A158591 -id:A158591 - cp's OEIS frontend

A247792 a(n) = 9*n^2 + 1.

1, 10, 37, 82, 145, 226, 325, 442, 577, 730, 901, 1090, 1297, 1522, 1765, 2026, 2305, 2602, 2917, 3250, 3601, 3970, 4357, 4762, 5185, 5626, 6085, 6562, 7057, 7570, 8101, 8650, 9217, 9802, 10405, 11026, 11665, 12322, 12997, 13690, 14401, 15130, 15877, 16642, 17425, 18226, 19045, 19882

Offset: 0

Karl V. Keller, Jr., Sep 23 2014

The odd numbers of the form 9n^2 + 1 are listed in A158591 (36n^2 + 1).
The even numbers of the form 9n^2 + 1 are given by 36x^2 - 36x + 10, x > 0.
Every integer n>0 give three perfect squares and consecutives from 2^2. The formulas for each value of n are: a(n)-6n, a(n)-1 and a(n)+6n. - Miquel Cerda, Sep 19 2016
These squares are, for n>0, A000290(3*n-1), 3*n and (3n+1) and the sum of them is 3*a(n) - 1. - Miquel Cerda, Sep 26 2016

			a(1) = (2^2 + 4^2)/2 = 3^2 + 1 = 10, a(2) = (5^2 + 7^2)/2 = 6^2 + 1 = 37, a(3) = (8^2 + 10^2)/2 = 9^2 + 1 = 82. - _Miquel Cerda_, Jun 25 2016

Karl V. Keller, Jr., Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016766, A158591 (36n^2 + 1), A156226 (primes of the form 9n^2 + 1).

Cf. also A000290.

[9*n^2+1: n in [0..60]]; // Vincenzo Librandi, Sep 27 2014

A247792:=n->9*n^2 + 1: seq(A247792(n), n=0..80); # Wesley Ivan Hurt, Jun 25 2016

(3Range[0, 49])^2 + 1 (* Alonso del Arte, Sep 24 2014 *)
CoefficientList[Series[(1 + 7 x + 10 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 27 2014 *)

a(n)=9*n^2+1 \\ Charles R Greathouse IV, Sep 26 2014

for n in range (0,100): print (9*n**2+1)

a(n) = (3n)^2 + 1 = 9n^2 + 1 = A016766(n) + 1.
G.f.: (1+7*x+10*x^2)/(1-x)^3. - Vincenzo Librandi, Sep 27 2014
a(n) = ((3n-1)^2 + (3n+1)^2)/2 = (A016790(n-1) + A016778(n))/2. - Miquel Cerda, Jun 25 2016
From Ilya Gutkovskiy, Jun 25 2016: (Start)
E.g.f.: (1 + 9*x + 9*x^2)*exp(x).
Dirichlet g.f.: 9*zeta(s-2) + zeta(s).
Sum_{n>=0} 1/a(n) = (3 + Pi*coth(Pi/3))/6. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Wesley Ivan Hurt, Jun 25 2016
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/3)*csch(Pi/3))/2. - Amiram Eldar, Jul 15 2020
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/3)*sinh(sqrt(2)*Pi/3).
Product_{n>=1} (1 - 1/a(n)) = (Pi/3)*csch(Pi/3). (End)

A158590 a(n) = 324*n^2 + 18.

Original entry on oeis.org

18, 342, 1314, 2934, 5202, 8118, 11682, 15894, 20754, 26262, 32418, 39222, 46674, 54774, 63522, 72918, 82962, 93654, 104994, 116982, 129618, 142902, 156834, 171414, 186642, 202518, 219042, 236214, 254034, 272502, 291618, 311382, 331794, 352854, 374562, 396918

Offset: 0

Vincenzo Librandi, Mar 22 2009

The identity (36*n^2 + 1)^2 - (324*n^2 + 18)*(2*n)^2 = 1 can be written as A158591(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158591.

I:=[18, 342, 1314]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 16 2012

LinearRecurrence[{3, -3, 1}, {18, 342, 1314}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
324 Range[0,40]^2+18 (* Harvey P. Dale, Nov 22 2018 *)

for(n=0, 40, print1(324*n^2 + 18", ")); \\ Vincenzo Librandi, Feb 16 2012

G.f.: -18*(1 + 16*x + 19*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 14 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) + 1)/36.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) + 1)/36. (End)
E.g.f.: 18*exp(x)*(1 + 18*x + 18*x^2). - Elmo R. Oliveira, Jan 15 2025

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

A158739 a(n) = 1296*n^2 + 36.

Original entry on oeis.org

36, 1332, 5220, 11700, 20772, 32436, 46692, 63540, 82980, 105012, 129636, 156852, 186660, 219060, 254052, 291636, 331812, 374580, 419940, 467892, 518436, 571572, 627300, 685620, 746532, 810036, 876132, 944820, 1016100, 1089972, 1166436, 1245492, 1327140, 1411380

Offset: 0

Vincenzo Librandi, Mar 25 2009

The identity (72*n^2 + 1)^2 - (1296*n^2 + 36)*(2*n)^2 = 1 can be written as A158740(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158591, A158740.

I:=[36, 1332, 5220]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 21 2012

A158739:=n->1296*n^2+36: seq(A158739(n), n=0..40); # Wesley Ivan Hurt, Nov 20 2014

LinearRecurrence[{3, -3, 1}, {36, 1332, 5220}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

for(n=0, 40, print1(1296*n^2 + 36", ")); \\ Vincenzo Librandi, Feb 21 2012

G.f.: -36*(1+34*x+37*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 22 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/6)*Pi/6 + 1)/72.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/6)*Pi/6 + 1)/72. (End)
From Elmo R. Oliveira, Jan 26 2025: (Start)
E.g.f.: 36*exp(x)*(1 + 36*x + 36*x^2).
a(n) = 36*A158591(n). (End)

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

A193252 Great rhombicuboctahedron with faces of centered polygons.

Original entry on oeis.org

1, 75, 365, 1015, 2169, 3971, 6565, 10095, 14705, 20539, 27741, 36455, 46825, 58995, 73109, 89311, 107745, 128555, 151885, 177879, 206681, 238435, 273285, 311375, 352849, 397851, 446525, 499015, 555465, 616019, 680821, 750015, 823745, 902155, 985389, 1073591

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a great rhombicosidodecahedron. Each iteration requires the addition of (n-2) edge units and (n-1) vertices to complete the centered polygon of each face: centered squares, centered octagons and centered hexagons.

Bruno Berselli, Table of n, a(n) for n = 1..1000
OEIS Wiki, (Centered polygons) pyramidal numbers
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences in 2*A158591.

Cf. A001844 (centered square numbers), A016754 (centered octagonal numbers), A003215 (centered hexagonal numbers).

Excel
```
=24*ROW()^3-36*ROW()^2+14*ROW()-1
```

List([1..40], n-> 24*n^3 -36*n^2 +14*n -1); # G. C. Greubel, Feb 26 2019

A069190:=func; [(2*n-1)*A069190(n): n in [1..40]];  // Bruno Berselli, Jul 21 2011

Table[24n^3-36n^2+14n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,75,365,1015},40] (* Harvey P. Dale, Jul 27 2011 *)

for(n=1,40, print1(24*n^3-36*n^2+14*n-1", "));  \\ Bruno Berselli, Jul 21 2011

[24*n^3 -36*n^2 +14*n -1 for n in (1..40)] # G. C. Greubel, Feb 26 2019

a(n) = 24*n^3 - 36*n^2 + 14*n - 1.
G.f.: x*(1+x)*(1+70*x+x^2)/(1-x)^4; a(n) = (2*n-1)*A069190(n). - Bruno Berselli, Jul 21 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=75, a(2)=365, a(3)=1015. - Harvey P. Dale, Jul 27 2011
a(n) = 72 * A000330(n-1) + A005408(n-1). - Bruce J. Nicholson, Feb 23 2019
E.g.f.: 1 + (-1 + 2*x + 36*x^2 + 24*x^3)*exp(x). - G. C. Greubel, Feb 26 2019

cp's OEIS Frontend

A247792 a(n) = 9*n^2 + 1.

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A158590 a(n) = 324*n^2 + 18.

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A158739 a(n) = 1296*n^2 + 36.

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A193252 Great rhombicuboctahedron with faces of centered polygons.

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