A195033 -id:A195033 - cp's OEIS frontend

A195034 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [21, 20, 29]. The edges of the spiral have length A195033.

0, 21, 41, 83, 123, 186, 246, 330, 410, 515, 615, 741, 861, 1008, 1148, 1316, 1476, 1665, 1845, 2055, 2255, 2486, 2706, 2958, 3198, 3471, 3731, 4025, 4305, 4620, 4920, 5256, 5576, 5933, 6273, 6651, 7011, 7410, 7790, 8210, 8610, 9051, 9471

Offset: 0

Omar E. Pol, Sep 12 2011

Zero together with partial sums of A195033.
The only primes in the sequence are 41 and 83 since a(n) = (1/2)*((2*n+(-1)^n+3)/4)*((82*n-43*(-1)^n+43)/4). - Bruno Berselli, Oct 12 2011
The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 29 (Cf. A195819). The vertices on the main diagonal are the numbers A195038 = (21+20)*A000217 = 41*A000217, where both 21 and 20 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 21, while the distance "b" between nearest edges that are parallel to the initial edge is 20, so the distance "c" between nearest vertices on the same axis is 29 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(21^2+20^2) = sqrt(441+400) = sqrt(841) = 29. - 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, A195032, A195033, A195036, A195038.

[(2*n*(41*n+83)-(2*n+43)*(-1)^n+43)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

LinearRecurrence[{1,2,-2,-1,1},{0,21,41,83,123},50] (* Harvey P. Dale, May 02 2012 *)

concat(0, Vec(x*(21+20*x)/((1+x)^2*(1-x)^3) + O(x^60))) \\ Michel Marcus, Mar 08 2016

From Bruno Berselli, Oct 12 2011: (Start)
G.f.: x*(21+20*x)/((1+x)^2*(1-x)^3).
a(n) = (2*n*(41*n+83)-(2*n+43)*(-1)^n+43)/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) = A142150(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

A195819 Multiples of 29.

Original entry on oeis.org

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 870, 899, 928, 957, 986, 1015, 1044, 1073, 1102, 1131, 1160, 1189, 1218, 1247, 1276, 1305, 1334

Offset: 0

Omar E. Pol, Oct 12 2011

Length of hypotenuses on the main diagonal of the Pythagorean spiral whose edges have length A195033 and whose vertices are the numbers A195034, if n >= 1.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004922, A004942, A008605, A010868, A135631, A195033, A195034.

29Range[0,50] (* Harvey P. Dale, Oct 25 2011 *)

a(n) = 29*n.
From Elmo R. Oliveira, Mar 21 2024: (Start)
G.f.: 29*x/(x-1)^2.
E.g.f.: 29*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A195031 Multiples of 5 and of 12 interleaved: a(2n-1) = 5n, a(2n) = 12n.

Original entry on oeis.org

5, 12, 10, 24, 15, 36, 20, 48, 25, 60, 30, 72, 35, 84, 40, 96, 45, 108, 50, 120, 55, 132, 60, 144, 65, 156, 70, 168, 75, 180, 80, 192, 85, 204, 90, 216, 95, 228, 100, 240, 105, 252, 110, 264, 115, 276, 120, 288, 125, 300, 130, 312, 135, 324, 140, 336, 145, 348

Offset: 1

Omar E. Pol, Sep 12 2011

First differences of A195032.

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 [5, 12, 13]. Zero together with partial sums give A195032, 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. A195019, A195032, A195033, A195035.

&cat[[5*n, 12*n]: n in [1..27]];  // Bruno Berselli, Sep 30 2011

With[{nn=30},Riffle[5Range[nn],12Range[nn]]] (* or *) LinearRecurrence[ {0,2,0,-1},{5,12,10,24},60] (* Harvey P. Dale, Aug 18 2012 *)

a(n)=(n+1)\2*if(n%2,5,12) \\ Charles R Greathouse IV, Oct 07 2015

From Bruno Berselli, Sep 30 2011:  (Start)
G.f.: x*(5+12*x)/((1-x)^2*(1+x)^2).
a(n) = ((17+7*(-1)^n)/2)*((2*n-(-1)^n+1)/4) = (17*n+(7*n-5)*(-1)^n+5)/4.
a(n)*a(n+1) = a(10*s), where s is A002620(n+1).
a(n) = 2*a(n-2) - a(n-4). (End)

More terms from Bruno Berselli, Sep 30 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)

A195038 41 times triangular numbers.

Original entry on oeis.org

0, 41, 123, 246, 410, 615, 861, 1148, 1476, 1845, 2255, 2706, 3198, 3731, 4305, 4920, 5576, 6273, 7011, 7790, 8610, 9471, 10373, 11316, 12300, 13325, 14391, 15498, 16646, 17835, 19065, 20336, 21648, 23001, 24395, 25830, 27306, 28823, 30381, 31980

Offset: 0

Omar E. Pol, Sep 12 2011

Related to the primitive Pythagorean triple [21, 20, 29].

Sequence found by reading the line from 0, in the direction 0, 41, ..., and the same line from 0, in the direction 0, 123, ..., in the Pythagorean spiral whose edges have length A195033 and whose vertices are the numbers A195034. This is the main diagonal of the square spiral.

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

Bisection of A195034.

Cf. A000217, A195033.

41*Accumulate[Range[0,40]] (* or *) LinearRecurrence[{3,-3,1},{0,41,123},40] (* Harvey P. Dale, Nov 20 2015 *)

a(n)=41*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (41*n^2 + 41*n)/2 = 41*n*(n+1)/2 = 41*A000217(n).
G.f.: -41*x/(x-1)^3. - R. J. Mathar, Oct 04 2014
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 41*exp(x)*x*(2 + x)/2.
a(n) = A195034(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

cp's OEIS Frontend

A195034 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [21, 20, 29]. The edges of the spiral have length A195033.

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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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A195819 Multiples of 29.

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A195031 Multiples of 5 and of 12 interleaved: a(2n-1) = 5n, a(2n) = 12n.

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

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A195038 41 times triangular numbers.

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