A116689 -id:A116689 - cp's OEIS frontend

A268685 a(n) = 3(n + 1)(n + 2)(3n + 1)(3n + 4)/4.

6, 126, 630, 1950, 4680, 9576, 17556, 29700, 47250, 71610, 104346, 147186, 202020, 270900, 356040, 459816, 584766, 733590, 909150, 1114470, 1352736, 1627296, 1941660, 2299500, 2704650, 3161106, 3673026, 4244730, 4880700, 5585580, 6364176, 7221456, 8162550

Offset: 0

Ilya Gutkovskiy, Feb 11 2016

a(n) is the total volume of the family of (n+1) rectangular prisms, where the k-th prism has dimensions (3k) X (3k-1) X (3k-2). - Wesley Ivan Hurt, Oct 02 2018

			a(0) = 1*2*3 = 6;
a(1) = 1*2*3 + 4*5*6 = 126;
a(2) = 1*2*3 + 4*5*6 + 7*8*9 = 630;
a(3) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 = 1950;
a(4) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 = 4680;
a(5) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 = 9576, etc.

Felix Fröhlich, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000027, A054776, A116689, A135036.

Trisection of A319014 and A319867.

[3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4: n in [0..40]]; // Vincenzo Librandi, Feb 11 2016

Table[3 (n + 1) (n + 2) (3 n + 1) ((3 n + 4)/4), {n, 0, 32}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {6, 126, 630, 1950, 4680}, 32]
CoefficientList[Series[6 (10 x^2 + 16 x + 1) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 11 2016 *)

a(n) = 3*(n+1)*(n+2)*(3*n+1)*(3*n+4)/4 \\ Felix Fröhlich, Jun 07 2016

G.f.: -6*(10*x^2 + 16*x + 1)/(x - 1)^5.
a(n) = Sum_{k = 0..n} (3*k + 1)(3*k + 2)(3*k + 3).
Sum {n>=0} 1/a(n) = 2*(sqrt(3)*Pi + 9*log(3) - 14)/15 = 0.1771878254287521...
a(n) mod 6 = 0.
a(n) = 6*A116689(n+1). - R. J. Mathar, Jun 07 2016
E.g.f.: 3*exp(x)*(8 + 160*x +256*x^2 + 96*x^3 + 9*x^4)/4. - Stefano Spezia, Apr 18 2023
Sum_{n>=0} (-1)^n/a(n) = 28/15 - 8*Pi/(15*sqrt(3)) - 16*log(2)/15. - Amiram Eldar, Apr 30 2023

A329599 Numbers that are sums of consecutive dodecahedral numbers (A006566).

Original entry on oeis.org

0, 1, 20, 21, 84, 104, 105, 220, 304, 324, 325, 455, 675, 759, 779, 780, 816, 1271, 1330, 1491, 1575, 1595, 1596, 2024, 2146, 2601, 2821, 2905, 2925, 2926, 3354, 4060, 4170, 4625, 4845, 4929, 4949, 4950, 5456, 6279, 6985, 7095, 7140, 7550, 7770, 7854, 7874, 7875, 9009, 9139, 9516

Offset: 1

Ilya Gutkovskiy, Nov 18 2019

nonn

Cf. A006566, A116689, A217843, A322468, A329072, A329597.

A323663 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is Sum_{j=1..n} binomial(j*k, k).

Original entry on oeis.org

1, 1, 3, 1, 7, 6, 1, 21, 22, 10, 1, 71, 105, 50, 15, 1, 253, 566, 325, 95, 21, 1, 925, 3256, 2386, 780, 161, 28, 1, 3433, 19489, 18760, 7231, 1596, 252, 36, 1, 12871, 119713, 154085, 71890, 17857, 2926, 372, 45, 1, 48621, 748342, 1303753, 747860, 214396, 38332, 4950, 525, 55

Offset: 1

Seiichi Manyama, Jan 23 2019

			Square array begins:
    1,   1,    1,     1,       1,        1, ...
    3,   7,   21,    71,     253,      925, ...
    6,  22,  105,   566,    3256,    19489, ...
   10,  50,  325,  2386,   18760,   154085, ...
   15,  95,  780,  7231,   71890,   747860, ...
   21, 161, 1596, 17857,  214396,  2695652, ...
   28, 252, 2926, 38332,  539028,  7941438, ...
   36, 372, 4950, 74292, 1197036, 20212950, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns 1-3 give A000217, A002412, A116689.
Rows 1-3 give A000012, A244174, A029848.
Main diagonal is A096131.
Cf. A060539.

cp's OEIS Frontend

A268685 a(n) = 3(n + 1)(n + 2)(3n + 1)(3n + 4)/4.

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A329599 Numbers that are sums of consecutive dodecahedral numbers (A006566).

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A323663 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is Sum_{j=1..n} binomial(j*k, k).

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cp's OEIS Frontend

A268685 a(n) = 3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4.

Views

Author

Keywords

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Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A329599 Numbers that are sums of consecutive dodecahedral numbers (A006566).

Views

Author

Keywords

Crossrefs

A323663 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is Sum_{j=1..n} binomial(j*k, k).

Views

Author

Keywords

Examples

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Crossrefs

A268685 a(n) = 3(n + 1)(n + 2)(3n + 1)(3n + 4)/4.