A283394 -id:A283394 - cp's OEIS frontend

A027468 9 times the triangular numbers A000217.

0, 9, 27, 54, 90, 135, 189, 252, 324, 405, 495, 594, 702, 819, 945, 1080, 1224, 1377, 1539, 1710, 1890, 2079, 2277, 2484, 2700, 2925, 3159, 3402, 3654, 3915, 4185, 4464, 4752, 5049, 5355, 5670, 5994, 6327, 6669, 7020, 7380, 7749, 8127, 8514, 8910, 9315

Offset: 0

Olivier Gérard

Staggered diagonal of triangular spiral in A051682, between (0,1,11) spoke and (0,8,25) spoke. - Paul Barry, Mar 15 2003
Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-2 fixed points. - Zerinvary Lajos, Oct 15 2006
Number of n permutations (n>=2) of 4 objects u, v, z, x with repetition allowed, containing n-2=0 u's. Example: if n=2 then n-2 =zero (0) u, a(1)=9 because we have vv, zz, xx, vx, xv, zx, xz, vz, zv. A027465 formatted as a triangular array: diagonal: 9, 27, 54, 90, 135, 189, 252, 324, ... . - Zerinvary Lajos, Aug 06 2008
a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is a multiple of 3. - Augustine O. Munagi, Dec 18 2008
Also sequence found by reading the line from 0, in the direction 0, 9, ..., and the same line from 0, in the direction 0, 27, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Axis perpendicular to A195147 in the same spiral. - Omar E. Pol, Sep 18 2011
Sum of the numbers from 4*n to 5*n. - Wesley Ivan Hurt, Nov 01 2014

			The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4 are 3+3+3, 6+6+6+3+3+3, 9+9+9+6+6+6+3+3+3, 12+12+12+9+9+9+6+6+6+3+3+3. - _Augustine O. Munagi_, Dec 18 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Augustine O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., Vol. 308, No. 12 (2008), pp. 2492-2501.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Leo Tavares, Illustration: Centroid Triangles.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, Vol. 7, No. 1 (2007), pp. 135-162.
D. Zvonkine, Home Page.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences.

Third diagonal of A027465.
Cf. A000459, A002378, A008585, A024966, A028895, A028896, A038764, A033996, A045943, A046092, A049598, A059073, A080855, A134171, A283394.
Cf. A060544.

[9*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, Dec 29 2012

[seq(9*binomial(n+1,2), n=0..50)]; # Zerinvary Lajos, Nov 24 2006

Table[(9/2)*n*(n+1), {n,0,50}] (* G. C. Greubel, Aug 22 2017 *)

PARI
```
a(n)=9*n*(n+1)/2
```

[9*binomial(n+1, 2) for n in (0..50)] # G. C. Greubel, May 20 2021

Numerators of sequence a[n, n-2] in (a[i, j])^2 where a[i, j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.
a(n) = (9/2)*n*(n+1).
a(n) = 9*C(n, 1) + 9*C(n, 2) (binomial transform of (0, 9, 9, 0, 0, ...)). - Paul Barry, Mar 15 2003
G.f.: 9*x/(1-x)^3.
a(-1-n) = a(n).
a(n) = 9*C(n+1,2), n>=0. - Zerinvary Lajos, Aug 06 2008
a(n) = a(n-1) + 9*n (with a(0)=0). - Vincenzo Librandi, Nov 19 2010
a(n) = A060544(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A218470(9*n+8). - Philippe Deléham, Mar 27 2013
E.g.f.: (9/2)*x*(x+2)*exp(x). - G. C. Greubel, Aug 22 2017
a(n) = A060544(n+1) - 1. See Centroid Triangles illustration. - Leo Tavares, Dec 27 2021
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 2/9. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(9/(2*Pi))*cos(sqrt(17)*Pi/6).
Product_{n>=1} (1 + 1/a(n)) = 9*sqrt(3)/(4*Pi). (End)

More terms from Patrick De Geest, Oct 15 1999

A038764 a(n) = (9n^2 + 3n + 2)/2.

Original entry on oeis.org

1, 7, 22, 46, 79, 121, 172, 232, 301, 379, 466, 562, 667, 781, 904, 1036, 1177, 1327, 1486, 1654, 1831, 2017, 2212, 2416, 2629, 2851, 3082, 3322, 3571, 3829, 4096, 4372, 4657, 4951, 5254, 5566, 5887, 6217, 6556, 6904, 7261, 7627, 8002, 8386, 8779, 9181

Offset: 0

N. J. A. Sloane, May 03 2000

Coefficients of x^2 of certain rook polynomials (for n>=1; see p. 18 of the Riordan paper). - Emeric Deutsch, Mar 08 2004

a(n) is also the least weight of self-conjugate partitions having n+1 different parts such that each part is congruent to 1 modulo 3. The first such self-conjugate partitions, corresponding to a(n) = 0, 1, 2, 3, are 1, 4+3, 7+4+4+4+3, 10+7+7+7+4+4+4+3. - Augustine O. Munagi, Dec 18 2008

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.

Colin Barker, Table of n, a(n) for n = 0..1000
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Reflection of A060544 in A081272.
Second column of A024462. Also = A064641(n+1, 2).
Shallow diagonal of triangular spiral in A051682.
Cf. A027468, A080855, A283394.
Partial sums of A122709.

LinearRecurrence[{3, -3, 1}, {1, 7, 22}, 50] (* Paolo Xausa, Jul 03 2025 *)

a(n)=n*(9*n+3)/2+1 \\ Charles R Greathouse IV, Jun 17 2017

Vec((1 + 2*x)^2 / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jan 22 2018

a = lambda n: hypergeometric([-n, -2], [1], 3)
print([simplify(a(n)) for n in range(46)]) # Peter Luschny, Nov 19 2014

a(n) = binomial(n,0) + 6*binomial(n,1) + 9*binomial(n,2).
From Paul Barry, Mar 15 2003: (Start)
G.f.: (1 + 2*x)^2/(1 - x)^3.
Binomial transform of (1, 6, 9, 0, 0, 0, ...). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. -  Colin Barker, Jan 22 2018
a(n) = a(n-1) + 3*(3*n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = hypergeometric([-n, -2], [1], 3). - Peter Luschny, Nov 19 2014
E.g.f.: exp(x)*(2 + 12*x + 9*x^2)/2. - Stefano Spezia, Mar 07 2023

More terms from James Sellers, May 03 2000

Entry revised by N. J. A. Sloane, Jan 23 2018

A080855 a(n) = (9n^2 - 3n + 2)/2.

Original entry on oeis.org

1, 4, 16, 37, 67, 106, 154, 211, 277, 352, 436, 529, 631, 742, 862, 991, 1129, 1276, 1432, 1597, 1771, 1954, 2146, 2347, 2557, 2776, 3004, 3241, 3487, 3742, 4006, 4279, 4561, 4852, 5152, 5461, 5779, 6106, 6442, 6787, 7141, 7504, 7876, 8257, 8647, 9046

Offset: 0

Paul Barry, Feb 23 2003

The old definition of this sequence was "Generalized polygonal numbers".
Row T(3,n) of A080853.
Equals binomial transform of [1, 3, 9, 0, 0, 0, ...] - Gary W. Adamson, Apr 30 2008
a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is congruent to 2 modulo 3. The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4, are 2+2, 5+5+2+2+2, 8+8+5+5+5+2+2+2, 11+11+8+8+8+5+5+5+2+2+2. - Augustine O. Munagi, Dec 18 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 3, a(n-1) = -coeff(charpoly(A,x), x^(n-2)). - Milan Janjic, Jan 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Leo Tavares, Illustration: Hexagonal Tri-Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A027468, A038764.
Cf. A283394 (see Crossrefs section).
Cf. A003215, A000217, A227776.

List([0..50],n->(9*n^2-3*n+2)/2); # Muniru A Asiru, Nov 02 2018

[(9*n^2 - 3*n +2)/2: n in [0..50]]; // G. C. Greubel, Nov 02 2018

seq((9*n^2-3*n+2)/2,n=0..50); # Muniru A Asiru, Nov 02 2018

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 500, 9}]; lst (* Zerinvary Lajos, Jul 11 2009 *)
Table[(9n^2-3n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1}, {1,4,16}, 50] (* Harvey P. Dale, Jul 24 2013 *)

a(n)=binomial(3*n,2)+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1 + x + 7*x^2)/(1 - x)^3.
a(n) = 9*n + a(n-1) - 6 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = n*A005448(n+1) - (n-1)*A005448(n), with A005448(0)=1. - Bruno Berselli, Jan 15 2013
a(0)=1, a(1)=4, a(2)=16; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 24 2013
a(n) = A152947(3*n+1). - Franck Maminirina Ramaharo, Jan 10 2018
E.g.f.: (2 + 6*x + 9*x^2)*exp(x)/2. - G. C. Greubel, Nov 02 2018
From Leo Tavares, Feb 20 2022: (Start)
a(n) = A003215(n-1) + 3*A000217(n). See Hexagonal Tri-Rays illustration in links.
a(n) = A227776(n) - 3*A000217(n). (End)

Definition replaced with the closed form by Bruno Berselli, Jan 15 2013

A081271 Vertical of triangular spiral in A051682.

Original entry on oeis.org

1, 13, 34, 64, 103, 151, 208, 274, 349, 433, 526, 628, 739, 859, 988, 1126, 1273, 1429, 1594, 1768, 1951, 2143, 2344, 2554, 2773, 3001, 3238, 3484, 3739, 4003, 4276, 4558, 4849, 5149, 5458, 5776, 6103, 6439, 6784, 7138, 7501, 7873, 8254, 8644, 9043, 9451

Offset: 0

Paul Barry, Mar 15 2003

Lies to the right of the y-axis of the triangle.

Binomial transform of (1, 12, 9, 0, 0, 0, ...).

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

Cf. A062741, A283394 (see Crossrefs section).

LinearRecurrence[{3,-3,1},{1,13,34},50] (* Harvey P. Dale, Aug 30 2025 *)

a(n)=(9*n^2+15*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (1 + 10*x - 2*x^2)/(1 - x)^3.
a(n) = binomial(n,0) + 12*binomial(n,1) + 9*binomial(n,2).
a(n) = (9*n^2 + 15*n + 2)/2.
a(0) = 1, a(n) = a(n-1) + 9*n + 3 for n > 0 - Gerald McGarvey, Aug 18 2004
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(1 + 12*x + 9*x^2/2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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A027468 9 times the triangular numbers A000217.

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A038764 a(n) = (9*n^2 + 3*n + 2)/2.

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A080855 a(n) = (9*n^2 - 3*n + 2)/2.

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A081271 Vertical of triangular spiral in A051682.

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Mathematica

PARI

Formula

A038764 a(n) = (9n^2 + 3n + 2)/2.

A080855 a(n) = (9n^2 - 3n + 2)/2.