A008531 -id:A008531 - cp's OEIS frontend

A222408 Partial sums of A008531, or crystal ball sequence for {A_4}* lattice.

1, 11, 61, 211, 551, 1201, 2311, 4061, 6661, 10351, 15401, 22111, 30811, 41861, 55651, 72601, 93161, 117811, 147061, 181451, 221551, 267961, 321311, 382261, 451501, 529751, 617761, 716311, 826211, 948301, 1083451

Offset: 0

N. J. A. Sloane, Feb 21 2013

Ben Thurston, 2-dimensional model of initial terms (cf. A001846)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008531, A001846.

LinearRecurrence[{5,-10,10,-5,1},{1,11,61,211,551},40] (* Harvey P. Dale, Aug 22 2025 *)

a(n)=5*n*(n+1)*(n^2+n+2)/4 \\ Charles R Greathouse IV, Mar 08 2013

a(n) = (4+10*n+15*n^2+10*n^3+5*n^4)/4. G.f.: -(x^4+6*x^3+16*x^2+6*x+1) / (x-1)^5. [Colin Barker, Mar 08 2013]

A226449 a(n) = n(5n^2-8*n+5)/2.

Original entry on oeis.org

0, 1, 9, 39, 106, 225, 411, 679, 1044, 1521, 2125, 2871, 3774, 4849, 6111, 7575, 9256, 11169, 13329, 15751, 18450, 21441, 24739, 28359, 32316, 36625, 41301, 46359, 51814, 57681, 63975, 70711, 77904, 85569, 93721, 102375, 111546, 121249, 131499, 142311, 153700

Offset: 0

Bruno Berselli, Jun 07 2013

Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:
A006003(n) = A000217(n) + n*A000217(n-1)      (b = triangular numbers);
A069778(n) = A000290(n+1) + (n+1)*A000290(n)  (b = square numbers);
A143690(n) = A000326(n+1) + (n+1)*A000326(n)  (b = pentagonal numbers);
A212133(n) = A000384(n) + n*A000384(n-1)      (b = hexagonal numbers);
a(n)       = A000566(n) + n*A000566(n-1)      (b = heptagonal numbers);
A226450(n) = A000567(n) + n*A000567(n-1)      (b = octagonal numbers);
A226451(n) = A001106(n) + n*A001106(n-1)      (b = nonagonal numbers);
A204674(n) = A001107(n+1) + (n+1)*A001107(n)  (b = decagonal numbers).

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

Cf. (see the comment) A000566, A006003, A069778, A143690, A204674, A212133, A226450, A226451.

Magma
```
[n*(5*n^2-8*n+5)/2: n in [0..40]];
```

I:=[0,1,9,39]; [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..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (5 n^2 - 8 n + 5)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,1,9,39},50] (* Harvey P. Dale, May 19 2017 *)

a(n)=n*(5*n^2-8*n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+5*x+9*x^2)/(1-x)^4.

a(n) - a(-n) = A008531(n) for n>0.

A008487 Expansion of (1-x^5) / (1-x)^5.

Original entry on oeis.org

1, 5, 15, 35, 70, 125, 205, 315, 460, 645, 875, 1155, 1490, 1885, 2345, 2875, 3480, 4165, 4935, 5795, 6750, 7805, 8965, 10235, 11620, 13125, 14755, 16515, 18410, 20445, 22625, 24955, 27440, 30085, 32895, 35875, 39030, 42365, 45885, 49595, 53500, 57605, 61915

Offset: 0

N. J. A. Sloane

Related to the 4-dimensional cyclotomic lattice Z[zeta_5] (or A_4^{*}).

Growth series of the affine Weyl group of type A4. - Paul E. Gunnells, Jan 06 2017

R. Bott, The geometry and the representation theory of compact Lie groups, in: Representation Theory of Lie Groups, in: London Math. Soc. Lecture Note Ser., vol. 34, Cambridge University Press, Cambridge, 1979, pp. 65-90.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.

Colin Barker, Table of n, a(n) for n = 0..1000
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A008498, A008531, A222408.

concatenation([1], List([1..50], n-> 5*n*(n^2 +5)/6)); # G. C. Greubel, Nov 07 2019

[1] cat [5*n*(n^2 +5)/6: n in [1..50]]; // G. C. Greubel, Nov 07 2019

1, seq(5*n*(n^2 +5)/6, n=1..50); # G. C. Greubel, Nov 07 2019

CoefficientList[Series[(1-x^5)/(1-x)^5, {x, 0, 50}], x] (* Stefano Spezia, Dec 30 2018 *)

Vec((1-x^5) / (1-x)^5+O(x^50)) \\ Charles R Greathouse IV, Sep 26 2012; corrected by Colin Barker, Jan 06 2017

[1]+[5*n*(n^2 +5)/6 for n in (1..50)] # G. C. Greubel, Nov 07 2019

a(n) is the sum of 5 consecutive tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6 = A000292(n) for n>0, a(0) = 1. a(n) = A000292(n-4) + A000292(n-3) + A000292(n-2) + A000292(n-1) + A000292(n) for n>0, a(0) = 1. - Alexander Adamchuk, May 20 2006
Equals binomial transform of [1, 4, 6, 4, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 29 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. - Colin Barker, Jan 06 2017
For n >= 1, a(n) = (5*n^3 + 25*n)/6. - Christopher Hohl, Dec 30 2018
E.g.f.: 1 + x*(30 + 15*x + 5*x^2)*exp(x)/6. - G. C. Greubel, Nov 07 2019

cp's OEIS Frontend

A222408 Partial sums of A008531, or crystal ball sequence for {A_4}* lattice.

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A226449 a(n) = n(5n^2-8*n+5)/2.

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A008487 Expansion of (1-x^5) / (1-x)^5.

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cp's OEIS Frontend

A222408 Partial sums of A008531, or crystal ball sequence for {A_4}* lattice.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A226449 a(n) = n*(5*n^2-8*n+5)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

A008487 Expansion of (1-x^5) / (1-x)^5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A226449 a(n) = n(5n^2-8*n+5)/2.