A050509 -id:A050509 - cp's OEIS frontend

A081267 Diagonal of triangular spiral in A051682.

1, 9, 26, 52, 87, 131, 184, 246, 317, 397, 486, 584, 691, 807, 932, 1066, 1209, 1361, 1522, 1692, 1871, 2059, 2256, 2462, 2677, 2901, 3134, 3376, 3627, 3887, 4156, 4434, 4721, 5017, 5322, 5636, 5959, 6291, 6632, 6982, 7341, 7709, 8086, 8472, 8867, 9271

Offset: 0

Paul Barry, Mar 15 2003

Binomial transform of (1, 8, 9, 0, 0, 0, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjić, Two Enumerative Functions
Richard J. Mathar, Bivariate generating functions enumerating non-bonding dominoes on rectangular boards, arXiv:2404.18806 (2024) Table 2.
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 7.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A050509, A051682, A081268.

Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.

[(9*n^2 + 7*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 14 2014

LinearRecurrence[{3,-3,1},{1,9,26},50] (* Harvey P. Dale, Aug 13 2014 *)
CoefficientList[Series[(1 + 6 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 14 2014 *)

a(n)=(9*n^2+7*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n, 0) + 8*C(n, 1) + 9*C(n, 2).
a(n) = (9*n^2 + 7*n + 2)/2.
G.f.: (1 + 6*x + 2*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 3 1 0 / 5 3 1]. M^n * [1 1 1] = [1 3n+1 a(n)]. - Gary W. Adamson, Dec 22 2004
a(n) = 9*n + a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = (n+1)*A000326(n+1) - (n)*A000326(n). - Bruno Berselli, Dec 10 2012
a(n) = A050509(n) - A050509(n-1). - Bill McEachen, Nov 01 2020
E.g.f.: exp(x)*(2 + 16*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A100177 Structured meta-prism numbers, the n-th number from a structured n-gonal prism number sequence.

Original entry on oeis.org

1, 4, 18, 64, 175, 396, 784, 1408, 2349, 3700, 5566, 8064, 11323, 15484, 20700, 27136, 34969, 44388, 55594, 68800, 84231, 102124, 122728, 146304, 173125, 203476, 237654, 275968, 318739, 366300, 418996, 477184, 541233, 611524, 688450, 772416, 863839, 963148, 1070784, 1187200, 1312861, 1448244

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

			There are no 1- or 2-gonal prisms, so 1 and (2n) are used as the first and second terms since all the sequences begin as such.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002411, A000578, A050509, A006597, A100176, A100177 - structured prisms; A006484 for other meta structured numbers; and A100145 for more on structured numbers.

Cf. A060354, A000124.

[(1/6)*(3*n^4-9*n^3+12*n^2): n in [1..50] ]; // Vincenzo Librandi, Aug 02 2011

Table[(3n^4-9n^3+12n^2)/6,{n,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,18,64,175},50] (* Harvey P. Dale, Nov 07 2017 *)

PARI
```
a(n)=(1/6)*(3*n^4-9*n^3+12*n^2);
```

a(n) = (1/6)*(3*n^4 - 9*n^3 + 12*n^2).
G.f.: x*(1 - x + 8*x^2 + 4*x^3)/(1-x)^5. - Colin Barker, Jun 08 2012
a(n) = A060354(n) * n = A000124(n-2) * n^2. - Bruce J. Nicholson, Jul 11 2018

A262000 a(n) = n^2(7n - 5)/2.

Original entry on oeis.org

0, 1, 18, 72, 184, 375, 666, 1078, 1632, 2349, 3250, 4356, 5688, 7267, 9114, 11250, 13696, 16473, 19602, 23104, 27000, 31311, 36058, 41262, 46944, 53125, 59826, 67068, 74872, 83259, 92250, 101866, 112128, 123057, 134674, 147000, 160056, 173863, 188442, 203814, 220000

Offset: 0

Bruno Berselli, Sep 08 2015

Also, structured enneagonal prism numbers.

			For n=8, a(8) = 8*(7*0+1)+8*(7*1+1)+8*(7*2+1)+8*(7*3+1)+8*(7*4+1)+8*(7*5+1)+8*(7*6+1)+8*(7*7+1) = 1632.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences with the formula n^2*(k*n - k + 2)/2: A000290 (k=0), A002411 (k=1), A000578 (k=2), A050509 (k=3), A015237 (k=4), A006597 (k=5), A100176 (k=6), this sequence (k=7), A103532 (k=8).

Cf. A001106, A069125.

Magma
```
[n^2*(7*n-5)/2: n in [0..40]];
```

Table[n^2 (7 n - 5)/2, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{0,1,18,72},50] (* Harvey P. Dale, Oct 04 2016 *)

PARI
```
vector(40, n, n--; n^2*(7*n-5)/2)
```
Sage
```
[n^2*(7*n-5)/2 for n in (0..40)]
```

G.f.: x*(1 + 14*x + 6*x^2)/(1 - x)^4.
a(n) = Sum_{i=0..n-1} n*(7*i+1) for n>0, a(0)=0.
a(n+1) + a(-n) = A069125(n+1).
Sum_{i>0} 1/a(i) = 1.082675669875907610300284768825... = (42*(log(14) + 2*(cos(Pi/7)*log(cos(3*Pi/14)) + log(sin(Pi/7))*sin(Pi/14) - log(cos(Pi/14)) * sin(3*Pi/14))) + 21*Pi*tan(3*Pi/14))/75 - Pi^2/15. - Vaclav Kotesovec, Oct 04 2016
From Elmo R. Oliveira, Aug 06 2025: (Start)
E.g.f.: exp(x)*x*(2 + 16*x + 7*x^2)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A192491 Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

Original entry on oeis.org

24, 240, 864, 2112, 4200, 7344, 11760, 17664, 25272, 34800, 46464, 60480, 77064, 96432, 118800, 144384, 173400, 206064, 242592, 283200, 328104, 377520, 431664, 490752, 555000, 624624, 699840, 780864, 867912, 961200

Offset: 1

Eric W. Weisstein, Jul 10 2011

Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000326, A000330, A000292, A017233, A008594, A140676, A046092.

a(n) = 12*n^2*(3*n-1).
a(n) = 24*A050509(n).
G.f.: 24*x*(2*x^2+6*x+1)/(x-1)^4. [Colin Barker, Nov 04 2012]
From Bruce J. Nicholson, Sep 18 2019: (Start)
a(n) = 24*n * A000326(n).
a(n) = 4*n^2 * A017233(n).
a(n) = 24*(n^3 + A000292(n-2) + A000330(n-2)).
a(n) = 24*(n^4 - (A008585(n) * A000330(n-1))).
a(n) = 6*A046092(n) + (A008594(n+1) * A140676(n-1)). (End)

cp's OEIS Frontend

A081267 Diagonal of triangular spiral in A051682.

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A100177 Structured meta-prism numbers, the n-th number from a structured n-gonal prism number sequence.

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Magma

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PARI

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A262000 a(n) = n^2*(7*n - 5)/2.

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Examples

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Magma

Mathematica

PARI

Sage

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A192491 Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

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Author

Keywords

Links

Crossrefs

Formula

A262000 a(n) = n^2(7n - 5)/2.