A195317 -id:A195317 - cp's OEIS frontend

A351354 Numbers k such that the k-th centered 40-gonal numbers (A195317) is a square.

1, 3, 7, 45, 117, 799, 2091, 14329, 37513, 257115, 673135, 4613733, 12078909, 82790071, 216747219, 1485607537, 3889371025, 26658145587, 69791931223, 478361013021, 1252365390981, 8583840088783, 22472785106427, 154030760585065, 403257766524697, 2763969850442379

Offset: 1

Lamine Ngom, Feb 08 2022

nonn

Corresponding square roots are listed in A351353.
3 and 7 are the unique primes in this sequence, a(2*n+1) and a(2*n) always sharing common factors that are closely linked to Fibonacci (A000045) and Lucas (A000032) numbers (detailed in formula section).
In addition, the ratio a(2*n+1)/a(2*n) converges to 2.618033988 ... = golden ratio squared: A104457.

			45 is in the sequence because the 45th centered 40-gonal number is 39601, which is a square: 199^2 = A000032(11)^2.
799 is in the sequence because the 799th centered 40-gonal number is 12752041, which is a square: 3571^2 = A000032(17)^2.

Index entries for linear recurrences with constant coefficients, signature (1,18,-18,-1,1).

Cf. A000032, A000045, A007310, A077259, A104457, A195317, A351353.

a[1] := 1: a[2] := 3: a[3] := 7: a[4] := 45: a[5] := 117:
for n from 6 to 30 do a[n] := a[n - 1] + 18*a[n - 2] - 18*a[n - 3] - a[n - 4] + a[n - 5]: od:
seq(a[n], n = 1 .. 30);

LinearRecurrence[{1, 18, -18, -1, 1}, {1, 3, 7, 45, 117}, 30] (* Amiram Eldar, Feb 08 2022 *)

a(n) = A077259(n-1) + 1.
a(1)=1, a(2)=3, a(3)=7, a(4)=45, a(5)=117 and a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5).
gcd(a(2*n+1), a(2*n)) = A000045(n)*(A000032(2*n) - 1)/2, if n is odd.
gcd(a(2*n+1), a(2*n)) = A000032(n)*(A000032(2*n) - 1)/2, if n is even.
A195317(a(n)) = A000032(A007310(n))^2 = A351353(n)^2.

A195148 Concentric 20-gonal numbers.

Original entry on oeis.org

0, 1, 20, 41, 80, 121, 180, 241, 320, 401, 500, 601, 720, 841, 980, 1121, 1280, 1441, 1620, 1801, 2000, 2201, 2420, 2641, 2880, 3121, 3380, 3641, 3920, 4201, 4500, 4801, 5120, 5441, 5780, 6121, 6480, 6841, 7220, 7601, 8000, 8401, 8820, 9241, 9680, 10121

Offset: 0

Omar E. Pol, Sep 17 2011

Concentric icosagonal numbers.

Sequence found by reading the line from 0, in the direction 0, 20, ..., and the same line from 1, in the direction 1, 41, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Main axis, perpendicular to A124080 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

A195322 and A195317 interleaved.
Cf. A032528, A077221, A195142, A195143, A195145, A195146, A195147, A195149.
Cf. A032527, A195048, A195049. Column 20 of A195040. - Omar E. Pol, Sep 29 2011

[5*n^2+2*(-1)^n-2: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

LinearRecurrence[{2,0,-2,1},{0,1,20,41},50] (* Harvey P. Dale, Apr 08 2016 *)

a(n)=5*n^2+2*(-1)^n-2 \\ Charles R Greathouse IV, Sep 28 2015

From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = 5*n^2 + 2*(-1)^n-2;
a(n) = -a(n-1) + 10*n^2 - 10*n + 1. (End)
G.f.: x*(1+18*x+x^2)/((1+x)*(1-x)^3). - Bruno Berselli, Sep 27 2011
Sum_{n>=1} 1/a(n) = Pi^2/120 + tan(Pi/sqrt(5))*Pi/(8*sqrt(5)). - Amiram Eldar, Jan 17 2023

A195322 a(n) = 20*n^2.

Original entry on oeis.org

0, 20, 80, 180, 320, 500, 720, 980, 1280, 1620, 2000, 2420, 2880, 3380, 3920, 4500, 5120, 5780, 6480, 7220, 8000, 8820, 9680, 10580, 11520, 12500, 13520, 14580, 15680, 16820, 18000, 19220, 20480, 21780, 23120, 24500, 25920, 27380, 28880, 30420, 32000, 33620, 35280

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 20, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semiaxis opposite to A195317 in the same spiral.
a(n) is the sum of all the integers less than 10*n which are not multiple of 2 or 5. a(2) = (1 + 3 + 7 + 9) + (11 + 13 + 17 + 19) = 20 + 60 = 80 = 20 * 2^2. (Link Crux Mathematicorum). - Bernard Schott, May 15 2017
Number of terms less than 10^k (k=0, 1, 2, ...): 1, 1, 3, 8, 23, 71, 224, 708, 2237, 7072, 22361, 70711, ... - Muniru A Asiru, Feb 01 2018

			From _Muniru A Asiru_, Feb 01 2018: (Start)
n=0, a(0) = 20*0^2 = 0.
n=1, a(1) = 20*1^2 = 20.
n=1, a(2) = 20*2^2 = 80.
n=1, a(3) = 20*3^2 = 180.
n=1, a(4) = 20*4^2 = 320.
...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Léo Sauvé, Problem 53, Crux Mathematicorum, Vol. 1, Nov. 1975, page 88.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195148.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195323.
Cf. A000290, A001105, A005899, A010014, A016742, A033429, A195162, A195317.
Cf. A008602.

List([0..10^3],n->20*n^2); # Muniru A Asiru, Feb 01 2018

[20*n^2: n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

a := n -> 20*n^2; seq(a(n), n=0..10^3); # Muniru A Asiru, Feb 01 2018

20 Range[0, 40]^2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 20, 80}, 50] (* Harvey P. Dale, Jan 18 2013 *)

a(n) = 20*n^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 20*A000290(n) = 10*A001105(n) = 5*A016742(n) = 4*A033429(n) = 2*A033583(n).
a(0)=0, a(1)=20, a(2)=80; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 18 2013
a(n) = A010014(n) - A005899(n) for n > 0. - R. J. Cano, Sep 29 2015
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 20*x*(1 + x)/(1-x)^3.
E.g.f.: 20*x*(1 + x)*exp(x).
a(n) = n*A008602(n) = A195148(2*n). (End)

A195314 Centered 28-gonal numbers.

Original entry on oeis.org

1, 29, 85, 169, 281, 421, 589, 785, 1009, 1261, 1541, 1849, 2185, 2549, 2941, 3361, 3809, 4285, 4789, 5321, 5881, 6469, 7085, 7729, 8401, 9101, 9829, 10585, 11369, 12181, 13021, 13889, 14785, 15709, 16661, 17641, 18649, 19685, 20749, 21841, 22961, 24109, 25285, 26489

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 29, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Semi-axis opposite to A144555 in the same spiral.

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

Bisection of A195145.
Cf. A003154, A069129, A069133, A069190, A195315, A195316, A195317, A195318.
Cf. A069127, A118277, A144555.

[(14*n^2-14*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[14n^2-14n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,29,85},50]

a(n)=14*n^2-14*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - 14*n + 1.
G.f.: -x*(1 + 26*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Oct 01 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5/7)*Pi/2)/(2*sqrt(35)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(14*x^2 + 1) - 1.
a(n) = 2*A069127(n) - 1. (End)

A195315 Centered 32-gonal numbers.

Original entry on oeis.org

1, 33, 97, 193, 321, 481, 673, 897, 1153, 1441, 1761, 2113, 2497, 2913, 3361, 3841, 4353, 4897, 5473, 6081, 6721, 7393, 8097, 8833, 9601, 10401, 11233, 12097, 12993, 13921, 14881, 15873, 16897, 17953, 19041, 20161, 21313, 22497, 23713, 24961, 26241, 27553, 28897, 30273

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Semi-axis opposite to A016802 in the same spiral.

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

Bisection of A195146.
Cf. A003154, A069129, A069133, A069190, A195314, A195316, A195317, A195318.
Cf. A016802, A074377.

[(16*n^2-16*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[16*n^2 - 16*n + 1, {n, 1, 41}] (* Amiram Eldar, Feb 11 2022 *)
LinearRecurrence[{3,-3,1},{1,33,97},50] (* Harvey P. Dale, Feb 11 2024 *)

a(n)=16*n^2-16*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 16*n^2 - 16*n + 1.
G.f.: -x*(1 + 30*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3)*Pi/4)/(8*sqrt(3)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(16*x^2 + 1) - 1.
a(n) = 2*A069129(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A195316 Centered 36-gonal numbers.

Original entry on oeis.org

1, 37, 109, 217, 361, 541, 757, 1009, 1297, 1621, 1981, 2377, 2809, 3277, 3781, 4321, 4897, 5509, 6157, 6841, 7561, 8317, 9109, 9937, 10801, 11701, 12637, 13609, 14617, 15661, 16741, 17857, 19009, 20197, 21421, 22681, 23977, 25309, 26677, 28081, 29521, 30997, 32509

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 37, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195321 in the same spiral.

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

Bisection of A195147.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195317, A195318.
Cf. A069131, A195160, A195321.

[(18*n^2-18*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[18*n^2 - 18*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

a(n)=18*n^2-18*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 18*n^2 - 18*n + 1.
G.f.: -x*(1 + 34*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7)*Pi/6)/(6*sqrt(7)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(18*x^2 + 1) - 1.
a(n) = 2*A069131(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A195318 Centered 44-gonal numbers.

Original entry on oeis.org

1, 45, 133, 265, 441, 661, 925, 1233, 1585, 1981, 2421, 2905, 3433, 4005, 4621, 5281, 5985, 6733, 7525, 8361, 9241, 10165, 11133, 12145, 13201, 14301, 15445, 16633, 17865, 19141, 20461, 21825, 23233, 24685, 26181, 27721, 29305, 30933, 32605, 34321, 36081, 37885, 39733

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 45, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195323 in the same spiral.

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

Bisection of A195149.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195316, A195317.
Cf. A069173, A195313, A195323.

[22*n^2 - 22*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 21 2011

Table[22n^2-22n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,45,133},50] (* Harvey P. Dale, Mar 16 2019 *)

a(n)=22*n^2-22*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 22*n^2 - 22*n + 1.
Sum_{n>=1} 1/a(n) = Pi*tan(3*Pi/(2*sqrt(11)))/(6*sqrt(11)). - Amiram Eldar, Feb 11 2022
G.f.: -x*(1+42*x+x^2)/(x-1)^3. - R. J. Mathar, May 07 2024
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(22*x^2 + 1) - 1.
a(n) = 2*A069173(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A010022 a(0) = 1, a(n) = 40*n^2 + 2 for n>0.

Original entry on oeis.org

1, 42, 162, 362, 642, 1002, 1442, 1962, 2562, 3242, 4002, 4842, 5762, 6762, 7842, 9002, 10242, 11562, 12962, 14442, 16002, 17642, 19362, 21162, 23042, 25002, 27042, 29162, 31362, 33642, 36002, 38442, 40962, 43562, 46242, 49002, 51842, 54762, 57762, 60842

Offset: 0

N. J. A. Sloane

First bisection of A005901. - Bruno Berselli, Feb 07 2012

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

Cf. A206399.

[1] cat [40*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 40 Range[39]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {42, 162, 362}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

G.f.: (1+x)*(1+38*x+x^2)/(1-x)^3; a(n) = A008253(4n). - Bruno Berselli, Feb 07 2012
E.g.f.: (x*(x+1)*40+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(5)/40*Pi*coth(Pi*sqrt(5)/10) = 1.03983104279172.. - R. J. Mathar, May 07 2024
a(n) = 2*A158493(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A195317(n)+A195317(n+1) = 2+10*A016742(n), n>0. - R. J. Mathar, May 07 2024

A193251 Small rhombicosidodecahedron with faces of centered polygons.

Original entry on oeis.org

1, 123, 605, 1687, 3609, 6611, 10933, 16815, 24497, 34219, 46221, 60743, 78025, 98307, 121829, 148831, 179553, 214235, 253117, 296439, 344441, 397363, 455445, 518927, 588049, 663051, 744173, 831655, 925737, 1026659, 1134661, 1249983, 1372865, 1503547, 1642269

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 small rhombicosidodecahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448), centered squares (A001844) and centered pentagons (A005891)]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
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).

Cf. A001844, A005448, A005891, A060747, A195317.

[40*n^3-60*n^2+22*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

A193251[n_] := (2*n - 1)*(20*(n - 1)*n + 1); Array[A193251, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 123, 605, 1687}, 50] (* Paolo Xausa, Aug 25 2025 *)

a(n)=40*n^3-60*n^2+22*n-1 \\ Charles R Greathouse IV, Oct 19 2022

a(n) = 40*n^3 - 60*n^2 + 22*n - 1.
G.f.: x*(1+x)*(x^2 + 118*x + 1)/(x-1)^4. - R. J. Mathar, Aug 26 2011
From Elmo R. Oliveira, Aug 22 2025: (Start)
E.g.f.: 1 + exp(x)*(-1 + 2*x + 60*x^2 + 40*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
a(n) = A060747(n)*A195317(n). (End)

A201786 Primes of the form 5*k^2 - 4.

Original entry on oeis.org

41, 241, 401, 601, 1801, 3121, 4201, 4801, 5441, 6121, 6841, 9241, 13001, 15121, 17401, 19841, 21121, 22441, 23801, 26641, 29641, 32801, 45121, 47041, 51001, 57241, 63841, 75641, 78121, 91121, 96601, 99401, 102241, 108041, 114001, 117041

Offset: 1

Vincenzo Librandi, Dec 05 2011

nonn

Also, primes of the form 20*k^2 + 20*k + 1. - Jan Rider, May 22 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A090562, A195317.

[a: n in [1..400] | IsPrime(a) where a is 5*n^2-4];

Mathematica
```
Select[Table[5n^2-4,{n,1,1000}],PrimeQ]
```

lista(nn) = for (n=1, nn, if (isprime(p=5*n^2-4), print1(p, ", "));); \\ Michel Marcus, May 22 2018

cp's OEIS Frontend

A351354 Numbers k such that the k-th centered 40-gonal numbers (A195317) is a square.

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A195148 Concentric 20-gonal numbers.

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A195322 a(n) = 20*n^2.

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A195314 Centered 28-gonal numbers.

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A195315 Centered 32-gonal numbers.

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A195316 Centered 36-gonal numbers.

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A195318 Centered 44-gonal numbers.

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