, 33, 36, 37, 39, 40, 43, 44, 46, 47, 50, 51, 53, 54, 57, 58, 60, 61, 64, 65, 67, 68, 71, 72, 74, 75, 78, 79, 81, 82, 85, 86, 88, 89, 92, 93, 95, 96, 99, 100, 102, 103, 106, 107, 109, 110

Offset: 1

N. J. A. Sloane

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

Cf. A002265, A047346, A047385.

A047380:=n->(14*n-11-3*I^(2*n)-(1+I)*I^(-n-1)-(1-I)*I^(n+1))/8: seq(A047380(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Table[(14n-11-3I^(2n)-(1+I)I^(-n-1)-(1-I)I^(n+1))/8, {n, 80}] (* Wesley Ivan Hurt, May 20 2016 *)
LinearRecurrence[{1,0,0,1,-1},{1,2,4,5,8},100] (* Harvey P. Dale, Jun 05 2016 *)

a(n)=(n-1)\4*7+[5,1,2,4][n%4+1] \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(1+x+2*x^2+x^3+2*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n - 11 - 3*i^(2*n) - (1+i)*i^(-n-1) - (1-i)*i^(n+1))/8 where i=sqrt(-1).
a(2n) = A047385(n), a(2n-1) = A047346(n). (End)

A047298 Numbers that are congruent to {1, 3, 4, 6} mod 7.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 11, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 45, 46, 48, 50, 52, 53, 55, 57, 59, 60, 62, 64, 66, 67, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 108, 109, 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. A047280, A047346.

[n : n in [0..150] | n mod 7 in [1, 3, 4, 6]]; // Wesley Ivan Hurt, May 22 2016

A047298:=n->I^(-n)*(I-1-7*I^n+14*n*I^n-(1+I)*I^(2*n)+I^(-n))/8: seq(A047298(n), n=1..100); # Wesley Ivan Hurt, May 22 2016

Table[I^(-n)*(I-1-7I^n+14n*I^n-(1+I)*I^(2n)+I^(-n))/8, {n, 80}] (* Wesley Ivan Hurt, May 22 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 4, 6, 8}, 80] (* Vincenzo Librandi, May 24 2016 *)

a(n) = ceiling(ceiling((7n + 2)/2)/2).
G.f.: x*(1+2*x+x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = i^(-n)*(i-1-7*i^n+14*n*i^n-(1+i)*i^(2n)+i^(-n))/8 where i=sqrt(-1).
a(2n) = A047280(n), a(2n-1) = A047346(n). (End)
E.g.f.: (4 + sin(x) - cos(x) + (7*x - 4)*sinh(x) + (7*x - 3)*cosh(x))/4. - Ilya Gutkovskiy, May 23 2016

More terms from Wesley Ivan Hurt, May 22 2016

A047351 Numbers that are congruent to {0, 1, 2, 4} mod 7.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 9, 11, 14, 15, 16, 18, 21, 22, 23, 25, 28, 29, 30, 32, 35, 36, 37, 39, 42, 43, 44, 46, 49, 50, 51, 53, 56, 57, 58, 60, 63, 64, 65, 67, 70, 71, 72, 74, 77, 78, 79, 81, 84, 85, 86, 88, 91, 92, 93, 95, 98, 99, 100, 102, 105, 106, 107, 109, 112

Offset: 1

N. J. A. Sloane

The set of values for m such that 7i+m is a perfect square (the quadratic residues of 7 including the trivial case of k*7). - Gary Detlefs, Mar 07 2010

The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4 etc. - Bruno Berselli, Dec 03 2012

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

Cf. A045373 (primes), A047346, A047352.

Complement of A047327.

[n : n in [0..150] | n mod 7 in [0, 1, 2, 4]]; // Wesley Ivan Hurt, Jun 01 2016

for i from 1 to 56 do if(i mod 4=0) then print(floor(7*i-3)/4)+1) else print(floor(7*i-3)/4)) fi od; # Gary Detlefs, Mar 07 2010
A047351:=n->n-3+(6*n+(2-I^(2*n))*(1-2*I^(n*(n+1)))+1)/8: seq(A047351(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,100], MemberQ[{0,1,2,4}, Mod[#,7]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,1,2,4,7}, 60] (* Harvey P. Dale, Jun 04 2013 *)

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

If n mod 4 = 0 then a(n) = floor((7*n-3)/4)+1, else a(n) = floor((7*n-3)/4). - Gary Detlefs, Mar 07 2010
G.f.: x^2*(1+x+2*x^2+3*x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
a(n) = n-3+(6*n+(2-(-1)^n)(1-2*i^(n(n+1)))+1)/8, where i=sqrt(-1). - Bruno Berselli, Dec 03 2012
a(0)=0, a(1)=1, a(2)=2, a(3)=4, a(4)=7, a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Harvey P. Dale, Jun 04 2013
a(2k) = A047346(k), a(2k-1) = A047352(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (12 + 3*sin(x) - cos(x) + (7*x - 10)*sinh(x) + (7*x - 11)*cosh(x))/4. - Ilya Gutkovskiy, Jun 02 2016

A047294 Numbers that are congruent to {1, 2, 4, 6} mod 7.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 78, 79, 81, 83, 85, 86, 88, 90, 92, 93, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047276, A047346.

I:=[1, 2, 4, 6, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Apr 27 2012

A047294:=n->ceil(floor((7*n-5)/2)/2): seq(A047294(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,100], MemberQ[{1,2,4,6}, Mod[#,7]]&] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{1,0,0,1,-1},{1,2,4,6,8},100] (* G. C. Greubel, Jun 01 2016 *)

x='x+O('x^100); Vec(x*(1+x+2*x^2+2*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

a(n) = ceiling(floor((7*n - 5)/2)/2).
From Colin Barker, Mar 14 2012: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
G.f.: x*(1 + x + 2*x^2 + 2*x^3 + x^4)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = (-9 -(-1)^n + (1+i)*(-i)^n + (1-i)*i^n + 14*n)/8 where i=sqrt(-1). - Colin Barker, May 14 2012
a(2k) = A047276(k), a(2k-1) = A047346(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (4 + sin(x) + cos(x) + (7*x - 4)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, Jun 01 2016

A047344 Numbers that are congruent to {0, 1, 3, 4} mod 7.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 10, 11, 14, 15, 17, 18, 21, 22, 24, 25, 28, 29, 31, 32, 35, 36, 38, 39, 42, 43, 45, 46, 49, 50, 52, 53, 56, 57, 59, 60, 63, 64, 66, 67, 70, 71, 73, 74, 77, 78, 80, 81, 84, 85, 87, 88, 91, 92, 94, 95, 98, 99, 101, 102, 105, 106, 108, 109

Offset: 1

N. J. A. Sloane

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

Cf. A030308, A047346, A047355.

[n : n in [0..150] | n mod 7 in [0, 1, 3, 4]]; // Wesley Ivan Hurt, May 23 2016

A047344:=n->(14*n-19-3*I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/8: seq(A047344(n), n=1..100); # Wesley Ivan Hurt, May 23 2016

Table[(14n-19-3*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},{0,1,3,4,7},80] (* Harvey P. Dale, May 06 2021 *)

forstep(n=0,200,[1,2,1,3],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=3 and b(k)=7*2^(k-2) for k>1. - Philippe Deléham, Oct 17 2011
G.f.: x^2*(1+2*x+x^2+3*x^3) / ( (1+x)*(x^2+1)*(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-19-3*i^(2n)-(1-i)*i^(-n)-(1+i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047346(n), a(2n-1) = A047355(n). (End)
E.g.f.: (12 + sin(x) - cos(x) + (7*x - 8)*sinh(x) + (7*x - 11)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

More terms from Wesley Ivan Hurt, May 23 2016

A047366 Numbers that are congruent to {1, 3, 4, 5} mod 7.

Original entry on oeis.org

1, 3, 4, 5, 8, 10, 11, 12, 15, 17, 18, 19, 22, 24, 25, 26, 29, 31, 32, 33, 36, 38, 39, 40, 43, 45, 46, 47, 50, 52, 53, 54, 57, 59, 60, 61, 64, 66, 67, 68, 71, 73, 74, 75, 78, 80, 81, 82, 85, 87, 88, 89, 92, 94, 95, 96, 99, 101, 102, 103, 106, 108, 109, 110

Offset: 1

N. J. A. Sloane

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

Cf. A047346, A047389.

[n : n in [0..150] | n mod 7 in [1, 3, 4, 5]]; // Wesley Ivan Hurt, May 24 2016

A047366:=n->(14*n-9-I^(2*n)-(3-I)*I^(-n)-(3+I)*I^n)/8: seq(A047366(n), n=1..100); # Wesley Ivan Hurt, May 24 2016

Table[(14n-9-I^(2n)-(3-I)*I^(-n)-(3+I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 24 2016 *)
Select[Range@ 120, MemberQ[{1, 3, 4, 5}, Mod[#, 7]] &] (* Michael De Vlieger, May 24 2016 *)

G.f.: x*(1+2*x+x^2+x^3+2*x^4) / ( (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) = (14n-9-i^(2n)-(3-i)*i^(-n)-(3+i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047389(k), a(2k-1) = A047346(k). (End)
E.g.f.: (8 + sin(x) - 3*cos(x) + (7*x - 4)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, May 25 2016

More terms from Wesley Ivan Hurt, May 24 2016

A271937 a(n) = (7/4)n^2 + (5/2)n + (7 + (-1)^n)/8.

Original entry on oeis.org

1, 5, 13, 24, 39, 57, 79, 104, 133, 165, 201, 240, 283, 329, 379, 432, 489, 549, 613, 680, 751, 825, 903, 984, 1069, 1157, 1249, 1344, 1443, 1545, 1651, 1760, 1873, 1989, 2109, 2232, 2359, 2489, 2623, 2760, 2901, 3045, 3193, 3344, 3499, 3657, 3819, 3984, 4153

Offset: 0

Vincenzo Librandi, Apr 20 2016

Let P be a polygon with vertices (0,0), (0,2), (1,1) and (0,3/2). The number of integer points in nP is counted by this quasi-polynomial (nP is the n-fold dilation of P). See Wikipedia in Links section.
From Bob Selcoe, Sep 10 2016: (Start)
a(n) = the number of partitions in reverse lexicographic order starting with n 3's followed by n 2's; i.e., the number of partitions summing to 5n such that no part > 3 and the number of 3's digits <= the number of 2's digits.
First differences are A047346(n+1); second differences are 4 when n is even and 3 when n is odd (i.e., A010702(n+1)); third differences are 1 when n is even and -1 when n is odd. (End)

			a(1) = 5; the 5 partitions are: {3,2}; {3,1,1}; {2,2,1}; {2,1,1,1}; {1,1,1,1,1}.
a(3) = 24: floor(8/2) + floor(11/2) + floor(14/2) + floor(17/2) = 4+5+7+8 = 24.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Examples of Ehrhart Quasi-Polynomials.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

First bisection (after 1) is A168235.
Second bisection is A135703 (without 0).
Cf. A006578, A047346.

[(7/4)*n^2+(5/2)*n+(7+(-1)^n)/8: n in [0..50]];

Table[(7/4) n^2 + (5/2) n + (7 + (-1)^n)/8, {n, 0, 50}]
LinearRecurrence[{2,0,-2,1},{1,5,13,24},50] (* Harvey P. Dale, Mar 23 2025 *)

Vec((1+3*x+3*x^2)/((1-x)^3*(1+x)) + O(x^99)) \\ Altug Alkan, Sep 10 2016

O.g.f.: (1 + 3*x + 3*x^2)/((1 - x)^3*(1 + x)).
E.g.f.: (7 + 34*x + 14*x^2)*exp(x)/8 + exp(-x)/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(2*k) = k*(7*k + 5) + 1, a(2*k+1) = (k + 1)*(7*k + 5).
From Bob Selcoe, Sep 10 2016 (Start):
a(n) = (n+1)^2 + A006578(n).
a(n) = a(n-1) + A047346(n+1).
a(n) = Sum_{j=0..n} floor((2n+3j+2)/2).
(End)

Edited and extended by Bruno Berselli, Apr 20 2016

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A047380 Numbers that are congruent to {1, 2, 4, 5} mod 7.

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A047298 Numbers that are congruent to {1, 3, 4, 6} mod 7.

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A047351 Numbers that are congruent to {0, 1, 2, 4} mod 7.

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A047294 Numbers that are congruent to {1, 2, 4, 6} mod 7.

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A047344 Numbers that are congruent to {0, 1, 3, 4} mod 7.

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A047366 Numbers that are congruent to {1, 3, 4, 5} mod 7.

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A271937 a(n) = (7/4)*n^2 + (5/2)*n + (7 + (-1)^n)/8.

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A271937 a(n) = (7/4)n^2 + (5/2)n + (7 + (-1)^n)/8.