A050441 -id:A050441 - cp's OEIS frontend

A237616 a(n) = n(n + 1)(5*n - 4)/2.

0, 1, 18, 66, 160, 315, 546, 868, 1296, 1845, 2530, 3366, 4368, 5551, 6930, 8520, 10336, 12393, 14706, 17290, 20160, 23331, 26818, 30636, 34800, 39325, 44226, 49518, 55216, 61335, 67890, 74896, 82368, 90321, 98770, 107730, 117216, 127243, 137826, 148980, 160720

Offset: 0

Bruno Berselli, Feb 10 2014

Also 17-gonal (or heptadecagonal) pyramidal numbers.

This sequence is related to A226489 by 2*a(n) = n*A226489(n) - Sum_{i=0..n-1} A226489(i).

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  16;
   3,  32,  31;
   4,  48,  62,  46;
   5,  64,  93,  92,  61;
   6,  80, 124, 138, 122,  76;
   7,  96, 155, 184, 183, 152,  91;
   8, 112, 186, 230, 244, 228, 182, 106;
   9, 128, 217, 276, 305, 304, 273, 212, 121;
  10, 144, 248, 322, 366, 380, 364, 318, 242, 136; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 15*r-14 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (fifteenth row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pyramidal Number.
Index to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051869, A104728.

Cf. sequences with formula n*(n+1)*(k*n-k+3)/6: A000217 (k=0), A000292 (k=1), A000330 (k=2), A002411 (k=3), A002412 (k=4), A002413 (k=5), A002414 (k=6), A007584 (k=7), A007585 (k=8), A007586 (k=9), A007587 (k=10), A050441 (k=11), A172073 (k=12), A177890 (k=13), A172076 (k=14), this sequence (k=15), A172078(k=16), A237617 (k=17), A172082 (k=18), A237618 (k=19), A172117(k=20), A256718 (k=21), A256716 (k=22), A256645 (k=23), A256646(k=24), A256647 (k=25), A256648 (k=26), A256649 (k=27), A256650(k=28).

List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(n+1)*(5*n-4)/2: n in [0..40]];
```

I:=[0,1,18,66]; [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, Feb 12 2014

seq(n*(n+1)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019

Table[n(n+1)(5n-4)/2, {n, 0, 40}]
CoefficientList[Series[x (1+14x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,66},50] (* Harvey P. Dale, Jan 11 2015 *)

a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1 + 14*x)/(1 - x)^4.
For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(15*i+1). More generally, the sequence with the closed form n*(n+1)*(k*n-k+3)/6 is also given by Sum_{i=0..n-1} (n-i)*(k*i+1) for n>0.
a(n) = A104728(A001844(n-1)) for n>0.
Sum_{n>=1} 1/a(n) = (2*sqrt(5*(5 + 2*sqrt(5)))*Pi + 10*sqrt(5)*arccoth(sqrt(5)) + 25*log(5) - 16)/72 = 1.086617842136293176... . - Vaclav Kotesovec, Dec 07 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*x*(2 + 16*x + 5*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

A180223 a(n) = (11n^2 - 7n)/2.

Original entry on oeis.org

0, 2, 15, 39, 74, 120, 177, 245, 324, 414, 515, 627, 750, 884, 1029, 1185, 1352, 1530, 1719, 1919, 2130, 2352, 2585, 2829, 3084, 3350, 3627, 3915, 4214, 4524, 4845, 5177, 5520, 5874, 6239, 6615, 7002, 7400, 7809, 8229, 8660

Offset: 0

Graziano Aglietti (mg5055(AT)mclink.it), Aug 16 2010

This sequence is related to A050441 by n*a(n) - Sum_{i=0..n-1} a(i) = 2*A050441(n). - Bruno Berselli, Aug 19 2010
Sum of n-th heptagonal number (A000566) and n-th octagonal number (A000567). - Bruno Berselli, Jun 11 2013
Create a triangle with T(r,1) = r^2 and T(r,c) = r^2 + r*c + c^2. The difference of the sum of the terms in row n and those in row n-1 is a(n). - J. M. Bergot, Jun 17 2013

B. Berselli, Table of n, a(n) for n = 0..10000.
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A050441, A000566, A000567, A226492.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=11). - Bruno Berselli, Jun 10 2013

List([0..30], n-> n*(11*n-7)/2); # G. C. Greubel, Sep 18 2019

[(11*n^2 - 7*n)/2: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

A180223:=n->(11*n^2 - 7*n)/2; seq(A180223(n), n=0..30); # Wesley Ivan Hurt, Feb 25 2014

Table[(11*n^2 - 7*n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Feb 25 2014 *)
LinearRecurrence[{3,-3,1},{0,2,15},50] (* Harvey P. Dale, Oct 10 2020 *)

PARI
```
a(n)=1/2*(11*n^2 - 7*n);
```

[n*(11*n-7)/2 for n in (0..30)] # G. C. Greubel, Sep 18 2019

G.f.: x*(2+9*x)/(1-x)^3. - Bruno Berselli, Aug 19 2010 - corrected in Apr 18 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with n>2. - Bruno Berselli, Aug 19 2010
a(n) = n + A226492(n). - Bruno Berselli, Jun 11 2013
E.g.f.: x*(4 + 11*x)*exp(x)/2. - G. C. Greubel, Aug 24 2015

A055268 a(n) = (11n + 4)C(n+3, 3)/4.

Original entry on oeis.org

1, 15, 65, 185, 420, 826, 1470, 2430, 3795, 5665, 8151, 11375, 15470, 20580, 26860, 34476, 43605, 54435, 67165, 82005, 99176, 118910, 141450, 167050, 195975, 228501, 264915, 305515, 350610, 400520, 455576, 516120, 582505, 655095, 734265

Offset: 0

Barry E. Williams, May 10 2000

a(n) is the number of compositions of n when there are 9 types of each natural number. - Milan Janjic, Aug 13 2010

Convolution of A000027 with A051865 (excluding 0). - Bruno Berselli, Dec 07 2012

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A050441.
Cf. A000292, A051865.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

List([0..30], n-> (11*n+4)*Binomial(n+3,3)/4 ); # G. C. Greubel, Jan 17 2020

/* A000027 convolved with A051865 (excluding 0): */ A051865:=func; [&+[(n-i+1)*A051865(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

seq( (11*n+4)*binomial(n+3,3)/4, n=0..30); # G. C. Greubel, Jan 17 2020

Table[11*Binomial[n+4,4] -10*Binomial[n+3,3], {n,0,30}] (* G. C. Greubel, Jan 17 2020 *)

a(n) = (11*n+4)*binomial(n+3, 3)/4; \\ Michel Marcus, Sep 07 2017

A055268_list, m = [], [11, 1, 1, 1, 1]
for _ in range(10**2):
    A055268_list.append(m[-1])
    for i in range(4):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[(11*n+4)*binomial(n+3,3)/4 for n in (0..30)] # G. C. Greubel, Jan 17 2020

G.f.: (1 + 10*x)/(1-x)^5. - R. J. Mathar, Oct 26 2011
From G. C. Greubel, Jan 17 2020:(Start)
a(n) = 11*binomial(n+4,4) - 10*binomial(n+3,3).
E.g.f.: (24 + 336*x + 432*x^2 + 136*x^3 + 11*x^4)*exp(x)/24. (End)

cp's OEIS Frontend

A237616 a(n) = n*(n + 1)*(5*n - 4)/2.

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A180223 a(n) = (11*n^2 - 7*n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A055268 a(n) = (11*n + 4)*C(n+3, 3)/4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A237616 a(n) = n(n + 1)(5*n - 4)/2.

A180223 a(n) = (11n^2 - 7n)/2.

A055268 a(n) = (11n + 4)C(n+3, 3)/4.