A192136 -id:A192136 - cp's OEIS frontend

A028895 5 times triangular numbers: a(n) = 5n(n+1)/2.

0, 5, 15, 30, 50, 75, 105, 140, 180, 225, 275, 330, 390, 455, 525, 600, 680, 765, 855, 950, 1050, 1155, 1265, 1380, 1500, 1625, 1755, 1890, 2030, 2175, 2325, 2480, 2640, 2805, 2975, 3150, 3330, 3515, 3705, 3900, 4100, 4305, 4515, 4730, 4950, 5175, 5405, 5640

Offset: 0

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

Sequence found by reading the line from 0, in the direction 0, 5, ... and the same line from 0, in the direction 0, 15, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Axis perpendicular to A195142 in the same spiral. - Omar E. Pol, Sep 18 2011
Bisection of A195014. Sequence found by reading the line from 0, in the direction 0, 5, ..., and the same line from 0, in the direction 0, 15, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is the main diagonal of the spiral. - Omar E. Pol, Sep 25 2011
a(n) = the Wiener index of the graph obtained by applying Mycielski's construction to the complete graph K(n) (n>=2). - Emeric Deutsch, Aug 29 2013
Sum of the numbers from 2*n to 3*n for n=0,1,2,... - Wesley Ivan Hurt, Nov 27 2015
Numbers k such that the concatenation k625 is a square, where also 625 is a square. - Bruno Berselli, Nov 07 2018
From Paul Curtz, Nov 29 2019: (Start)
Main column of the pentagonal spiral for n (A001477):
                         50
                      49 30 31
                   48 29 15 16 32
                47 28 14  5  6 17 33
             46 27 13  4  0  1  7 18 34
               45 26 12  3  2  8 19 35
                 44 25 11 10  9 20 36
                   43 24 23 22 21 37
                    42 41 40 39 38
(End)

D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Rangaswami Balakrishnan and S. Francis Raj, The Wiener number of powers of the Mycielskian, Discussiones Math. Graph Theory, Vol. 30, No. 3 (2010), pp. 489-498 (see Theorem 2.1).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A001068, A005891, A028896, A046092, A055998, A085787, A130520, A195013, A195014, A195142.
Cf. index to numbers of the form n*(d*n+10-d)/2 in A140090.
Cf. A000566, A005475, A005476, A033583, A085787, A147875, A192136, A326725 (all in the spiral).

[5*n*(n+1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

[seq(5*binomial(n,2), n=1..45)]; # Zerinvary Lajos, Nov 24 2006

Table[Sum[i + 2*n - 1, {i, 2, n}], {n, 45}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[5 n (n + 1)/2, {n, 0, 50}] (* Bruno Berselli, Sep 23 2016 *)
5 Accumulate[Range[0,60]] (* Harvey P. Dale, Jun 17 2025 *)

a(n)=5*n*(n+1)/2 \\ Charles R Greathouse IV, Sep 24 2015

def A028895(n): return 5*n*(n+1)>>1 # Chai Wah Wu, Aug 07 2025

G.f.: 5*x/(1-x)^3.
a(n) = 5*n*(n+1)/2 = 5*A000217(n).
a(n+1) = 5*n+a(n). - Vincenzo Librandi, Aug 05 2010
a(n) = A005891(n) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A130520(5n+4). - Philippe Deléham, Mar 26 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Wesley Ivan Hurt, Nov 27 2015
a(n) = Sum_{i=0..n} A001068(4i). - Wesley Ivan Hurt, May 06 2016
E.g.f.: 5*x*(2 + x)*exp(x)/2. - Ilya Gutkovskiy, May 06 2016
a(n) = A055998(3*n) - A055998(2*n). - Bruno Berselli, Sep 23 2016
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2/5)*(2*log(2) - 1). (End)
Product_{n>=1} (1 - 1/a(n)) = -(5/(2*Pi))*cos(sqrt(13/5)*Pi/2). - Amiram Eldar, Feb 21 2023

A211849 T(n,k)=Number of nonnegative integer arrays of length n+2k+1 with new values 0 upwards introduced in order, no three adjacent elements all unequal, and containing the value k+1.

Original entry on oeis.org

1, 1, 5, 1, 7, 19, 1, 9, 35, 63, 1, 11, 56, 147, 196, 1, 13, 82, 286, 561, 588, 1, 15, 113, 494, 1302, 2013, 1731, 1, 17, 149, 785, 2619, 5486, 6936, 5049, 1, 19, 190, 1173, 4755, 12713, 21897, 23244, 14689, 1, 21, 236, 1672, 7996, 26163, 57913, 84003, 76434

Offset: 1

R. H. Hardin Apr 22 2012

Table starts
....1.....1.....1......1......1.......1.......1.......1........1........1
....5.....7.....9.....11.....13......15......17......19.......21.......23
...19....35....56.....82....113.....149.....190.....236......287......343
...63...147...286....494....785....1173....1672....2296.....3059.....3975
..196...561..1302...2619...4755....7996...12671...19152....27854....39235
..588..2013..5486..12713..26163...49210...86275..142968...226230...344475
.1731..6936.21897..57913.134164..280751..542235..981675..1685165..2766870
.5049.23244.84003.251481.651814.1510267.3200979.6311155.11721555.20705130

			Some solutions for n=3 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....0....1....0....1....1....0....0....0....1....1....1....1....1....1
..1....0....1....1....1....1....1....0....1....1....1....1....1....1....1....0
..2....0....1....2....1....2....2....1....1....1....1....2....2....0....2....0
..2....2....2....2....1....2....1....1....2....2....2....2....2....0....2....2
..3....0....2....2....2....3....1....2....2....2....1....2....0....2....3....2
..3....0....3....3....2....3....3....2....3....3....1....3....0....2....3....3
..3....3....3....3....3....4....3....3....2....3....3....2....3....3....4....2
..4....3....4....4....3....4....4....3....2....3....3....2....3....3....4....2
..4....4....3....4....4....1....4....4....4....4....4....4....4....4....5....4
..4....4....3....4....4....1....4....4....4....4....4....4....4....4....5....4
..5....5....5....5....5....5....5....5....5....5....5....5....5....5....4....5

R. H. Hardin, Table of n, a(n) for n = 1..9999

Row 3 is A192136(n+2)

A203551 a(n) = n*(5n^2 + 3n + 4) / 6.

Original entry on oeis.org

0, 2, 10, 29, 64, 120, 202, 315, 464, 654, 890, 1177, 1520, 1924, 2394, 2935, 3552, 4250, 5034, 5909, 6880, 7952, 9130, 10419, 11824, 13350, 15002, 16785, 18704, 20764, 22970, 25327, 27840, 30514, 33354, 36365, 39552, 42920, 46474, 50219

Offset: 0

Michael Somos, Jan 02 2012

			G.f. = 2*x + 10*x^2 + 29*x^3 + 64*x^4 + 120*x^5 + 202*x^6 + 315*x^7 + 464*x^8 + ...

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

Cf. A203552.

I:=[0, 2, 10, 29]; [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, Jan 07 2012

LinearRecurrence[{4,-6,4,-1},{0,2,10,29},40] (* Vincenzo Librandi, Jan 07 2012 *)
Table[n (5n^2+3n+4)/6,{n,0,40}] (* Harvey P. Dale, Mar 24 2022 *)

PARI
```
{a(n) = n * (5*n^2 + 3*n + 4) / 6};
```

a(n) = Sum_{k = 1..n} A(-k, k-n-1) where A(i, j) = i^2 + i*j + j^2 + i + j + 1.
G.f.: x * (2 + 2*x + x^2) / (1 - x)^4.
a(n) = -A203552(-n) for all n in Z.
a(n)-a(n-1) = A192136(n). - Bruno Berselli, Jan 03 2012
E.g.f.: x*(5*x^2 + 18*x + 12)*exp(x)/6. - G. C. Greubel, Aug 12 2018

cp's OEIS Frontend

A028895 5 times triangular numbers: a(n) = 5n(n+1)/2.

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A211849 T(n,k)=Number of nonnegative integer arrays of length n+2k+1 with new values 0 upwards introduced in order, no three adjacent elements all unequal, and containing the value k+1.

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A203551 a(n) = n*(5n^2 + 3n + 4) / 6.

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cp's OEIS Frontend

A028895 5 times triangular numbers: a(n) = 5*n*(n+1)/2.

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Author

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References

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Magma

Maple

Mathematica

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Python

Formula

A211849 T(n,k)=Number of nonnegative integer arrays of length n+2k+1 with new values 0 upwards introduced in order, no three adjacent elements all unequal, and containing the value k+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A203551 a(n) = n*(5n^2 + 3n + 4) / 6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A028895 5 times triangular numbers: a(n) = 5n(n+1)/2.