A195031 -id:A195031 - cp's OEIS frontend

A195032 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [5, 12, 13]. The edges of the spiral have length A195031.

0, 5, 17, 27, 51, 66, 102, 122, 170, 195, 255, 285, 357, 392, 476, 516, 612, 657, 765, 815, 935, 990, 1122, 1182, 1326, 1391, 1547, 1617, 1785, 1860, 2040, 2120, 2312, 2397, 2601, 2691, 2907, 3002, 3230, 3330, 3570, 3675, 3927, 4037, 4301, 4416, 4692

Offset: 0

Omar E. Pol, Sep 12 2011

Zero together with partial sums of A195031.

The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 13 (cf. A008595). The vertices on the main diagonal are the numbers A195037 = (5+12)*A000217 = 17*A000217, where both 5 and 12 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 5, while the distance "b" between nearest edges that are parallel to the initial edge is 12, so the distance "c" between nearest vertices on the same axis is 13 because from the Pythagorean theorem we can write c = (a^2 + b^2)^(1/2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. - Omar E. Pol, Oct 12 2011

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. A195020, A195031, A195034, A195036, A195037.

[(2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

a[n_] := (2 n (17 n + 27) + (14 n - 3)*(-1)^n + 3)/16; Array[a, 50, 0] (* Amiram Eldar, Nov 23 2018 *)

vector(50, n, n--; (2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16) \\ G. C. Greubel, Nov 23 2018

[(2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16 for n in range(50)] # G. C. Greubel, Nov 23 2018

From Bruno Berselli, Oct 13 2011: (Start)
G.f.: x*(5 + 12*x)/((1 + x)^2*(1 - x)^3).
a(n) = (1/2)*((2*n + (-1)^n + 3)/4)*((34*n - 3*(-1)^n+3)/4) = (2*n*(17*n + 27) + (14*n - 3)*(-1)^n + 3)/16.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
E.g.f.: (1/16)*((3 + 88*x + 34*x^2)*exp(x) - (3 + 14*x)*exp(-x)). - Franck Maminirina Ramaharo, Nov 23 2018

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

A195033 Multiples of 21 and of 20 interleaved: a(2n-1) = 21n, a(2n) = 20n.

Original entry on oeis.org

21, 20, 42, 40, 63, 60, 84, 80, 105, 100, 126, 120, 147, 140, 168, 160, 189, 180, 210, 200, 231, 220, 252, 240, 273, 260, 294, 280, 315, 300, 336, 320, 357, 340, 378, 360, 399, 380, 420, 400, 441, 420, 462, 440, 483, 460, 504, 480, 525, 500, 546, 520, 567, 540

Offset: 1

Omar E. Pol, Sep 12 2011

First differences of A195034.

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 [21, 20, 29]. Zero together with partial sums give A195034, 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. A010693, A010718, A029578.

Cf. A195019, A195031, A195034, A195035.

&cat[[21*n,20*n]: n in [1..27]];  // Bruno Berselli, Sep 29 2011

Riffle[21*#, 20*#] & [Range[50]] (* Paolo Xausa, Mar 21 2024 *)

a(n)=(n+1)\2*if(n%2,21,20) \\ Charles R Greathouse IV, Oct 07 2015

From Bruno Berselli, Sep 29 2011:  (Start)
G.f.: x*(21+20*x)/((1-x)^2*(1+x)^2).
a(n) = A010693(n)*A010718(n)*A029578(n+1) = (41*n-(n+21)*(-1)^n+21)/4.
a(n) = 2*a(n-2) - a(n-4). (End)

More terms from Bruno Berselli, Sep 29 2011

A195035 Multiples of 15 and of 8 interleaved: a(2n-1) = 15n, a(2n) = 8n.

Original entry on oeis.org

15, 8, 30, 16, 45, 24, 60, 32, 75, 40, 90, 48, 105, 56, 120, 64, 135, 72, 150, 80, 165, 88, 180, 96, 195, 104, 210, 112, 225, 120, 240, 128, 255, 136, 270, 144, 285, 152, 300, 160, 315, 168, 330, 176, 345, 184, 360, 192, 375, 200, 390, 208, 405, 216

Offset: 1

Omar E. Pol, Sep 12 2011

First differences of A195036.

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 [15, 8, 17]. Zero together with partial sums give A195036; 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. A004526, A010686, A010706.

Cf. A195019, A195031, A195033, A195036.

&cat[[15*n, 8*n]: n in [1..27]];  // Bruno Berselli, Sep 30 2011

Riffle[15*#, 8*#] & [Range[50]] (* Paolo Xausa, Mar 21 2024 *)

a(n)=(n+1)\2*if(n%2,15,8) \\ Charles R Greathouse IV, Oct 07 2015

From Bruno Berselli, Sep 30 2011: (Start)
G.f.: x*(15+8*x)/((1-x)^2*(1+x)^2).
a(n) = A010686(n)*A010706(n-1)*A004526(n+1) = (23*n-(7*n+15)*(-1)^n+15)/4.
a(n) = 2*a(n-2) - a(n-4).
a(-n) = -a(A014681(n-1)). (End)

A195037 17 times triangular numbers.

Original entry on oeis.org

0, 17, 51, 102, 170, 255, 357, 476, 612, 765, 935, 1122, 1326, 1547, 1785, 2040, 2312, 2601, 2907, 3230, 3570, 3927, 4301, 4692, 5100, 5525, 5967, 6426, 6902, 7395, 7905, 8432, 8976, 9537, 10115, 10710, 11322, 11951, 12597, 13260, 13940, 14637, 15351, 16082, 16830

Offset: 0

Omar E. Pol, Sep 12 2011

Related to the primitive Pythagorean triple [5, 12, 13].
Sequence found by reading the line from 0, in the direction 0, 17, ..., and the same line from 0, in the direction 0, 51, ..., in the Pythagorean spiral whose edges have length A195031 and whose vertices are the numbers A195032. This is the main diagonal of the square spiral.
Sum of the numbers from 8*n to 9*n. - Wesley Ivan Hurt, Dec 23 2015

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195032.

Cf. A000217, A195031.

[17*n*(n+1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Dec 23 2015

A195037:=n->17*n*(n+1)/2: seq(A195037(n), n=0..60); # Wesley Ivan Hurt, Dec 23 2015

17*Accumulate[Range[0,50]] (* Harvey P. Dale, May 31 2014 *)
Table[17*n*(n + 1)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Dec 23 2015 *)

a(n)=17*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (17*n^2 + 17*n)/2 = 17*n*(n+1)/2 = 17*A000217(n).
From Wesley Ivan Hurt, Dec 23 2015: (Start)
G.f.: 17*x/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = Sum_{i=8*n..9*n} i. (End)
From Amiram Eldar, Feb 22 2023: (Start)
Sum_{n>=1} 1/a(n) = 2/17.
Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/17.
Product_{n>=1} (1 - 1/a(n)) = -(17/(2*Pi))*cos(5*Pi/(2*sqrt(17))).
Product_{n>=1} (1 + 1/a(n)) = (17/(2*Pi))*cos(3*Pi/(2*sqrt(17))). (End)
From Elmo R. Oliveira, Dec 25 2024: (Start)
E.g.f.: 17*exp(x)*x*(2 + x)/2.
a(n) = A195032(2*n). (End)

cp's OEIS Frontend

A195032 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [5, 12, 13]. The edges of the spiral have length A195031.

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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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A195033 Multiples of 21 and of 20 interleaved: a(2n-1) = 21n, a(2n) = 20n.

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A195035 Multiples of 15 and of 8 interleaved: a(2n-1) = 15n, a(2n) = 8n.

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A195037 17 times triangular numbers.

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