A004466 -id:A004466 - cp's OEIS frontend

A063521 a(n) = n(7n^2-4)/3.

0, 1, 16, 59, 144, 285, 496, 791, 1184, 1689, 2320, 3091, 4016, 5109, 6384, 7855, 9536, 11441, 13584, 15979, 18640, 21581, 24816, 28359, 32224, 36425, 40976, 45891, 51184, 56869, 62960, 69471, 76416, 83809, 91664, 99995, 108816, 118141, 127984, 138359, 149280, 160761

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(14*n^3-8*n), n>0: structured heptagonal anti-diamond numbers (vertex structure 15) (Cf. A100186 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

A063521:=n->n*(7*n^2-4)/3; seq(A063521(k), k=0..100); # Wesley Ivan Hurt, Oct 24 2013

lst={};Do[AppendTo[lst, n*(7*n^2-4)/3], {n, 1, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[x*(1+12*x+x^2)/(1-x)^4, {x, 0, 50}], x] (* G. C. Greubel, Sep 01 2017 *)

a(n) = { n*(7*n^2 - 4)/3 } \\ Harry J. Smith, Aug 25 2009

G.f.: x*(1+12*x+x^2)/(1-x)^4. - Colin Barker, Jan 10 2012

E.g.f.: (x/3)*(3 + 21*x + 7*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A004126 a(n) = n(7n^2 - 1)/6.

Original entry on oeis.org

0, 1, 9, 31, 74, 145, 251, 399, 596, 849, 1165, 1551, 2014, 2561, 3199, 3935, 4776, 5729, 6801, 7999, 9330, 10801, 12419, 14191, 16124, 18225, 20501, 22959, 25606, 28449, 31495, 34751, 38224, 41921, 45849, 50015, 54426, 59089, 64011

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers.
Sum of n triangular numbers starting from T(n), where T = A000217. E.g., a(4) = T(4) + T(5) + T(6) + T(7) = 10 + 15 + 21 + 28 = 74. - Amarnath Murthy, Jul 16 2004
Also as a(n) = (1/6)*(7*n^3-n), n>0: structured heptagonal diamond numbers (vertex structure 8). Cf. A100179 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Partial sums of A069099, centered heptagonal numbers (A000566). - Jonathan Vos Post, Mar 16 2006
Binomial transform of (0, 1, 7, 7, 0, 0, 0, ...) and third partial sum of (0, 1, 6, 7, 7, 7, ...). - Gary W. Adamson, Oct 05 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A000217, A000566, A016993, A069099.
Cf. A000447, A000292.

[n*(7*n^2-1)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

seq(binomial(2*n+1,3)-binomial(n+1,3), n=0..38); # Zerinvary Lajos, Jan 21 2007

Table[n (7 n^2 - 1)/6, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

makelist(n*(7*n^2-1)/6,n,0,30); /* Martin Ettl, Jan 08 2013 */

vector(100, n, n--; n*(7*n^2 - 1)/6) \\ Altug Alkan, Oct 06 2015

a(n) = C(2*n+1,3)-C(n+1,3), n>=0. - Zerinvary Lajos, Jan 21 2007
a(n) = A000447(n) - A000292(n). - Zerinvary Lajos, Jan 21 2007
G.f.: x*(1+5*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
E.g.f.: (x/6)*(7*x^2 + 21*x + 6)*exp(x). - G. C. Greubel, Oct 05 2015
a(n) = Sum_{i = n..2*n-1} A000217(i). - Bruno Berselli, Sep 06 2017
a(n) = n^3 + Sum_{k=0..n-1} k*(k+1)/2. Alternately, a(n) = A000578(n) + A000292(n-1) for n>0. - Bruno Berselli, May 23 2018

A063522 a(n) = n(5n^2 - 3)/2.

Original entry on oeis.org

0, 1, 17, 63, 154, 305, 531, 847, 1268, 1809, 2485, 3311, 4302, 5473, 6839, 8415, 10216, 12257, 14553, 17119, 19970, 23121, 26587, 30383, 34524, 39025, 43901, 49167, 54838, 60929, 67455, 74431, 81872, 89793, 98209, 107135, 116586, 126577, 137123, 148239, 159940

Offset: 0

N. J. A. Sloane, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

(1/12)*t*(n^3 - n) + n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Bisections: A160674, A160699.

[n*(5*n^2 -3)/2: n in [0..30]]; // G. C. Greubel, May 02 2018

lst={};Do[AppendTo[lst, LegendreP[3, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[x*(1 + 13*x + x^2)/(1-x)^4, {x, 0, 50}], x] (* G. C. Greubel, Sep 01 2017 *)
LinearRecurrence[{4,-6,4,-1},{0,1,17,63},40] (* Harvey P. Dale, Sep 06 2023 *)

a(n) = { n*(5*n^2 - 3)/2 } \\ Harry J. Smith, Aug 25 2009

G.f.: x*(1 + 13*x + x^2)/(1-x)^4. - Colin Barker, Jan 10 2012

E.g.f.: (x/2)*(2 + 15*x + 5*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A004467 a(n) = n(11n^2 - 5)/6.

Original entry on oeis.org

0, 1, 13, 47, 114, 225, 391, 623, 932, 1329, 1825, 2431, 3158, 4017, 5019, 6175, 7496, 8993, 10677, 12559, 14650, 16961, 19503, 22287, 25324, 28625, 32201, 36063, 40222, 44689, 49475, 54591, 60048

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers, that is: centered hendecagonal pyramidal numbers (see Deza paper in References).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

[n*(11*n^2-5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

Table[n(11n^2-5)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,13,47},80] (* Harvey P. Dale, Sep 22 2013 *)

a(n)=n*(11*n^2-5)/6 \\ Charles R Greathouse IV, Sep 28 2011

G.f.: x*(1+9*x+x^2)/(1-x)^4. - Colin Barker, Jan 08 2012
a(0)=0, a(1)=1, a(2)=13, a(3)=47; for n>3, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2013
E.g.f.: (x/6)*(6 + 33*x + 11*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A062025 a(n) = n(13n^2 - 7)/6.

Original entry on oeis.org

0, 1, 15, 55, 134, 265, 461, 735, 1100, 1569, 2155, 2871, 3730, 4745, 5929, 7295, 8856, 10625, 12615, 14839, 17310, 20041, 23045, 26335, 29924, 33825, 38051, 42615, 47530, 52809, 58465, 64511, 70960, 77825, 85119, 92855, 101046, 109705, 118845, 128479, 138620, 149281

Offset: 0

N. J. A. Sloane, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Table[n(13n^2-7)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n) = n*(13*n^2 - 7)/6 \\ Harry J. Smith, Jul 29 2009

From G. C. Greubel, Sep 01 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (x + 11*x^2 + x^3)/(1 - x)^4.
E.g.f.: (x/6)*(6 + 39*x + 13*x^2)*exp(x). (End)

A063523 a(n) = n(8n^2 - 5)/3.

Original entry on oeis.org

0, 1, 18, 67, 164, 325, 566, 903, 1352, 1929, 2650, 3531, 4588, 5837, 7294, 8975, 10896, 13073, 15522, 18259, 21300, 24661, 28358, 32407, 36824, 41625, 46826, 52443, 58492, 64989, 71950, 79391, 87328, 95777, 104754, 114275, 124356, 135013, 146262, 158119, 170600

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(16*n^3-10*n), n>0: structured octagonal anti-diamond numbers (vertex structure 17) (Cf. A100187 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Table[n(8n^2-5)/3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,67},81] (* or *) CoefficientList[ Series[ (x+14 x^2+x^3)/(x-1)^4,{x,0,80}],x] (* Harvey P. Dale, Jul 11 2011 *)

a(n) = n*(8*n^2 - 5)/3 \\ Harry J. Smith, Aug 25 2009

a(0)=0, a(1)=1, a(2)=18, a(3)=67, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 11 2011
G.f.: (x+14*x^2+x^3)/(x-1)^4. - Harvey P. Dale, Jul 11 2011
E.g.f.: (x/3)*(3 + 24*x + 8*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A100188 Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.

Original entry on oeis.org

1, 6, 27, 84, 205, 426, 791, 1352, 2169, 3310, 4851, 6876, 9477, 12754, 16815, 21776, 27761, 34902, 43339, 53220, 64701, 77946, 93127, 110424, 130025, 152126, 176931, 204652, 235509, 269730, 307551, 349216

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

			There are no 1- or 2-gonal anti-diamonds, so 1 and (2n+2) are the first and second terms since all the sequences begin as such.

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

Cf. A000578, A000447, A004466, A007588, A063521, A062523 - "polar" structured anti-diamonds; A100189 - "equatorial" structured meta-anti-diamond numbers; A006484 for other structured meta numbers; and A100145 for more on structured numbers.

[(1/6)*(2*n^4-2*n^2+6*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Table[(2n^4-2n^2+6n)/6,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {1,6,27,84,205},40] (* Harvey P. Dale, May 11 2016 *)

vector(40, n, (n^4 -n^2 +3*n)/3) \\ G. C. Greubel, Nov 08 2018

a(n) = (1/6)*(2*n^4 - 2*n^2 + 6*n).
G.f.: x*(1 + x + 7*x^2 - x^3)/(1-x)^5. - Colin Barker, Apr 16 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(1)=1, a(2)=6, a(3)=27, a(4)=84, a(5)=205. - Harvey P. Dale, May 11 2016
E.g.f.: (3*x + 6*x^2 + 6*x^3 + x^4)*exp(x)/3. - G. C. Greubel, Nov 08 2018

A051673 Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.

Original entry on oeis.org

0, 1, 12, 47, 120, 245, 436, 707, 1072, 1545, 2140, 2871, 3752, 4797, 6020, 7435, 9056, 10897, 12972, 15295, 17880, 20741, 23892, 27347, 31120, 35225, 39676, 44487, 49672, 55245, 61220, 67611, 74432, 81697, 89420, 97615, 106296, 115477, 125172

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Also as a(n) = (1/6)*(14*n^3 - 12*n^2 + 4*n), n>0: structured cubeoctahedral numbers (vertex structure 7); and structured pentagonal anti-diamond numbers (vertex structure 7) (cf. A004466 = alternate vertex) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = binomial transform of [1, 11, 24, 14, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009
This is prime for a(3) = 47. The subsequence of semiprimes begins: 707, 7435, 10897, 20741, 115477, 341797, 825091, 897097, no more through a(100). - Jonathan Vos Post, May 27 2010

			a(51) = 51*(51*(7*51-6)+2)/3 = 304351 = 17 * 17903 is semiprime. - _Jonathan Vos Post_, May 27 2010

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

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

Cf. A002378, A004466, A005915, A051662, A100145, A100188.

Cf. A214659, A214661.

[n*(n*(7*n-6)+2)/3: n in [0..50]]; // Vincenzo Librandi, May 12 2011

A051673:=n->n*(n*(7*n-6)+2)/3; seq(A051673(n), n=0..40); # Wesley Ivan Hurt, Feb 02 2014

Table[n^3+4Sum[i^2,{i,0,n-1}],{n,0,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,1,12,47},40] (* Harvey P. Dale, Jul 22 2011 *)

a(n)=n*(n*(7*n-6)+2)/3 \\ Charles R Greathouse IV, Oct 07 2015

[n*(7*n^2-6*n+2)/3 for n in range(51)] # G. C. Greubel, Mar 10 2024

a(n) = n*(n*(7*n-6) + 2)/3.
G.f.: x*(1+8*x+5*x^2)/(1-x)^4. - Bruno Berselli, May 12 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=12, a(3)=47. - Harvey P. Dale, Jul 22 2011
From Reinhard Zumkeller, Jul 25 2012: (Start)
a(n) = A214659(n) - A002378(n).
a(n) = Sum_{k=1..n} A214661(n, k), for n > 0 (row sums). (End)
E.g.f.: (x/3)*(3 + 15*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 10 2024

Corrected by T. D. Noe, Nov 01 2006, Nov 08 2006

A166340 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)p(n, x)/x, p(n, x) = xD( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x(1+x)/(1-x)^3, and p(3, x) = x(1+8*x+x^2)/(1-x)^4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 19, 19, 1, 1, 42, 114, 42, 1, 1, 89, 510, 510, 89, 1, 1, 184, 1975, 4080, 1975, 184, 1, 1, 375, 7029, 26195, 26195, 7029, 375, 1, 1, 758, 23712, 146954, 261950, 146954, 23712, 758, 1, 1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 1

Offset: 1

Roger L. Bagula, Oct 12 2009

			Triangle begins as:
  1;
  1,    1;
  1,    8,     1;
  1,   19,    19,      1;
  1,   42,   114,     42,       1;
  1,   89,   510,    510,      89,       1;
  1,  184,  1975,   4080,    1975,     184,      1;
  1,  375,  7029,  26195,   26195,    7029,    375,     1;
  1,  758, 23712, 146954,  261950,  146954,  23712,   758,    1;
  1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 1;

Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A166341, A166343, A166344, A166345, A166346, A166349.

Cf. A004466, A123125.

(* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+8*x+x^2)/(1-x)^4;
p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
(* Second program *)
b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
t[n_, k_, m_]:= t[n, k]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
T[n_, k_, m_]:= T[n, k, m]= If[k==1, 1, t[n-1, k, m] - t[n-1, k-1, m]];
Table[T[n, k, 2], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)

def b(n,k,m):
    if (n<2): return 1
    elif (k==0): return 0
    else: return k^(n-1)*((m+3)*k^2 - m)/3
@CachedFunction
def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
def A166340(n,k): return 1 if (k==1) else t(n-1,k,2) - t(n-1,k-1,2)
flatten([[A166340(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4.
From G. C. Greubel, Mar 11 2022: (Start)
T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A004466(k), b(n, 0) = 1, and b(1, k) = 1.
T(n, n-k) = T(n, k). (End)

Edited by G. C. Greubel, Mar 11 2022

A264853 a(n) = n(n + 1)(5n^2 + 5n - 4)/12.

Original entry on oeis.org

0, 1, 13, 56, 160, 365, 721, 1288, 2136, 3345, 5005, 7216, 10088, 13741, 18305, 23920, 30736, 38913, 48621, 60040, 73360, 88781, 106513, 126776, 149800, 175825, 205101, 237888, 274456, 315085, 360065, 409696, 464288, 524161, 589645, 661080, 738816, 823213, 914641

Offset: 0

Ilya Gutkovskiy, Nov 26 2015

Partial sums of centered 10-gonal (or decagonal) pyramidal numbers.

Subsequence of A204221. In fact, a(n) is of the form (k^2-1)/15 for k = 5*n*(n+1)/2-1. - Bruno Berselli, Nov 27 2015

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004466 (first differences), A201106 (partial sums), A204221.

Cf. similar sequences listed in A264854.

[n*(n+1)*(5*n^2+5*n-4)/12: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Table[n (n + 1) (5 n^2 + 5 n - 4)/12, {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,13,56,160},40] (* Harvey P. Dale, Aug 14 2017 *)

a(n)=n*(n+1)*(5*n^2+5*n-4)/12 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 + 8*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A004466(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

cp's OEIS Frontend

A063521 a(n) = n*(7*n^2-4)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A004126 a(n) = n*(7*n^2 - 1)/6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Formula

A063522 a(n) = n*(5*n^2 - 3)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A004467 a(n) = n*(11*n^2 - 5)/6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A062025 a(n) = n*(13*n^2 - 7)/6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A063523 a(n) = n*(8*n^2 - 5)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A100188 Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.

Views

Author

Keywords

Examples

Links

Crossrefs

A063521 a(n) = n(7n^2-4)/3.

A004126 a(n) = n(7n^2 - 1)/6.

A063522 a(n) = n(5n^2 - 3)/2.

A004467 a(n) = n(11n^2 - 5)/6.

A062025 a(n) = n(13n^2 - 7)/6.

A063523 a(n) = n(8n^2 - 5)/3.

A166340 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)p(n, x)/x, p(n, x) = xD( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x(1+x)/(1-x)^3, and p(3, x) = x(1+8*x+x^2)/(1-x)^4, read by rows.

A264853 a(n) = n(n + 1)(5n^2 + 5n - 4)/12.