A195020 -id:A195020 - cp's OEIS frontend

A195021 a(n) = n(14n - 11).

0, 3, 34, 93, 180, 295, 438, 609, 808, 1035, 1290, 1573, 1884, 2223, 2590, 2985, 3408, 3859, 4338, 4845, 5380, 5943, 6534, 7153, 7800, 8475, 9178, 9909, 10668, 11455, 12270, 13113, 13984, 14883, 15810, 16765, 17748, 18759, 19798, 20865, 21960, 23083, 24234, 25413

Offset: 0

Omar E. Pol, Sep 07 2011

Sequence found by reading the first two vertices [0, 3] together with the line from 34, in the direction 34, 93, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020, which is related to the primitive Pythagorean triple [3, 4, 5]. For another version see A195030.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A144555, A185019, A193053, A195019, A195020, A195023, A195024, A195025, A195030, A195320, A198017.

Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=28: see Comments lines of A226492.

[14*n^2 - 11*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[n*(14*n - 11), {n, 0, 50}] (* Paolo Xausa, Jan 15 2025 *)

a(n)=n*(14*n-11) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - 11*n.
From Colin Barker, Apr 09 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(3+25*x)/(1-x)^3. (End)
E.g.f.: exp(x)*x*(3 + 14*x). - Elmo R. Oliveira, Dec 30 2024

Edited by Bruno Berselli, Oct 18 2011

A195023 a(n) = 14n^2 - 4n.

Original entry on oeis.org

0, 10, 48, 114, 208, 330, 480, 658, 864, 1098, 1360, 1650, 1968, 2314, 2688, 3090, 3520, 3978, 4464, 4978, 5520, 6090, 6688, 7314, 7968, 8650, 9360, 10098, 10864, 11658, 12480, 13330, 14208, 15114, 16048, 17010, 18000, 19018, 20064, 21138, 22240, 23370, 24528, 25714

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 10, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-axis of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A135703, A144555, A152760, A193053, A195019, A195020, A195024, A195320, A198017.

[14*n^2 - 4*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[14n^2-4n,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,10,48},50] (* Harvey P. Dale, Sep 05 2012 *)

a(n)=14*n^2-4*n \\ Charles R Greathouse IV, Apr 10 2012

a(n) = 2*A135703(n). - Bruno Berselli, Oct 13 2011
From Colin Barker, Apr 09 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*x*(5+9*x)/(1-x)^3. (End)
E.g.f.: 2*exp(x)*x*(5 + 7*x). - Elmo R. Oliveira, Dec 30 2024

Corrected by Vincenzo Librandi, Oct 14 2011

A195024 a(n) = n(14n - 1).

Original entry on oeis.org

0, 13, 54, 123, 220, 345, 498, 679, 888, 1125, 1390, 1683, 2004, 2353, 2730, 3135, 3568, 4029, 4518, 5035, 5580, 6153, 6754, 7383, 8040, 8725, 9438, 10179, 10948, 11745, 12570, 13423, 14304, 15213, 16150, 17115, 18108, 19129, 20178, 21255, 22360, 23493, 24654, 25843

Offset: 0

Omar E. Pol, Oct 13 2011

Related to the primitive Pythagorean triple [3, 4, 5].
Sequence found by reading the line from 0, in the direction 0, 13, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral.
Also sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the generalized 9-gonal numbers A118277. - Omar E. Pol, Jul 28 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A118277, A144555, A152760, A185019, A193053, A195019, A195020, A195320, A198017.

[14*n^2 - n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[n(14n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,13,54},50] (* Harvey P. Dale, Jul 28 2012 *)

a(n)=n*(14*n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - n.
From Colin Barker, Apr 09 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(13+15*x)/(1-x)^3. (End)
E.g.f.: exp(x)*x*(13 + 14*x). - Elmo R. Oliveira, Jan 12 2025

A152760 4 times 9-gonal numbers: a(n) = 2n(7*n-5).

Original entry on oeis.org

0, 4, 36, 96, 184, 300, 444, 616, 816, 1044, 1300, 1584, 1896, 2236, 2604, 3000, 3424, 3876, 4356, 4864, 5400, 5964, 6556, 7176, 7824, 8500, 9204, 9936, 10696, 11484, 12300, 13144, 14016, 14916, 15844, 16800, 17784, 18796, 19836, 20904, 22000, 23124

Offset: 0

Omar E. Pol, Dec 14 2008

Sequence found by reading the line from 0, in the direction 0, 4, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. The square spiral is related to the primitive Pythagorean triple [3, 4, 5]. - Omar E. Pol, Oct 13 2011

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

Cf. A001106, A139268, A152759, A195019, A195020, A195021.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,4,8!,28}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
4*PolygonalNumber[9,Range[0,50]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{0,4,36},50] (* Harvey P. Dale, Aug 26 2019 *)

a(n)=2*n*(7*n-5) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - 10*n = 4*A001106(n) = 2*A139268(n).
a(n) = a(n-1) + 28*n - 24 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
From Colin Barker, Apr 09 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 4*x*(1+6*x)/(1-x)^3. (End)
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 2*exp(x)*x*(2 + 7*x).
a(n) = n + A195021(n). (End)

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.

Original entry on oeis.org

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)

A158482 a(n) = 14*n^2 + 1.

Original entry on oeis.org

15, 57, 127, 225, 351, 505, 687, 897, 1135, 1401, 1695, 2017, 2367, 2745, 3151, 3585, 4047, 4537, 5055, 5601, 6175, 6777, 7407, 8065, 8751, 9465, 10207, 10977, 11775, 12601, 13455, 14337, 15247, 16185, 17151, 18145, 19167, 20217, 21295, 22401

Offset: 1

Vincenzo Librandi, Mar 20 2009

The identity (14*n^2 + 1)^2 - (49*n^2 + 7)*(2*n)^2 = 1 can be written as a(n)^2 - A158481(n)*A005843(n)^2 = 1.

Sequence found by reading the line from 15, in the direction 15, 57, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A118277, A158481, A195019, A195020.

I:=[15, 57, 127]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

LinearRecurrence[{3,-3,1},{15,57,127},50]

PARI
```
a(n) = 14*n^2+1;
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: x*(15+12*x+x^2)/(1-x)^3.
From Amiram Eldar, Feb 05 2021: (Start)
Sum_{n>=0} 1/a(n) = (1 - (Pi/sqrt(14))*coth(Pi/sqrt(14)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(14))*csch(Pi/sqrt(14)))/2.
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(14))*sinh(Pi/sqrt(7)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(14))*csch(Pi/sqrt(14)). (End)

A158485 a(n) = 14*n^2 - 1.

Original entry on oeis.org

13, 55, 125, 223, 349, 503, 685, 895, 1133, 1399, 1693, 2015, 2365, 2743, 3149, 3583, 4045, 4535, 5053, 5599, 6173, 6775, 7405, 8063, 8749, 9463, 10205, 10975, 11773, 12599, 13453, 14335, 15245, 16183, 17149, 18143, 19165, 20215, 21293, 22399

Offset: 1

Vincenzo Librandi, Mar 20 2009

The identity (14*n^2-1)^2-(49*n^2-7)*(2*n)^2=1 can be written as a(n)^2-A158484(n)*A005843(n)^2=1.

Sequence found by reading the line from 13, in the direction 13, 55,..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158484.

I:=[13, 55, 125]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

LinearRecurrence[{3,-3,1},{13,55,125},50]

PARI
```
a(n) = 14*n^2-1
```

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
G.f: x*(-13-16*x+x^2)/(x-1)^3.
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(14))*cot(Pi/sqrt(14)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(14))*csc(Pi/sqrt(14)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(14))*csc(Pi/sqrt(14)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(14))*sin(Pi/sqrt(7))/sqrt(2). (End)

A195025 a(n) = n(14n + 3).

Original entry on oeis.org

0, 17, 62, 135, 236, 365, 522, 707, 920, 1161, 1430, 1727, 2052, 2405, 2786, 3195, 3632, 4097, 4590, 5111, 5660, 6237, 6842, 7475, 8136, 8825, 9542, 10287, 11060, 11861, 12690, 13547, 14432, 15345, 16286, 17255, 18252, 19277, 20330, 21411, 22520, 23657, 24822, 26015

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 17, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-axis of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

a(k) is a square for k = (3/56)*((449 + 120*sqrt(14))^n + (449 - 120*sqrt(14))^n - 2). - Bruno Berselli, Oct 18 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A144555, A152760, A185019, A193053, A195019, A195020, A195023, A195024, A195320.

[14*n^2 +3*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[n(14n+3),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,17,62},50] (* Harvey P. Dale, Jul 17 2023 *)

a(n)=n*(14*n+3) \\ Charles R Greathouse IV, Oct 07 2015

a(n) =  14*n^2 + 3*n.
G.f.: x*(17+11*x)/(1-x)^3. - Bruno Berselli, Oct 18 2011
From Elmo R. Oliveira, Dec 30 2024: (Start)
E.g.f.: exp(x)*x*(17 + 14*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Name suggested by Bruno Berselli, Oct 13 2011

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.

Original entry on oeis.org

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

A195036 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [15, 8, 17]. The edges of the spiral have length A195035.

Original entry on oeis.org

0, 15, 23, 53, 69, 114, 138, 198, 230, 305, 345, 435, 483, 588, 644, 764, 828, 963, 1035, 1185, 1265, 1430, 1518, 1698, 1794, 1989, 2093, 2303, 2415, 2640, 2760, 3000, 3128, 3383, 3519, 3789, 3933, 4218, 4370, 4670, 4830, 5145, 5313, 5643, 5819, 6164, 6348

Offset: 0

Omar E. Pol, Sep 12 2011

Zero together with partial sums of A195035.
The only primes in the sequence are 23 and 53 since a(n) = (1/2)*((2*n+(-1)^n+3)/4)*((46*n-37*(-1)^n+37)/4). - Bruno Berselli, Sep 30 2011
The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 17 (Cf. A008599). The vertices on the main diagonal are the numbers A195039 = (15+8)*A000217 = 23*A000217, where both 15 and 8 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 15, while the distance "b" between nearest edges that are parallel to the initial edge is 8, so the distance "c" between nearest vertices on the same axis is 17 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(15^2+8^2) = sqrt(225+64) = sqrt(289) = 17. - 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, A195034, A195035, A195039.

[(2*n*(23*n+53)-(14*n+37)*(-1)^n+37)/16: n in [0..46]];  // Bruno Berselli, Sep 30 2011

a(n)=(2*n*(23*n+53)-(14*n+37)*(-1)^n+37)/16 \\ 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)^3).
a(n) = (2*n*(23*n+53) - (14*n+37)*(-1)^n + 37)/16.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)

More terms from Bruno Berselli, Sep 30 2011

cp's OEIS Frontend

A195021 a(n) = n*(14*n - 11).

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A195023 a(n) = 14*n^2 - 4*n.

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A195024 a(n) = n*(14*n - 1).

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A152760 4 times 9-gonal numbers: a(n) = 2*n*(7*n-5).

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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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A158482 a(n) = 14*n^2 + 1.

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A158485 a(n) = 14*n^2 - 1.

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A195021 a(n) = n(14n - 11).

A195023 a(n) = 14n^2 - 4n.

A195024 a(n) = n(14n - 1).

A152760 4 times 9-gonal numbers: a(n) = 2n(7*n-5).

A195025 a(n) = n(14n + 3).