A229079 -id:A229079 - cp's OEIS frontend

A099721 a(n) = n^2(2n+1).

0, 3, 20, 63, 144, 275, 468, 735, 1088, 1539, 2100, 2783, 3600, 4563, 5684, 6975, 8448, 10115, 11988, 14079, 16400, 18963, 21780, 24863, 28224, 31875, 35828, 40095, 44688, 49619, 54900, 60543, 66560, 72963, 79764, 86975, 94608, 102675, 111188, 120159, 129600

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 07 2004

For a right triangle with sides of lengths 8*n^3 + 12*n^2 + 8*n + 2, 4*n^4 + 8*n^3 + 4*n^2, and 4*n^4 + 8*n^3 + 12*n^2 + 8*n + 2, dividing the area by the perimeter gives a(n). - J. M. Bergot, Jul 30 2013
This sequence is the difference between the centered icosahedral (or cuboctahedral) numbers (A005902(n)) and the centered octagonal pyramidal numbers (A000447(n+1)). - Peter M. Chema, Jan 09 2016
a(n) is the sum of the integers in the closed interval (n-1)*n to n*(n+1). - J. M. Bergot, Apr 19 2017

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

Row n=3 of A229079.
Cf. A000578, A001093, A011379, A015237, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568, A098547.
Cf. A000290, A000447, A002378, A005408, A005902, A024196.

[n^2*(2*n+1): n in [0..50]]; // Vincenzo Librandi, May 01 2011

A099721 := proc(n) n^2*(2*n+1) ; end proc:
seq(A099721(n),n=0..10) ;

a[n_]:=2*n^3+n^2; (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,3,20,63},40] (* Harvey P. Dale, Aug 19 2022 *)

a(n) = ceil(sum(i=n^2-(n-1), n^2+(n-1), if(!issquare(4*i+1), (2*i+1+sqrt(4*i+1))/2, 0))); \\ Michel Marcus, Nov 14 2014, after Richard R. Forberg

G.f.: x*(3 + 8*x + x^2)/(x-1)^4.
a(n) = A024196(n) - A024196(n-1). - Philippe Deléham, May 07 2012
a(n) = ceiling(Sum_{i=n^2-(n-1)..n^2+(n-1)} s(i)), for n > 0 and integer i, where s(i) are the real solutions to x = i + sqrt(x), and the summation range excludes the integer solutions which occur where i is an oblong number (A002378). The fractional portion of the summation converges to 2/3 for large n. If s(i) is replaced with i, then the summation equals n^2*(2*n-1) = A015237. - Richard R. Forberg, Oct 15 2014
a(n) = A005902(n) - A000447(n+1). - Peter M. Chema, Jan 09 2016
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + 4*log(2) - 4.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - Pi - 2*log(2) + 4. (End)
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: x*(1 + 2*x)*(3 + x)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A000290(n)*A005408(n). (End)

A066373 a(n) = (3n-2)2^(n-3).

Original entry on oeis.org

2, 7, 20, 52, 128, 304, 704, 1600, 3584, 7936, 17408, 37888, 81920, 176128, 376832, 802816, 1703936, 3604480, 7602176, 15990784, 33554432, 70254592, 146800640, 306184192, 637534208, 1325400064, 2751463424, 5704253440, 11811160064, 24427626496, 50465865728, 104152956928

Offset: 2

N. J. A. Sloane, Jan 04 2002

An elephant sequence, see A175654. For the corner squares 16 A[5] vectors, with decimal values between 59 and 440, lead to this sequence (with a leading 1 added). For the central square these vectors lead to the companion sequence A098156 (without a(1)). - Johannes W. Meijer, Aug 15 2010

a(n) is the total number of 1's in runs of 1's of length >= 2 over all binary words with n bits. - Félix Balado, Jan 15 2024

Harry J. Smith, Table of n, a(n) for n = 2..200
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Column k=2 of A229079.

Cf. A016789, A098156, A130129, A288834.

seq((3*n-2)*2^(n-3),n=2..30); # Emeric Deutsch, Jul 23 2006

Array[(3 # - 2)*2^(# - 3) &, 28, 2] (* or *)
Drop[CoefficientList[Series[x^2*(2 - x)/(1 - 2 x)^2, {x, 0, 29}], x], 2] (* Michael De Vlieger, Jun 30 2018 *)

a(n) = { (3*n - 2)*2^(n - 3) } /* Harry J. Smith, Feb 11 2010 */

G.f.: x^2*(2-x)/(1-2x)^2. - Emeric Deutsch, Jul 23 2006
a(n) = 2*a(n-1) +3*2^(n-3). - Vincenzo Librandi, Mar 20 2011
a(n+1) - a(n) = A098156(n). - R. J. Mathar, Apr 25 2013
From Paul Curtz, Jun 29 2018: (Start)
a(n) = A130129(n-2) - A130129(n-3) for n >= 2.
Binomial transform of A016789.
Inverse binomial transform of A288834.
Also the main diagonal of the difference table of m -> (-1)^m*(m+2).
    2,  -3,   4,  -5, ...
   -5,   7,  -9,  11, ...
   12, -16,  20, -24, ...
  -28,  36, -44,  52, ... . (End)

A229078 Number of ascending runs in {1,...,n}^n.

Original entry on oeis.org

0, 1, 7, 63, 736, 10625, 182736, 3647119, 82837504, 2109289329, 59500000000, 1841557146671, 62041198952448, 2259914256880657, 88499197217837056, 3707501605224609375, 165444235911082541056, 7834451891982365825441, 392371124973096027488256

Offset: 0

Alois P. Heinz, Sep 12 2013

nonn

			a(1) = 1: [1].
a(2) = 7 = 2+2+1+2: [1,1], [2,1], [1,2], [2,2].

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lambert W-Function
Wikipedia, Lambert W function

Main diagonal of A229079.

Cf. A062023 (nondescending runs), A066274.

a:= n-> `if`(n=0, 0, n^(n-1)*(n*(n+2)-1)/2):
seq(a(n), n=0..25);

a(n) = n^(n-1)*(n*(n+2)-1)/2 for n>0, a(0) = 0.
E.g.f.: 1/2*W(-x)*(W(-x)^3+W(-x)^2-W(-x)-2)/(1+W(-x))^3, W(x) Lambert's function (principal branch).
a(n) = A062023(n) + A066274(n) for n>0.

A229147 a(n) = n^4(3n+2).

Original entry on oeis.org

0, 5, 128, 891, 3584, 10625, 25920, 55223, 106496, 190269, 320000, 512435, 787968, 1171001, 1690304, 2379375, 3276800, 4426613, 5878656, 7688939, 9920000, 12641265, 15929408, 19868711, 24551424, 30078125, 36558080, 44109603, 52860416, 62948009, 74520000

Offset: 0

Alois P. Heinz, Sep 15 2013

Number of ascending runs in {1,...,n}^5.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row n=5 of A229079.

Cf. A000583, A016789.

a:= n-> n^4*(3*n+2):
seq(a(n), n=0..40);

Table[n^4 (3n+2),{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{0,5,128,891,3584,10625},40] (* Harvey P. Dale, Aug 14 2015 *)

G.f.: (x^4+58*x^3+198*x^2+98*x+5)*x/(x-1)^6.
a(0)=0, a(1)=5, a(2)=128, a(3)=891, a(4)=3584, a(5)=10625, a(n)= 6*a(n-1)- 15*a(n-2)+ 20*a(n-3)- 15*a(n-4)+ 6*a(n-5)- a(n-6). - Harvey P. Dale, Aug 14 2015
E.g.f.: exp(x)*x*(5 + 59*x + 87*x^2 + 32*x^3 + 3*x^4). - Stefano Spezia, Jul 17 2024

A229146 a(n) = n^3(5n+3)/2.

Original entry on oeis.org

0, 4, 52, 243, 736, 1750, 3564, 6517, 11008, 17496, 26500, 38599, 54432, 74698, 100156, 131625, 169984, 216172, 271188, 336091, 412000, 500094, 601612, 717853, 850176, 1000000, 1168804, 1358127, 1569568, 1804786, 2065500, 2353489, 2670592, 3018708, 3399796

Offset: 0

Alois P. Heinz, Sep 15 2013

Number of ascending runs in {1,...,n}^4.

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

Row n=4 of A229079.

a:= n-> n^3*(5*n+3)/2:
seq(a(n), n=0..40);

Table[n^3(5n+3)/2,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,4,52,243,736},40] (* Harvey P. Dale, Apr 29 2022 *)

G.f.: -(x^3+23*x^2+32*x+4)*x/(x-1)^5.

A229148 a(n) = n^5(7n+5)/2.

Original entry on oeis.org

0, 6, 304, 3159, 16896, 62500, 182736, 453789, 999424, 2007666, 3750000, 6603091, 11073024, 17822064, 27697936, 41765625, 61341696, 88031134, 123766704, 170850831, 232000000, 310391676, 409713744, 534216469, 688766976, 878906250, 1110908656, 1391843979

Offset: 0

Alois P. Heinz, Sep 15 2013

Number of ascending runs in {1,...,n}^6.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Row n=6 of A229079.

a:= n-> n^5*(7*n+5)/2:
seq(a(n), n=0..40);

Table[n^5(7*n+5)/2,{n,0,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,6,304,3159,16896,62500,182736},30] (* Harvey P. Dale, Apr 07 2018 *)

G.f.: -(x^5+137*x^4+957*x^3+1157*x^2+262*x+6)*x/(x-1)^7.

A229149 a(n) = n^6(4n+3).

Original entry on oeis.org

0, 7, 704, 10935, 77824, 359375, 1259712, 3647119, 9175040, 20726199, 43000000, 83263367, 152285184, 265474495, 444242624, 717609375, 1124073472, 1713767399, 2550916800, 3716624599, 5312000000, 7461652527, 10317571264, 14063409455, 18919194624, 25146484375

Offset: 0

Alois P. Heinz, Sep 15 2013

Number of ascending runs in {1,...,n}^7.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Row n=7 of A229079.

a:= n-> n^6*(4*n+3):
seq(a(n), n=0..40);

Table[n^6 (4n+3),{n,0,40}] (* Harvey P. Dale, Oct 10 2023 *)

G.f.: (x^6+312*x^5+4029*x^4+9664*x^3+5499*x^2+648*x+7)*x/(x-1)^8.

A229150 a(n) = n^7(9n+7)/2.

Original entry on oeis.org

0, 8, 1600, 37179, 352256, 2031250, 8538048, 28824005, 82837504, 210450636, 485000000, 1032820063, 2060328960, 3890408054, 7009998016, 12131015625, 20266876928, 32827093840, 51732592704, 79554584771, 119680000000, 176506677018, 255671683520, 364316322829

Offset: 0

Alois P. Heinz, Sep 15 2013

Number of ascending runs in {1,...,n}^8.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row n=8 of A229079.

a:= n-> n^7*(9*n+7)/2:
seq(a(n), n=0..40);

A229150[n_]:=n^7(9n+7)/2;Array[A229150,30,0] (* Paolo Xausa, Oct 24 2023 *)

G.f.: -(x^7+695*x^6+15570*x^5+65998*x^4+74573*x^3+23067*x^2+1528*x+8)*x/ (x-1)^9.

A229151 a(n) = n^8(5n+4).

Original entry on oeis.org

0, 9, 3584, 124659, 1572864, 11328125, 57106944, 224827239, 738197504, 2109289329, 5400000000, 12647173979, 27518828544, 56285419749, 109208390144, 202468359375, 360777252864, 620842412249, 1035876294144, 1681372741059, 2662400000000, 4122691670349

Offset: 0

Alois P. Heinz, Sep 15 2013

Number of ascending runs in {1,...,n}^9.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Row n=9 of A229079.

a:= n-> n^8*(5*n+4):
seq(a(n), n=0..40);

Table[n^8 (5n+4),{n,0,30}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,9,3584,124659,1572864,11328125,57106944,224827239,738197504,2109289329},30] (* Harvey P. Dale, Feb 11 2015 *)

G.f.: (x^8 +1526*x^7 +56856*x^6 +395866*x^5 +780950*x^4 +486474*x^3 +89224*x^2 +3494*x+9)*x / (x-1)^10.

a(0)=0, a(1)=9, a(2)=3584, a(3)=124659, a(4)=1572864, a(5)=11328125, a(6)=57106944, a(7)=224827239, a(8)=738197504, a(9)=2109289329, a(n)= 10*a(n-1)- 45*a(n-2)+120*a(n-3)-210*a(n-4)+252*a(n-5)-210*a(n-6)+120*a(n-7)- 45*a(n-8)+ 10*a(n-9)-a(n-10). - Harvey P. Dale, Feb 11 2015

A229152 a(n) = n^9(11n+9)/2.

Original entry on oeis.org

0, 10, 7936, 413343, 6946816, 62500000, 377913600, 1735205101, 6509559808, 20920706406, 59500000000, 153266599915, 363764514816, 805941952348, 1683875312896, 3344572265625, 6356551598080, 11621611896706, 20530186553088, 35172959057911, 58624000000000

Offset: 0

Alois P. Heinz, Sep 15 2013

Number of ascending runs in {1,...,n}^10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Row n=10 of A229079.

a:= n-> n^9*(11*n+9)/2:
seq(a(n), n=0..40);

A229152[n_]:=n^9(11n+9)/2;Array[A229152,30,0] (* Paolo Xausa, Oct 23 2023 *)

G.f.: -(x^9 +3317*x^8 +199643*x^7 +2172239*x^6 +6901145*x^5 +7512749*x^4 +2834873*x^3 +326597*x^2 +7826*x +10)*x / (x-1)^11.

cp's OEIS Frontend

A099721 a(n) = n^2*(2*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A066373 a(n) = (3*n-2)*2^(n-3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A229078 Number of ascending runs in {1,...,n}^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A229147 a(n) = n^4*(3*n+2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A229146 a(n) = n^3*(5*n+3)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A229148 a(n) = n^5*(7*n+5)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A229149 a(n) = n^6*(4*n+3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A099721 a(n) = n^2(2n+1).

A066373 a(n) = (3n-2)2^(n-3).

A229147 a(n) = n^4(3n+2).

A229146 a(n) = n^3(5n+3)/2.

A229148 a(n) = n^5(7n+5)/2.

A229149 a(n) = n^6(4n+3).

A229150 a(n) = n^7(9n+7)/2.

A229151 a(n) = n^8(5n+4).

A229152 a(n) = n^9(11n+9)/2.