A047355 -id:A047355 - cp's OEIS frontend

A195020 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The edges of the spiral have length A195019.

0, 3, 7, 13, 21, 30, 42, 54, 70, 85, 105, 123, 147, 168, 196, 220, 252, 279, 315, 345, 385, 418, 462, 498, 546, 585, 637, 679, 735, 780, 840, 888, 952, 1003, 1071, 1125, 1197, 1254, 1330, 1390, 1470, 1533, 1617, 1683, 1771, 1840, 1932, 2004, 2100

Offset: 0

Omar E. Pol, Sep 07 2011 - Sep 12 2011

Zero together with the partial sums of A195019.
The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives A008587. The vertices on the main diagonal are the numbers A024966 = (3+4)*A000217 = 7*A000217, where both 3 and 4 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 3, while the distance "b" between nearest edges that are parallel to the initial edge is 4, so the distance "c" between nearest vertices on the same axis is 5 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(3^2+4^2) = sqrt(9+16) = sqrt(25) = 5.
Let an array have m(0,n)=m(n,0)=n*(n-1)/2 and m(n,n)=n*(n+1)/2. The first n+1 terms in row(n) are the numbers in the closed interval m(0,n) to m(n,n). The terms in column(n) are the same from m(n,0) to m(n,n). The first few antidiagonals are 0; 0,0; 1,1,1; 3,2,2,3; 6,4,3,4,6; 10,7,5,5,7,10. a(n) is the difference between the sum of the terms in the n+1 X n+1 matrices and those in the n X n matrices. - J. M. Bergot, Jul 05 2013 [The first five rows are: 0,0,1,3,6; 0,1,2,4,7; 1,2,3,5,8; 3,4,5,6,9; 6,7,8,9,10]

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ron Knott, Pythagorean triangles and triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A024966, A008585, A008586, A001106, A022264, A024966, A033572, A144555, A158482, A158485, A195018, A195032, A195034, A195036.

[(2*n*(7*n+13)+(2*n-5)*(-1)^n+5)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

With[{r = Range[50]}, Join[{0}, Accumulate[Riffle[3*r, 4*r]]]] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 3, 7, 13, 21}, 100] (* Paolo Xausa, Feb 09 2024 *)

From Bruno Berselli, Oct 13 2011: (Start)
G.f.: x*(3+4*x)/((1+x)^2*(1-x)^3).
a(n) = (1/2)*A004526(n+2)*A047335(n+1) = (2*n*(7*n+13) + (2*n-5)*(-1)^n+5)/16.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) - a(n-2) = A047355(n+1). (End)

A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 12, 16, 15, 20, 18, 24, 21, 28, 24, 32, 27, 36, 30, 40, 33, 44, 36, 48, 39, 52, 42, 56, 45, 60, 48, 64, 51, 68, 54, 72, 57, 76, 60, 80, 63, 84, 66, 88, 69, 92, 72, 96, 75, 100, 78, 104, 81, 108, 84, 112, 87, 116, 90, 120, 93, 124, 96, 128

Offset: 1

Omar E. Pol, Sep 07 2011, Sep 12 2011

First differences of A195020.

a(n) is also the length of the n-th edge of a square spiral in which the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The spiral contains infinitely many Pythagorean triples in which the hypotenuses are the positives A008587. Zero together with partial sums give A195020; the vertices of the spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean triangles and triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008585, A008586, A026741, A195020-A195029, A195031, A195033, A195035.

[((n-3)*(-1)^n+7*n+3)/4: n in [1..60]]; // Vincenzo Librandi, Sep 12 2011

Table[((n-3)*(-1)^n + 7*n + 3)/4, {n,1,50}] (* G. C. Greubel, Aug 19 2017 *)

a(n)=(n+1)\2*(4-n%2)  \\ M. F. Hasler, Sep 08 2011

pair(3*n, 4*n).
a(2*n-1) = 3*n, a(2*n) = 4*n. - M. F. Hasler, Sep 08 2011
G.f.: x*(3+4*x) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Sep 09 2011
From Bruno Berselli, Sep 12 2011: (Start)
a(n) = ((n-3)*(-1)^n + 7*n + 3)/4.
a(n) + a(n+1) = A047355(n+2). (End)
E.g.f.: (1/4)*((3 + 7*x)*exp(x) - (3 + x)*exp(-x)). - G. C. Greubel, Aug 19 2017

A010702 Period 2: repeat (3,4).

Original entry on oeis.org

3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3

Offset: 0

N. J. A. Sloane

Continued fraction expansion of A176102. - R. J. Mathar, Mar 08 2012

Also decimal expansion of 34/99. - Nicolas Bělohoubek, Nov 12 2021

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

Cf. A047355 (partial sums), A176102.

Cf. A010695, A010696.

a010702 = (+ 3) . (`mod` 2)
a010702_list = cycle [3,4]  -- Reinhard Zumkeller, Jul 05 2012

[3 + (n mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2014

&cat[[3,4]: n in [0..50]]; // Vincenzo Librandi, Aug 01 2015

A010702:=n->3+(n mod 2): seq(A010702(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2014

3 + Mod[Range[0, 100], 2] (* Wesley Ivan Hurt, Jul 24 2014 *)
PadRight[{}, 100, {3, 4}] (* Vincenzo Librandi, Aug 01 2015 *)

a(n)=3+n%2 \\ Charles R Greathouse IV, Dec 21 2011

def A010702(n): return 3 + (n & 1) # Chai Wah Wu, May 25 2022

G.f.: (3+4*x)/(1-x^2). - Jaume Oliver Lafont, Mar 20 2009
a(n) = floor((n+1)*7/2) - floor((n)*7/2). - Hailey R. Olafson, Jul 23 2014
a(n) = 3 + (n mod 2) = 4 - ((n+1) mod 2). - Wesley Ivan Hurt, Jul 24 2014
From Nicolas Bělohoubek, Nov 12 2021: (Start)
a(n) = 12/a(n-1). See also A010696.
a(n) = 7 - a(n-1). See also A010695. (End)
a(n) = (7-(-1)^n)/2. - Aaron J Grech, Jul 28 2024

A030123 Most likely total for a roll of n 6-sided dice, choosing the smallest if there is a choice.

Original entry on oeis.org

0, 1, 7, 10, 14, 17, 21, 24, 28, 31, 35, 38, 42, 45, 49, 52, 56, 59, 63, 66, 70, 73, 77, 80, 84, 87, 91, 94, 98, 101, 105, 108, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 154, 157, 161, 164, 168, 171, 175, 178, 182, 185, 189, 192

Offset: 0

Eric W. Weisstein

In fact ceiling(7n/2) is just as likely as floor(7n/2), so sequence could equally well be A047345. - Henry Bottomley, Jan 19 2001. a(1) is the only exception to this rule. - Dmitry Kamenetsky, Nov 03 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (corrected by Sean A. Irvine, Jan 18 2019)
Eric Weisstein's World of Mathematics, Dice.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047355.

I:=[7,10,14]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..60]]; // Vincenzo Librandi, Oct 19 2013

A030123:=n->floor(7*n/2): seq(A030123(n), n=2..100); # Wesley Ivan Hurt, Jan 23 2017

CoefficientList[Series[-(3 x^2 - 3 x - 7)/((x - 1)^2 (x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 19 2013 *)

a(n)=7*n\2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = floor(7*n/2) for n >= 2.
From Colin Barker, Jun 09 2013: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 5.
G.f.: x - x^2 * (3*x^2-3*x-7) / ((x-1)^2*(x+1)). (End)

a(0) and a(1) added by Dmitry Kamenetsky, Nov 03 2017

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

A198268 Round(n*sqrt(12)).

Original entry on oeis.org

0, 3, 7, 10, 14, 17, 21, 24, 28, 31, 35, 38, 42, 45, 48, 52, 55, 59, 62, 66, 69, 73, 76, 80, 83, 87, 90, 94, 97, 100, 104, 107, 111, 114, 118, 121, 125, 128, 132, 135, 139, 142, 145, 149, 152, 156, 159, 163, 166, 170, 173, 177, 180, 184, 187

Offset: 0

Vincenzo Librandi, Oct 24 2011

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

Cf. A047355, A194028.

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

A047283 Numbers that are congruent to {0, 1, 3, 6} mod 7.

Original entry on oeis.org

0, 1, 3, 6, 7, 8, 10, 13, 14, 15, 17, 20, 21, 22, 24, 27, 28, 29, 31, 34, 35, 36, 38, 41, 42, 43, 45, 48, 49, 50, 52, 55, 56, 57, 59, 62, 63, 64, 66, 69, 70, 71, 73, 76, 77, 78, 80, 83, 84, 85, 87, 90, 91, 92, 94, 97, 98, 99, 101, 104, 105, 106, 108, 111

Offset: 1

N. J. A. Sloane

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

Cf. A047336, A047355.

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

A047283:=n->(14*n-15+I^(2*n)+(3+I)*I^(-n)+(3-I)*I^n)/8: seq(A047283(n), n=1..100); # Wesley Ivan Hurt, May 22 2016

Select[Range[0,100], MemberQ[{0,1,3,6}, Mod[#,7]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,1,3,6,7}, 60] (* Harvey P. Dale, Mar 09 2012 *)

G.f.: x^2*(1+2*x+3*x^2+x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Harvey P. Dale, Mar 09 2012
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = (14n-15+i^(2n)+(3+i)*i^(-n)+(3-i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047336(n), a(2n-1) = A047355(n). (End)

More terms from Wesley Ivan Hurt, May 22 2016

A047285 Numbers that are congruent to {0, 2, 3, 6} mod 7.

Original entry on oeis.org

0, 2, 3, 6, 7, 9, 10, 13, 14, 16, 17, 20, 21, 23, 24, 27, 28, 30, 31, 34, 35, 37, 38, 41, 42, 44, 45, 48, 49, 51, 52, 55, 56, 58, 59, 62, 63, 65, 66, 69, 70, 72, 73, 76, 77, 79, 80, 83, 84, 86, 87, 90, 91, 93, 94, 97, 98, 100, 101, 104, 105, 107, 108, 111

Offset: 1

N. J. A. Sloane

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

Cf. A047276, A047355.

[n : n in [0..150] | n mod 7 in [0, 2, 3, 6]]; // Wesley Ivan Hurt, Jun 02 2016

A047285:=n->(14*n-13+3*I^(2*n)+(1+I)*I^(-n)+(1-I)*I^n)/8: seq(A047285(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Table[(14n-13+3*I^(2n)+(1+I)*I^(-n)+(1-I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 02 2016 *)
Select[Range[0,120],MemberQ[{0,2,3,6},Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,0,1,-1},{0,2,3,6,7},100] (* Harvey P. Dale, Jul 12 2020 *)

G.f.: x^2*(2+x+3*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-13+3*i^(2*n)+(1+i)*i^(-n)+(1-i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047276(k), a(2k-1) = A047355(k). (End)

A047340 Numbers that are congruent to {0, 2, 3, 4} mod 7.

Original entry on oeis.org

0, 2, 3, 4, 7, 9, 10, 11, 14, 16, 17, 18, 21, 23, 24, 25, 28, 30, 31, 32, 35, 37, 38, 39, 42, 44, 45, 46, 49, 51, 52, 53, 56, 58, 59, 60, 63, 65, 66, 67, 70, 72, 73, 74, 77, 79, 80, 81, 84, 86, 87, 88, 91, 93, 94, 95, 98, 100, 101, 102, 105, 107, 108, 109, 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. A047348, A047355.

[n : n in [0..150] | n mod 7 in [0,2,3,4]]; // Vincenzo Librandi, Feb 17 2014

A047340:=n->(14*n-17-I^(2*n)-(3-I)*I^(-n)-(3+I)*I^n)/8: seq(A047340(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Select[Range[0,100],MemberQ[{0,2,3,4},Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,0,1,-1},{0,2,3,4,7},100] (* Harvey P. Dale, Feb 16 2014 *)
CoefficientList[Series[x (2 + x + x^2 + 3 x^3)/((1 + x) (1 + x^2) (x - 1)^2), {x, 0, 200}], x] (* Vincenzo Librandi, Feb 17 2014 *)

G.f.: x^2*(2+x+x^2+3*x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 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-17-i^(2n)-(3-i)*i^(-n)-(3+i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047348(n), a(2n-1) = A047355(n). (End)

More terms from Vincenzo Librandi, Feb 17 2014

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

cp's OEIS Frontend

A195020 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The edges of the spiral have length A195019.

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A195019 Multiples of 3 and of 4 interleaved: a(2*n-1) = 3*n, a(2*n) = 4*n.

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A010702 Period 2: repeat (3,4).

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A030123 Most likely total for a roll of n 6-sided dice, choosing the smallest if there is a choice.

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

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A198268 Round(n*sqrt(12)).

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Magma

A047283 Numbers that are congruent to {0, 1, 3, 6} mod 7.

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A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.