A081438 -id:A081438 - cp's OEIS frontend

A081437 Diagonal in array of n-gonal numbers A081422.

1, 10, 33, 76, 145, 246, 385, 568, 801, 1090, 1441, 1860, 2353, 2926, 3585, 4336, 5185, 6138, 7201, 8380, 9681, 11110, 12673, 14376, 16225, 18226, 20385, 22708, 25201, 27870, 30721, 33760, 36993, 40426, 44065, 47916, 51985, 56278, 60801, 65560

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
For q a prime power, a(q-1) = q^3 + q^2 - q is the number of pairs of commuting nilpotent 2*2 matrices with coefficients in GF(q). (Proof: the zero matrix commutes with all q^2 nilpotent matrices, there are q^2-1 nonzero nilpotent matrices, all conjugate, each commuting with q nilpotent matrices.) - Mark Wildon, Jun 20 2017
Also the cyclomatic number (= circuit rank) of the n+1 X n+1 rook graph. - Eric W. Weisstein, Jun 20 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081435, A081436, A081437, A081438.

Equals A027620(n-1) + 1.

List([0..40], n-> (n+1)^3+n*(n+1)); # G. C. Greubel, Aug 14 2019

[n^3+4*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:=n->sum(n*k, k=0..n):seq(a(n)+sum(n*k, k=2..n), n=1..40); # Zerinvary Lajos, Jun 10 2008
a:=n->sum(-2+sum(2+sum(2, j=1..n),j=1..n),j=1..n):seq(a(n)/2,n=1..40); # Zerinvary Lajos, Dec 06 2008

Table[n^3 + 4 n^2 + 4n + 1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 10, 33, 76}, 40] (* Harvey P. Dale, Jan 24 2012 *)
CoefficientList[Series[(1 + 5 x - 7 x^2 + x^3)/(1 - x)^5, {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (n+1)^3+n*(n+1)) \\ G. C. Greubel, Aug 14 2019

[(n+1)^3+n*(n+1) for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = n^3 + 4*n^2 + 4*n + 1.
G.f.: (1 +5*x -7*x^2 +x^3)/(1-x)^5.
a(0)=1, a(1)=10, a(2)=33, a(3)=76; for n>3, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Harvey P. Dale, Jan 24 2012
E.g.f.: (1 +9*x +7*x^2 +x^3)*exp(x). - G. C. Greubel, Aug 14 2019

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

A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

Original entry on oeis.org

1, 1, 6, 22, 55, 111, 196, 316, 477, 685, 946, 1266, 1651, 2107, 2640, 3256, 3961, 4761, 5662, 6670, 7791, 9031, 10396, 11892, 13525, 15301, 17226, 19306, 21547, 23955, 26536, 29296, 32241, 35377, 38710, 42246, 45991, 49951, 54132, 58540, 63181

Offset: 0

Paul Barry, Mar 21 2003

Diagonal in array of n-gonal numbers A081422.

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

Cf. A005564, A081422, A081436, A081437, A081438.

List([0..40], n-> (2*n^3-n^2-n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:= n-> (2*n^3-n^2-n+2)/2: seq(a(n), n=0..50); # Zerinvary Lajos, Sep 13 2006

Table[(2n^3-n^2-n+2)/2,{n,0,40}] (* Harvey P. Dale, May 29 2012 *)
CoefficientList[Series[(1 - 4 x + 11 x^2 - 8 x^3) / (1 - x)^5, {x, 0, 50}],x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (2*n^3-n^2-n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

G.f.: (1 -4*x +11*x^2 -8*x^3)/(1-x)^5.
a(n) = (n + 1)*(2*n^2 - 3*n + 2)/2 = (n-1)*A005564(n+1) - n*A005564(n), where A005564(0..2) = 0, -1, 0. - Bruno Berselli, May 19 2015
E.g.f.: (2 + 5*x^2 + 2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A177058 a(n) = n^3 - 3n^2 + 3.

Original entry on oeis.org

3, 1, -1, 3, 19, 53, 111, 199, 323, 489, 703, 971, 1299, 1693, 2159, 2703, 3331, 4049, 4863, 5779, 6803, 7941, 9199, 10583, 12099, 13753, 15551, 17499, 19603, 21869, 24303, 26911, 29699, 32673, 35839, 39203, 42771, 46549, 50543, 54759, 59203

Offset: 0

Vincenzo Librandi, May 27 2010

For n>2, fourth diagonal of A162611.

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

Cf. A162611, A081438.

Table[n^3-3n^2+3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{3,1,-1,3},50] (* Harvey P. Dale, May 15 2020 *)

a(n)=n^3-3*n^2+3 \\ Charles R Greathouse IV, Jan 11 2012

From Bruno Berselli, Jun 04 2010: (Start)
G.f.: (3-11*x+13*x^2+x^3)/(1-x)^4.
a(n)-4*a(n-1)+6*a(n-2)-4*a(n-3)+a(n-4) = 0, with n>3.
a(n)+a(n-1) = 2*A081438(n-3), with n>2. (End)
G.f.: 3+x+x^2*G(0) where G(k) = 1 - x*(k+1)*(k+1)*(k+4)/(1 - 1/(1 - (k+1)*(k+1)*(k+4)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012

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A081437 Diagonal in array of n-gonal numbers A081422.

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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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A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

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A177058 a(n) = n^3 - 3n^2 + 3.

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