A152744 -id:A152744 - cp's OEIS frontend

A152743 6 times pentagonal numbers: a(n) = 3n(3*n-1).

0, 6, 30, 72, 132, 210, 306, 420, 552, 702, 870, 1056, 1260, 1482, 1722, 1980, 2256, 2550, 2862, 3192, 3540, 3906, 4290, 4692, 5112, 5550, 6006, 6480, 6972, 7482, 8010, 8556, 9120, 9702, 10302, 10920, 11556, 12210, 12882, 13572, 14280, 15006, 15750, 16512, 17292

Offset: 0

Omar E. Pol, Dec 12 2008

a(n) is also the Wiener index of the windmill graph D(4,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e. a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(3,n), D(5,n), and D(6,n) see A033991, A028994, and A180577, respectively. - Emeric Deutsch, Sep 21 2010

a(n+1) gives the number of edges in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter consists of 18 external edges. There are 6 edges shared by the perimeter hexagons. The core hexagon has 6 edges. a(2) is the total number of edges, i.e. 18 + 6 + 6 = 30. - Ivan N. Ianakiev, Mar 10 2015

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ivan N. Ianakiev, Hexagon-like honeycomb built from regular congruent hexagons.
Eric Weisstein's World of Mathematics, Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A152734, A152744, A049450, A062741, A033991, A028994, A180577.

[ 3*n*(3*n-1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

A152743:=n->3*n*(3*n-1); seq(A152743(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Table[3n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,6,30},40] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[-6x (2x+1)/(x-1)^3,{x,0,40}],x] (* Robert G. Wilson v, Mar 10 2015 *)

a(n)=3*n*(3*n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9n^2 - 3n = A000326(n)*6.
a(n) = A049450(n)*3 = A062741(n)*2. - Omar E. Pol, Dec 15 2008
a(n) = a(n-1) + 18*n - 12 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: -((6*x*(2*x+1))/(x-1)^3). - Harvey P. Dale, Jun 30 2011
E.g.f.: 3*x*(2+3*x)*exp(x). - G. C. Greubel, Sep 01 2018
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi*sqrt(3) - 6*log(2))/9. (End)

Converted reference to link by Omar E. Pol, Oct 07 2010

A316466 a(n) = 2n(7*n - 3).

Original entry on oeis.org

0, 8, 44, 108, 200, 320, 468, 644, 848, 1080, 1340, 1628, 1944, 2288, 2660, 3060, 3488, 3944, 4428, 4940, 5480, 6048, 6644, 7268, 7920, 8600, 9308, 10044, 10808, 11600, 12420, 13268, 14144, 15048, 15980, 16940, 17928, 18944, 19988, 21060, 22160, 23288, 24444, 25628, 26840

Offset: 0

Bruno Berselli, Jul 04 2018

This is the case k = 9 of Sum_{i = 2..k} P(i,n) = (k - 1)*n*((k - 2)*n - (k - 6))/4, where P(k,n) = n*((k - 2)*n - (k - 4))/2 (see Crossrefs for similar sequences and "Square array in A139600" in Links section).

14*x + 9 is a square for x = a(n) or x = a(-n).

Colin Barker, Table of n, a(n) for n = 0..1000
Bruno Berselli, Square array in A139600.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A139600, A218471.

Similar sequences (see the first comment): A000096 (k = 3), A045943 (k = 4), A049451 (k = 5), A033429 (k = 6), A167469 (k = 7), A152744 (k = 8), this sequence (k = 9), A152994 (k = 10).

GAP
```
List([0..50], n -> 2*n*(7*n-3));
```
Julia
```
[2*n*(7*n-3) for n in 0:50] |> println
```
Magma
```
[2*n*(7*n-3): n in [0..50]];
```

Table[2 n (7 n - 3), {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,8,44},50] (* Harvey P. Dale, Jan 24 2021 *)

Maxima
```
makelist(2*n*(7*n-3), n, 0, 50);
```
PARI
```
vector(50, n, n--; 2*n*(7*n-3))
```

concat(0, Vec(4*x*(2 + 5*x)/(1 - x)^3 + O(x^40))) \\ Colin Barker, Jul 05 2018

Python
```
[2*n*(7*n-3) for n in range(50)]
```
Sage
```
[2*n*(7*n-3) for n in (0..50)]
```

O.g.f.: 4*x*(2 + 5*x)/(1 - x)^3.
E.g.f.: 2*x*(4 + 7*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 4*A218471(n).

A139607 a(n) = 21*n + 7.

Original entry on oeis.org

7, 28, 49, 70, 91, 112, 133, 154, 175, 196, 217, 238, 259, 280, 301, 322, 343, 364, 385, 406, 427, 448, 469, 490, 511, 532, 553, 574, 595, 616, 637, 658, 679, 700, 721, 742, 763, 784, 805, 826, 847, 868, 889, 910, 931, 952, 973, 994

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 7th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

7th transversal numbers (or 7-transversal numbers): (A000217(7)-7)*n + 7.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000217, A008603, A016777, A057145, A139600, A152744.

[21*n+7: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[7, 1500, 21] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
21*Range[0,50]+7 (* or *) LinearRecurrence[{2,-1},{7,28},50] (* Harvey P. Dale, Feb 23 2020 *)

a(n)=21*n+7 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,7).
G.f.: 7*(1+2*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 12 2024: (Start)
E.g.f.: 7*exp(x)*(1 + 3*x).
a(n) = 7*A016777(n) = A008603(n) + 7 = A152744(n+1) - A152744(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A154894 Fibonacci numbers whose number of proper divisors is prime.

Original entry on oeis.org

8, 21, 34, 55, 377, 610, 987, 4181, 10946, 17711, 46368, 75025, 121393, 1346269, 3524578, 5702887, 9227465, 24157817, 39088169, 63245986, 165580141, 701408733, 1134903170, 1836311903, 7778742049, 12586269025, 20365011074, 53316291173

Offset: 1

Omar E. Pol, Jan 25 2009

nonn

			8 is member because the number of proper divisors of 8 is 3, a prime number.

Amiram Eldar, Table of n, a(n) for n = 1..494

Cf. A000045, A152744.

Fibonacci[Select[Range[60], PrimeQ[DivisorSigma[0, Fibonacci[ # ]] - 1] &]] (* Stefan Steinerberger, Jan 31 2009 *)

More terms from Stefan Steinerberger, Jan 31 2009

cp's OEIS Frontend

A152743 6 times pentagonal numbers: a(n) = 3*n*(3*n-1).

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A316466 a(n) = 2*n*(7*n - 3).

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A139607 a(n) = 21*n + 7.

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Magma

Mathematica

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A154894 Fibonacci numbers whose number of proper divisors is prime.

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Programs

Mathematica

Extensions

A152743 6 times pentagonal numbers: a(n) = 3n(3*n-1).

A316466 a(n) = 2n(7*n - 3).