A060354 -id:A060354 - cp's OEIS frontend

A101357 Partial sums of A060354.

0, 1, 3, 9, 25, 60, 126, 238, 414, 675, 1045, 1551, 2223, 3094, 4200, 5580, 7276, 9333, 11799, 14725, 18165, 22176, 26818, 32154, 38250, 45175, 53001, 61803, 71659, 82650, 94860, 108376, 123288, 139689, 157675, 177345, 198801, 222148, 247494

Offset: 0

Jonathan Vos Post, Dec 25 2004

The Ca4 triangle sums of A139600 are given by the terms of this sequence. For the definitions of the Ca4 and other triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Polygonal number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A060354, A000332.

[(n^4-2*n^3+3*n^2+6*n)/8: n in [0..40]]; // Vincenzo Librandi, Aug 06 2011

Table[Sum[(i*(i - 2)^2 + i^2)/2, {i, 0, n}], {n, 0, 38}]
Accumulate[Table[(n (n-2)^2+n^2)/2,{n,0,50}]] (* Harvey P. Dale, Aug 05 2011 *)

a(n)=(n^4-2*n^3+3*n^2+6*n)/8 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = Sum_{i=0..n} (i(i-2)^2 + i^2)/2.
a(n) = A004255(n), n > 0. - R. J. Mathar, Sep 02 2008
a(n) = binomial(n+3,4) - 2*binomial(n+2,4) + 4*binomial(n+1,4).
a(n) = (n^4 - 2*n^3 + 3*n^2 + 6*n)/8. - Johannes W. Meijer, Apr 29 2011
G.f.: -x*(4*x^2 - 2*x + 1) / (x-1)^5. - Colin Barker, Apr 29 2013

More terms from Joshua Zucker, May 12 2006

Edited by Stefan Steinerberger, Aug 01 2007

A271319 Number of distinct prime factors of the n-th n-gonal number (A060354).

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 4, 2, 2, 3, 4, 3, 3, 3, 3, 3, 2, 4, 3, 3, 2, 5, 3, 2, 4, 3, 4, 3, 2, 4, 4, 4, 3, 4, 3, 3, 3, 3, 4, 3, 2, 4, 4, 4, 2, 4, 4, 3, 4, 3, 3, 4, 4, 3, 5, 2, 3, 4, 4, 4, 4, 4, 3, 4, 2, 4, 5, 3, 3, 4, 3, 4, 3, 4, 4, 5, 4, 3, 4

Offset: 2

Colin Barker, Apr 04 2016

nonn

			a(7) = 2 because A060354(7) = 112 = 2^4 * 7^1.

Colin Barker, Table of n, a(n) for n = 2..1000

Cf. A001221, A060354, A271320, A271321, A271322.

Table[PrimeNu[PolygonalNumber[n,n]],{n,2,90}] (* Harvey P. Dale, Sep 24 2023 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
a(n) = omega(pg(n, n))
vector(100, n, n++; a(n))

a(n) = A001221(A060354(n)).

A271320 Number of prime factors, with multiplicity, of the n-th n-gonal number (A060354).

Original entry on oeis.org

1, 2, 4, 2, 3, 5, 5, 3, 3, 3, 7, 2, 3, 5, 6, 3, 4, 4, 6, 3, 3, 5, 6, 3, 4, 5, 9, 2, 5, 4, 7, 4, 4, 4, 7, 2, 4, 9, 7, 3, 4, 3, 7, 4, 3, 5, 7, 3, 5, 4, 7, 2, 6, 6, 6, 4, 3, 3, 9, 4, 3, 7, 8, 3, 4, 4, 7, 4, 4, 6, 8, 2, 4, 6, 7, 3, 4, 4, 8, 6, 4, 4, 8, 4, 3, 6

Offset: 2

Colin Barker, Apr 04 2016

nonn

			a(7) = 5 because A060354(7) = 112 = 2^4 * 7^1.

Colin Barker, Table of n, a(n) for n = 2..1000

Cf. A001222, A060354, A271319, A271321, A271322.

PrimeOmega/@Table[PolygonalNumber[n,n],{n,90}] (* Harvey P. Dale, Feb 08 2025 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
a(n) = bigomega(pg(n, n))
vector(100, n, n++; a(n))

a(n) = A001222(A060354(n)).

A271321 Smallest prime factor of the n-th n-gonal number (A060354).

Original entry on oeis.org

2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 13, 2, 2, 2, 11, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 29, 2, 2, 2, 3, 2, 2, 2, 37, 2, 2, 2, 11, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 53, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 73, 2, 2, 2, 7, 2, 2, 2, 3, 2, 2, 2, 5

Offset: 2

Colin Barker, Apr 04 2016

nonn

			a(5) = 5 because A060354(5) = 35 = 5 * 7.

Colin Barker, Table of n, a(n) for n = 2..1000

Cf. A020639, A060354, A271319, A271320, A271322.

Table[FactorInteger[PolygonalNumber[n,n]][[1,1]],{n,2,90}] (* Harvey P. Dale, Feb 26 2022 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
spf(m) = factorint(m)[1, 1] \\ Smallest prime factor
a(n) = spf(pg(n, n))
vector(100, n, n++; a(n))

a(n) = A020639(A060354(n)).

A271322 Largest prime factor of the n-th n-gonal number (A060354).

Original entry on oeis.org

2, 3, 2, 7, 11, 7, 11, 29, 37, 23, 7, 67, 79, 23, 53, 17, 137, 19, 43, 191, 211, 29, 127, 277, 43, 163, 11, 379, 37, 109, 233, 71, 23, 281, 149, 631, 29, 13, 53, 71, 821, 431, 113, 947, 991, 47, 541, 1129, 107, 613, 29, 1327, 197, 179, 743, 67, 1597, 827

Offset: 2

Colin Barker, Apr 04 2016

nonn

			a(5) = 7 because A060354(5) = 35 = 5 * 7.

Colin Barker, Table of n, a(n) for n = 2..1000

Cf. A006530, A060354, A271319, A271320, A271321.

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
lpf(m) = vecmax(factorint(m)[, 1]) \\ Largest prime factor
a(n) = lpf(pg(n, n))
vector(100, n, n++; a(n))

a(n) = A006530(A060354(n)).

A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 7, 1, 4, 7, 10, 13, 1, 5, 9, 13, 17, 21, 1, 6, 11, 16, 21, 26, 31, 1, 7, 13, 19, 25, 31, 37, 43, 1, 8, 15, 22, 29, 36, 43, 50, 57, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81

Offset: 1

Omar E. Pol, Jul 09 2009

Equals A081493 when first column is removed. - Georg Fischer, Jul 25 2023

			Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  5,  7;
  1,  4,  7, 10, 13;
  1,  5,  9, 13, 17, 21;
  1,  6, 11, 16, 21, 26, 31;

Cf. A002061, A159797, A159798, A162610, A161611, A162612, A162613.

Cf. A060354 (row sums), A081493 (without first column).

Table[NestList[#+(n-2)&,1,n-1],{n,20}]//Flatten (* Harvey P. Dale, Oct 23 2017 *)

T(n,n) = A002061(n-1).

T(n,k) = A076110(n-1,k) = 1+(n-2)*(k-1). - R. J. Mathar, Mar 30 2023

A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451

Offset: 0

Paul Barry, Mar 21 2003

			The array starts
  1  1  3 10 ...
  1  2  6 16 ...
  1  3  9 22 ...
  1  4 12 28 ...
The triangle starts
  1;
  1,  1;
  1,  2,  3;
  1,  3,  6, 10;
  1,  4,  9, 16, 25;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number

Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.

Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019

[[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018

Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)

tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

[[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019

Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.

T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015

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

A100119 a(n) = n-th centered n-gonal number.

Original entry on oeis.org

1, 2, 7, 19, 41, 76, 127, 197, 289, 406, 551, 727, 937, 1184, 1471, 1801, 2177, 2602, 3079, 3611, 4201, 4852, 5567, 6349, 7201, 8126, 9127, 10207, 11369, 12616, 13951, 15377, 16897, 18514, 20231, 22051, 23977, 26012, 28159, 30421, 32801, 35302

Offset: 0

Jonathan Vos Post, Dec 26 2004

a(n) is n times the n-th triangular number plus 1. - Thomas M. Green, Nov 16 2009
From Gary W. Adamson, Jul 31 2010: (Start)
Equals (1, 2, 3, 4, ...) convolved with (1, 0, 4, 7, 10, 13, ...).
Example: a(5) = 76 = (6, 5, 4, 3, 2, 1) dot (1, 0, 4, 7, 10, 13) = (6 + 0 + 16 + 21 + 20 + 13). (End)

			a(2) = 2*3 + 1 = 7, a(3) = 3*6 + 1 = 19, a(4) = 4*10 + 1 = 41. - _Thomas M. Green_, Nov 16 2009

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

See also A101357 (Cumulative sums of the n-th n-gonal numbers).
A diagonal of A101321.
Cf. A060354, A098547, A101357.
Cf. A006003, A005920.

I:=[1, 2, 7, 19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 25 2012

Table[(n^3+n^2)/2+1,{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
LinearRecurrence[{4,-6,4,-1},{1,2,7,19},40] (* Vincenzo Librandi, Jun 25 2012 *)

a(n) = n^2*(n+1)/2+1; \\ Altug Alkan, Sep 21 2018

a(n) = 1 + n*(n + n^2)/2 = 1 + (1/2)*n^2 + (1/2) * n^3 = 1 + mean(n^2, n^3). - Joshua Zucker, May 03 2006
Equals A002411(n) + 1. - Olivier Gérard, Jun 20 2007
G.f.: (1 - 2*x + 5*x^2 - x^3) / (x-1)^4. - R. J. Mathar, Apr 04 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 25 2012
a(n) = (A098547(n)+1)/2. - Richard Turk, Jul 18 2017
a(n) = A060354(n+2) - A000290(n+1) = A006003(n+1) - A005563(n) and for n>0 A005920(n) - A068601(n+1). - Bruce J. Nicholson, Jun 23 2018

Corrected and extended by Joshua Zucker, May 03 2006

A188947 a(n) = n^3 - 2n^2 + 2n + 1.

Original entry on oeis.org

2, 5, 16, 41, 86, 157, 260, 401, 586, 821, 1112, 1465, 1886, 2381, 2956, 3617, 4370, 5221, 6176, 7241, 8422, 9725, 11156, 12721, 14426, 16277, 18280, 20441, 22766, 25261, 27932, 30785, 33826, 37061, 40496, 44137, 47990, 52061, 56356, 60881, 65642, 70645

Offset: 1

Adeniji, Adenike, Apr 14 2011

The original definition was "Identity difference partial one - one transformation semigroup is a semigroup having the property that the difference between max im(alpha) and min im(alpha) is not greater than 1. This is denoted by S = IDI_n for each n." [Needs editing.]

For all n >= 3, a(n) expressed in base n has the three digits n-2, 2, and 1; for example, a(16) in hexadecimal is "E21". For all n >= 3, a(n+1) expressed in base n is "1112". For all n >= 7, a(n+2) expressed in base n is "1465". - Mathew Englander, Jan 07 2021

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

Cf. A027444, A053698, A056106 (first differences), A060354, A162607, A188377, A188716.

[n^3 - 2*n^2 + 2*n + 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A188947[n_] := n^3 - 2*n^2 + 2*n + 1; Table[A188947[n], {n, 1, 42}] (* Robert P. P. McKone, Jan 31 2021 *)

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

a(n) = (n+1) + n*(n-1)^2 = n^3 - 2*n^2 + 2*n + 1 = 1 + A053698(n-1).
G.f.: ( -x*(-2 + 3*x - 8*x^2 + x^3) ) / ( (x-1)^4 ). - R. J. Mathar, Apr 14 2011
a(n) = A060354(n) + A162607(n+1). - Lechoslaw Ratajczak, Sep 24 2020
E.g.f.: exp(x)*(1 + x)*(1 + x^2) - 1. - Stefano Spezia, Apr 10 2022

Edited by N. J. A. Sloane, Apr 23 2011

cp's OEIS Frontend

A101357 Partial sums of A060354.

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A271319 Number of distinct prime factors of the n-th n-gonal number (A060354).

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A271320 Number of prime factors, with multiplicity, of the n-th n-gonal number (A060354).

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A271321 Smallest prime factor of the n-th n-gonal number (A060354).

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A271322 Largest prime factor of the n-th n-gonal number (A060354).

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A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

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A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

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A188947 a(n) = n^3 - 2n^2 + 2n + 1.