A071237 -id:A071237 - cp's OEIS frontend

A027445 a(n) = n^4 + n^3 + n^2 + n^1.

0, 4, 30, 120, 340, 780, 1554, 2800, 4680, 7380, 11110, 16104, 22620, 30940, 41370, 54240, 69904, 88740, 111150, 137560, 168420, 204204, 245410, 292560, 346200, 406900, 475254, 551880, 637420, 732540, 837930, 954304, 1082400, 1222980, 1376830, 1544760, 1727604

Offset: 0

Patrick De Geest

a(A047203(n)) mod 10 = 0; a(A016861(n)) mod 10 = 4. - Reinhard Zumkeller, Oct 23 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Equals 2 * A071237(n).

Column k=4 of A228275.

List([0..50],n->n^4+n^3+n^2+n); # Muniru A Asiru, Jul 15 2018

[n^4 + n^3 + n^2 + n^1: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

seq(n^4+n^3+n^2+n,n=0..50); # Muniru A Asiru, Jul 15 2018

lst={};Do[AppendTo[lst, n^4+n^3+n^2+n], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 20 2008 *)
Table[Total[n^Range[4]],{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,4,30,120,340},40] (* Harvey P. Dale, Jul 01 2017 *)

a(n)=n^4+n^3+n^2+n^1 \\ Charles R Greathouse IV, Oct 21 2022

A155121 a(n) = 2n(1 + n + n^2 + n^3) - 3.

Original entry on oeis.org

-3, 5, 57, 237, 677, 1557, 3105, 5597, 9357, 14757, 22217, 32205, 45237, 61877, 82737, 108477, 139805, 177477, 222297, 275117, 336837, 408405, 490817, 585117, 692397, 813797, 950505, 1103757, 1274837, 1465077, 1675857

Offset: 0

Roger L. Bagula, Jan 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A024003, A071237, A142463, A155120.

[2*n*(1+n+n^2+n^3)-3: n in [0..40] ]; // Vincenzo Librandi, May 23 2011

seq( -3 +2*n +2*n^2 +2*n^3 +2*n^4, n=0..40); # G. C. Greubel, Mar 25 2021

Table[-3 +2n +2n^2 +2n^3 +2n^4, {n, 0, 30}]

[-3 +2*n +2*n^2 +2*n^3 +2*n^4 for n in (0..40)] # G. C. Greubel, Mar 25 2021

a(n) = 2*n*(1 + n + n^2 + n^3) - 3.
G.f.: (3 - 20*x - 2*x^2 - 32*x^3 + 3*x^4)/(x-1)^5.
From Bruno Berselli, Dec 16 2010: (Start)
a(n) = 4*A071237(n) - 3.
a(n) = 2*A024003(n)/(1-n) - 5 (n>1). (End)
E.g.f.: (-3 + 8*x + 22*x^2 + 14*x^3 + 2*x^4)*exp(x). - G. C. Greubel, Mar 25 2021

A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

Original entry on oeis.org

2, 5, 5, 10, 10, 10, 17, 17, 17, 17, 26, 26, 26, 26, 26, 37, 37, 37, 37, 37, 37, 50, 50, 50, 50, 50, 50, 50, 65, 65, 65, 65, 65, 65, 65, 65, 82, 82, 82, 82, 82, 82, 82, 82, 82, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101

Offset: 1

Craig Knecht, Dec 02 2015

The natural numbers can sequentially fill a right- or left-handed equilateral triangle. Componentwise addition of the values of these two triangles produces the present triangle.
The row sums for this triangle give A034262.
The difference between the right- and left-handed triangles produces A049581.

			Displayed as a triangle:
   2;
   5  5;
  10 10 10;
  17 17 17 17;
  26 26 26 26 26;
  37 37 37 37 37 37;
  ...

Craig Knecht, Sum of right and left triangles.
Craig Knecht, Difference between right and left triangles A049581.

Column k=1 gives A002522.
Cf. A049581 (difference of triangles), A034262 (row sum of triangle), A069894 (center column).
Cf. A071237.

seq(seq(n^2+1,k=1..n),n=1..10); # Georg Fischer, Oct 01 2021

T(n,k) = n^2 + 1 for k = 1..n and n >= 1. - Georg Fischer, Oct 01 2021

Sum_{k=1..n} k * T(n,k) = A071237(n). - Alois P. Heinz, Oct 01 2021

Row 6 with T(6,k)=37 inserted by Georg Fischer, Oct 01 2021

cp's OEIS Frontend

A027445 a(n) = n^4 + n^3 + n^2 + n^1.

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A155121 a(n) = 2n(1 + n + n^2 + n^3) - 3.

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A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

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cp's OEIS Frontend

A027445 a(n) = n^4 + n^3 + n^2 + n^1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

A155121 a(n) = 2*n*(1 + n + n^2 + n^3) - 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A155121 a(n) = 2n(1 + n + n^2 + n^3) - 3.