A047356 -id:A047356 - cp's OEIS frontend

A047361 Numbers that are congruent to {0, 1, 2, 3} mod 7.

0, 1, 2, 3, 7, 8, 9, 10, 14, 15, 16, 17, 21, 22, 23, 24, 28, 29, 30, 31, 35, 36, 37, 38, 42, 43, 44, 45, 49, 50, 51, 52, 56, 57, 58, 59, 63, 64, 65, 66, 70, 71, 72, 73, 77, 78, 79, 80, 84, 85, 86, 87, 91, 92, 93, 94, 98, 99, 100, 101, 105, 106, 107, 108, 112

Offset: 1

N. J. A. Sloane

Nonnegative m for which floor(2*m/7) = 2*floor(m/7). [Bruno Berselli, Dec 03 2015]

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

Cf. A047352, A047356.

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

A047361:=n->(14*n-23-3*I^(2*n)-(3-3*I)*I^(-n)-(3+3*I)*I^n)/8: seq(A047361(n), n=1..100); # Wesley Ivan Hurt, May 23 2016

Flatten[#+{0,1,2,3}&/@(7*Range[0,20])] (* Harvey P. Dale, Jan 17 2013 *)

concat(0, Vec(x^2*(1+x+x^2+4*x^3)/((1+x)*(x^2+1)*(x-1)^2) + O(x^100))) \\ Altug Alkan, Dec 09 2015

a(n) = 7*floor(n/4) + (n mod 4), with offset 0 and a(0) = 0. - Gary Detlefs, Mar 09 2010
G.f.: x^2*(1+x+x^2+4*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) = (14*n-23-3*i^(2*n)-(3-3*i)*i^(-n)-(3+3*i)*i^n)/8, where i=sqrt(-1).
a(2k) = A047356(k), a(2k-1) = A047352(k). (End)
E.g.f.: (16 + 3*(sin(x) - cos(x)) + (7*x - 10)*sinh(x) + (7*x - 13)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

More terms from Wesley Ivan Hurt, May 23 2016

A047286 Numbers that are congruent to {1, 2, 3, 6} mod 7.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 10, 13, 15, 16, 17, 20, 22, 23, 24, 27, 29, 30, 31, 34, 36, 37, 38, 41, 43, 44, 45, 48, 50, 51, 52, 55, 57, 58, 59, 62, 64, 65, 66, 69, 71, 72, 73, 76, 78, 79, 80, 83, 85, 86, 87, 90, 92, 93, 94, 97, 99, 100, 101, 104, 106, 107, 108, 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, A047356.

[n : n in [0..150] | n mod 7 in [1, 2, 3, 6]]; // Wesley Ivan Hurt, May 22 2016

A047286:=n->(14*n-11+I^(2*n)+(1+3*I)*I^(-n)+(1-3*I)*I^n)/8: seq(A047286(n), n=1..100); # Wesley Ivan Hurt, May 22 2016

Table[(14n-11+I^(2n)+(1+3I)*I^(-n)+(1-3I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 22 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 6, 8}, 80] (* Vincenzo Librandi, May 24 2016 *)

G.f.: x*(1+x+x^2+3*x^3+x^4) / ( (1+x)*(1+x^2)*(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) = (14n-11+i^(2n)+(1+3i)*i^(-n)+(1-3i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047276(n), a(2n-1) = A047356(n). (End)
E.g.f.: (4 + 3*sin(x) + cos(x) + (7*x - 6)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, May 23 2016

More terms from Wesley Ivan Hurt, May 22 2016

A047338 Numbers that are congruent to {1, 2, 3, 4} mod 7.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18, 22, 23, 24, 25, 29, 30, 31, 32, 36, 37, 38, 39, 43, 44, 45, 46, 50, 51, 52, 53, 57, 58, 59, 60, 64, 65, 66, 67, 71, 72, 73, 74, 78, 79, 80, 81, 85, 86, 87, 88, 92, 93, 94, 95, 99, 100, 101, 102, 106, 107, 108, 109

Offset: 1

N. J. A. Sloane

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

Cf. A047348, A047356.

[n : n in [0..150] | n mod 7 in [1, 2, 3, 4]]; // Wesley Ivan Hurt, May 23 2016

A047338:=n->(14*n-15-3*I^(2*n)-(3-3*I)*I^(-n)-(3+3*I)*I^n)/8: seq(A047338(n), n=1..100); # Wesley Ivan Hurt, May 23 2016

Table[(14n-15-3*I^(2n)-(3-3*I)*I^(-n)-(3+3*I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 23 2016 *)

G.f.: x*(1+x+x^2+x^3+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-15-3*i^(2n)-(3-3*i)*i^(-n)-(3+3*i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047348(n), a(2n-1) = A047356(n). (End)
E.g.f.: (12 + 3*(sin(x) - cos(x)) + (7*x - 6)*sinh(x) + (7*x - 9)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

More terms from Wesley Ivan Hurt, May 23 2016

A047372 Numbers that are congruent to {1, 2, 3, 5} mod 7.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 19, 22, 23, 24, 26, 29, 30, 31, 33, 36, 37, 38, 40, 43, 44, 45, 47, 50, 51, 52, 54, 57, 58, 59, 61, 64, 65, 66, 68, 71, 72, 73, 75, 78, 79, 80, 82, 85, 86, 87, 89, 92, 93, 94, 96, 99, 100, 101, 103, 106, 107, 108, 110

Offset: 1

N. J. A. Sloane

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

Cf. A047356, A047385.

[n : n in [0..150] | n mod 7 in [1, 2, 3, 5]]; // Wesley Ivan Hurt, Jun 04 2016

A047372:=n->(14*n+(3*I-1)*(-I)^n-(3*I+1)*I^n-(-1)^n-13)/8: seq(A047372(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016

Flatten[7Range[0,12]+n/.n->{1,2,3,5}] (* Harvey P. Dale, Dec 13 2010 *)

From Bruno Berselli, Dec 01 2010: (Start)
G.f.: x*(1+x+x^2+2*x^3+2*x^4) / ((1-x)^2*(1+x+x^2+x^3)).
a(n) = (14*n+(3*i-1)*(-i)^n-(3*i+1)*i^n-(-1)^n-13)/8, i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047385(k), a(2k-1) = A047356(k). (End)
E.g.f.: (8 + 3*sin(x) - cos(x) + (7*x - 6)*sinh(x) + 7*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, Jun 04 2016

A065450 Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.

Original entry on oeis.org

1, 2, 10, 22, 37, 54, 76, 100, 129, 160, 196, 234, 277, 322, 372, 424, 481, 540, 604, 670, 741, 814, 892, 972, 1057, 1144, 1236, 1330, 1429, 1530, 1636, 1744, 1857, 1972, 2092, 2214, 2341, 2470, 2604, 2740, 2881, 3024, 3172, 3322, 3477, 3634, 3796, 3960

Offset: 0

Bodo Zinser, Nov 18 2001

The first conjecture is true: Partial sums of A047356 = b(n) = (14*(n*(n+1)) + 2*n + 5 + 3*(-1)^n )/8, since A047356(n)=(14*n+1+3*(-1)^n)/4. And b(n) has g.f. (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). The difference a(n) - b(n) = 0,2,2,0,0,0,0,0,0..., which has g.f. 2*x^2 + 2*x. Then (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1) + 2*x^2 + 2*x = (-2*x^6 + 2*x^5 + 4*x^4 - 4*x^3 + 2*x^2 + 4*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). - Vim Wenders, Apr 16 2008

Cf. A098498.

Conjectures: G.f.: [1+6x^2+4x^3-4x^4-2x^5+2x^6]/[(1+x)*(1-x)^3]. For n>3, partial sums of A047356. - Ralf Stephan, Mar 06 2004

The second conjecture "For n>3, partial sums of A047356" is also true. From the last possible move, we can either move back to the second last possible move or to b(n)=A047883(n) new squares. So a(n) = a(n-2)+b(n). For n>6, b(n)=7(n-1)+4=A017029(n-1). But a number of the form 7n+4 is naturally the sum of two consecutive terms in A047356 (4=1+3,11=3+8,18=8+10, ...). The conjecture follows. - Vim Wenders, Apr 12 2008

More terms from Don Reble, Nov 28 2001

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A047361 Numbers that are congruent to {0, 1, 2, 3} mod 7.

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A047286 Numbers that are congruent to {1, 2, 3, 6} mod 7.

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A047338 Numbers that are congruent to {1, 2, 3, 4} mod 7.

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A047372 Numbers that are congruent to {1, 2, 3, 5} mod 7.

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A065450 Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.

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