A006597 -id:A006597 - cp's OEIS frontend

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

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

A220083 a(n) = (15n^2 + 9n + 2)/2.

Original entry on oeis.org

1, 13, 40, 82, 139, 211, 298, 400, 517, 649, 796, 958, 1135, 1327, 1534, 1756, 1993, 2245, 2512, 2794, 3091, 3403, 3730, 4072, 4429, 4801, 5188, 5590, 6007, 6439, 6886, 7348, 7825, 8317, 8824, 9346, 9883, 10435, 11002, 11584, 12181, 12793, 13420, 14062

Offset: 0

Bruno Berselli, Dec 10 2012

Sequence related to the heptagonal numbers (A000566) by a(n) = n*A000566(n)-(n-1)*A000566(n-1).
Other similar sequences:
A005408(m) = (m+1)*A001477(m+1)-m*A001477(m), A001477 = nonn. integers;
A000326(m) = m*A000217(n)-(m-1)*A000217(m-1), A000217 = triangular numbers;
A003215(m) = (m+1)*A000290(m+1)-n*A000290(m), A000290 = square numbers;
A081267(m) = (m+1)*A000326(m+1)-n*A000326(m), A000326 = pentagonal numbers;
A080859(m) = (m+1)*A000384(m+1)-n*A000384(m), A000384 = hexagonal numbers;
A214675(m) = m*A000567(m)-(m-1)*A000567(m-1), A000567 = octagonal numbers.

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

Cf. A000326, A003215, A005408, A006597, A080859, A081267, A214675.

/* By first comment: */  A000566:=func; [n*A000566(n)-(n-1)*A000566(n-1): n in [1..45]];

[(15*n^2 + 9*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 18 2013

A220083:=n->(15*n^2 + 9*n + 2)/2; seq(A220083(n), n=0..100); # Wesley Ivan Hurt, Nov 14 2013

Table[(15 n^2 + 9 n + 2)/2, {n, 0, 45}]
CoefficientList[Series[(1 + 10 x + 4 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

A220083(n):=(15*n^2 + 9*n + 2)/2$ makelist(A220083(n),n,0,20); /* Martin Ettl, Dec 11 2012 */

a(n)=(15*n^2+9*n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+10*x+4*x^2)/(1-x)^3.
Sum( a(i), i=0..n ) = A006597(n+1).
a(n) + a(-n) = A010005(n) for n>0.

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)

A006592 a(n) = 10n^3 - 6n^2.

Original entry on oeis.org

0, 4, 56, 216, 544, 1100, 1944, 3136, 4736, 6804, 9400, 12584, 16416, 20956, 26264, 32400, 39424, 47396, 56376, 66424, 77600, 89964, 103576, 118496, 134784, 152500, 171704, 192456, 214816, 238844, 264600, 292144, 321536, 352836, 386104, 421400, 458784, 498316, 540056

Offset: 0

N. J. A. Sloane

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 29.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

[10*n^3-6*n^2: n in [0..40]]; // Vincenzo Librandi, Jul 20 2011

Table[10n^3-6n^2,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,4,56,216},50] (* Harvey P. Dale, Aug 13 2012 *)

PARI
```
a(n)=10*n^3-6*n^2;
```

a(n) = 4 * A006597(n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=4, a(2)=56, a(3)=216. - Harvey P. Dale, Aug 13 2012
From G. C. Greubel, Oct 18 2018: (Start)
G.f.: 4*(x + 10*x^2 + 4*x^3)/(1 - x)^4.
E.g.f.: 2*x*(2 + 12*x + 5*x^2)*exp(x). (End)

Name corrected by Arkadiusz Wesolowski, Jul 20 2011

A329530 a(n) = n * (7*binomial(n, 2) + 1).

Original entry on oeis.org

0, 1, 16, 66, 172, 355, 636, 1036, 1576, 2277, 3160, 4246, 5556, 7111, 8932, 11040, 13456, 16201, 19296, 22762, 26620, 30891, 35596, 40756, 46392, 52525, 59176, 66366, 74116, 82447, 91380, 100936, 111136, 122001, 133552, 145810, 158796, 172531, 187036, 202332, 218440

Offset: 0

Ilya Gutkovskiy, Nov 15 2019

Centered heptagonal prism numbers.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 144.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Centered m-gonal prism numbers: A100175 (m = 3), A059722 (m = 4), A006564 (m = 5), A005915 (m = 6), this sequence (m = 7), A139757 (m = 8), A006566 (m = 9).

Cf. A000217, A002413, A006597, A024966, A069099.

Table[n (7 Binomial[n, 2] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 + 12 x + 8 x^2)/(1 - x)^4, {x, 0, nmax}], x]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 16, 66}, 41]

G.f.: x * (1 + 12*x + 8*x^2) / (1 - x)^4.
E.g.f.: exp(x) * x * (2 + 14*x + 7*x^2) / 2.
a(n) = n * (7*n^2 - 7*n + 2) / 2.
a(n) = n * (7*A000217(n-1) + 1).
a(n) = n * A069099(n).

cp's OEIS Frontend

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

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A220083 a(n) = (15*n^2 + 9*n + 2)/2.

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Magma

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

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Magma

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Sage

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A006592 a(n) = 10*n^3 - 6*n^2.

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Magma

Mathematica

PARI

Formula

Extensions

A329530 a(n) = n * (7*binomial(n, 2) + 1).

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Mathematica

Formula

A220083 a(n) = (15n^2 + 9n + 2)/2.

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

A006592 a(n) = 10n^3 - 6n^2.