A195019 -id:A195019 - cp's OEIS frontend

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

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)

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

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

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)

A033572 a(n) = (2n+1)(7*n+1).

Original entry on oeis.org

1, 24, 75, 154, 261, 396, 559, 750, 969, 1216, 1491, 1794, 2125, 2484, 2871, 3286, 3729, 4200, 4699, 5226, 5781, 6364, 6975, 7614, 8281, 8976, 9699, 10450, 11229, 12036, 12871, 13734, 14625, 15544, 16491, 17466, 18469, 19500, 20559, 21646, 22761, 23904, 25075, 26274, 27501, 28756

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 1, in the direction 1, 24,..., 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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A001106.

List([0..50], n-> (2*n+1)*(7*n+1)); # G. C. Greubel, Oct 12 2019

[(2*n+1)*(7*n+1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

seq((2*n+1)*(7*n+1), n=0..50); # G. C. Greubel, Oct 12 2019

Table[(2*n+1)*(7*n+1), {n, 0, 50}] (* G. C. Greubel, Oct 12 2019 *)
LinearRecurrence[{3,-3,1},{1,24,75},50] (* Harvey P. Dale, Apr 19 2023 *)

a(n)=(2*n+1)*(7*n+1) \\ Charles R Greathouse IV, Jun 17 2017

[(2*n+1)*(7*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

a(n) = a(n-1) + 28*n - 5 for n>0, a(0)=1. - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Oct 12 2019: (Start)
G.f.: (1 + 21*x + 6*x^2)/(1-x)^3.
E.g.f.: (1 + 23*x + 14*x^2)*exp(x). (End)
Sum 1/a(n) = -gamma/5 -2*log(2)/5 -psi(1/7)/5 = 1.0800940432405839438217..., gamma=A001620, psi(1/7) = -A354627. - R. J. Mathar, May 07 2024

A195026 a(n) = 7n(2*n + 1).

Original entry on oeis.org

0, 21, 70, 147, 252, 385, 546, 735, 952, 1197, 1470, 1771, 2100, 2457, 2842, 3255, 3696, 4165, 4662, 5187, 5740, 6321, 6930, 7567, 8232, 8925, 9646, 10395, 11172, 11977, 12810, 13671, 14560, 15477, 16422, 17395, 18396, 19425, 20482, 21567, 22680, 23821, 24990

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 21, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. Semi-diagonal opposite to A195320 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

Sum of the numbers from 6*n to 8*n. - Wesley Ivan Hurt, Dec 23 2015

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

Cf. A014105, A144555, A152760, A195019, A195020, A195021, A195023, A195024, A195025, A195320.

Cf. A185019, A193053, A198017.

[14*n^2 +7*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

A195026:=n->7*n*(2*n+1): seq(A195026(n), n=0..50); # Wesley Ivan Hurt, Dec 23 2015

Table[7*n*(2*n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Dec 23 2015 *)
LinearRecurrence[{3,-3,1},{0,21,70},50] (* Harvey P. Dale, Apr 26 2017 *)

a(n)=7*n*(2*n+1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 14*n^2 + 7*n.
a(n) = 7*A014105(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) for n > 2.
G.f.: 7*x*(3+x)/(1-x)^3. (End)
a(n) = Sum_{i=6*n..8*n} i. - Wesley Ivan Hurt, Dec 23 2015
E.g.f.: 7*exp(x)*x*(3 + 2*x). - Elmo R. Oliveira, Dec 29 2024

A195030 a(n) = (n-2)(14n-39) for n > 2, otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 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

Offset: 0

Omar E. Pol, Oct 18 2011

Union of [1, 2] and A195021.

Sequence found by reading the line from 0, in the direction 0, 1,..., 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..1000
Bruno Berselli, Illustration of initial terms: An origin of A195030
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A195019, A195020, A195021, A185019, A193053, A198017.

[0,1,2] cat[(n-2)*(14*n-39): n in [3..50]]; // Vincenzo Librandi, Jul 10 2012

Join[{0,1,2},Table[(n-2)*(14*n-39),{n,3,50}]] (* Vincenzo Librandi, Jul 10 2012 *)

a(n)=if(n,([0,1,0; 0,0,1; 1,-3,3]^n*[0;1;2])[1,1],0) \\ Charles R Greathouse IV, Oct 16 2015

G.f.: x*(1-x+30*x^3-2*x^4)/(1-x)^3. - Bruno Berselli, Oct 18 2011

Both sequence (based on A195021) and definition suggested by Bruno Berselli, Oct 18 2011

cp's OEIS Frontend

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

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

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

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Magma

Mathematica

PARI

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

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Programs

Magma

Mathematica

PARI

Formula

Extensions

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

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

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

A033572 a(n) = (2n+1)(7*n+1).

A195026 a(n) = 7n(2*n + 1).

A195030 a(n) = (n-2)(14n-39) for n > 2, otherwise a(n) = n.