A026040 -id:A026040 - cp's OEIS frontend

A026041 a(n) = d(n)/2, where d = A026040.

12, 20, 32, 49, 72, 102, 140, 187, 244, 312, 392, 485, 592, 714, 852, 1007, 1180, 1372, 1584, 1817, 2072, 2350, 2652, 2979, 3332, 3712, 4120, 4557, 5024, 5522, 6052, 6615, 7212, 7844, 8512, 9217, 9960, 10742, 11564, 12427, 13332, 14280, 15272, 16309, 17392, 18522, 19700, 20927, 22204, 23532, 24912, 26345

Offset: 4

Clark Kimberling

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

[n*(n^2-3*n+14)/6: n in [4..60]]; // Vincenzo Librandi, Oct 17 2013

CoefficientList[Series[-(7 x^3 - 24 x^2 + 28 x - 12)/(x - 1)^4, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 17 2013 *)

a(n)=n*(n^2-3*n+14)/6 \\ Charles R Greathouse IV, Oct 18 2022

For n>4, a(n) = a(n-1)+A022856(n+2). - Levi R. Self (levi.r.self(AT)gmail.com), Aug 04 2007
G.f.: -x^4*(7*x^3-24*x^2+28*x-12)/(x-1)^4. - Colin Barker, Oct 07 2012
a(n) = n*(n^2-3*n+14)/6. - Vincenzo Librandi, Oct 17 2013

A026042 a(n) = (d(n)-r(n))/5, where d = A026040 and r is the periodic sequence with fundamental period (4,0,4,3,4).

Original entry on oeis.org

4, 8, 12, 19, 28, 40, 56, 74, 97, 124, 156, 194, 236, 285, 340, 402, 472, 548, 633, 726, 828, 940, 1060, 1191, 1332, 1484, 1648, 1822, 2009, 2208, 2420, 2646, 2884, 3137, 3404, 3686, 3984, 4296, 4625, 4970, 5332, 5712, 6108, 6523, 6956, 7408, 7880, 8370, 8881, 9412, 9964, 10538, 11132, 11749

Offset: 1

Clark Kimberling

nonn

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1,-3,3,-1).

A152856 [From Richard Choulet, Dec 14 2008]

a(n)=(n + 4)*(n^2 + 5*n + 18)/15 - 3/5 - (1/2 - 3/10*5^(1/2))/5*cos(2*n*Pi/5) + (1/10*(4*(5 + 5^(1/2))^(1/2) - (5 - 5^(1/2))^(1/2))*2^(1/2))/5*sin(2*n*Pi/5) - (3/10*5^(1/2) + 1/2)/5*cos(4*n*Pi/5) + (1/10*((5 + 5^(1/2))^(1/2) + 4*(5 - 5^(1/2))^(1/2))*2^(1/2))/5*sin(4*n*Pi/5) [From Richard Choulet, Dec 14 2008]

G.f. x*( 4-4*x+3*x^3-x^4-3*x^5+5*x^6-2*x^7 ) / ( (x^4+x^3+x^2+x+1)*(x-1)^4 ). - R. J. Mathar, Jun 22 2013

A094414 Triangle T read by rows: dot product <1,2,...,r> * .

Original entry on oeis.org

1, 5, 4, 14, 11, 11, 30, 24, 22, 24, 55, 45, 40, 40, 45, 91, 76, 67, 64, 67, 76, 140, 119, 105, 98, 98, 105, 119, 204, 176, 156, 144, 140, 144, 156, 176, 285, 249, 222, 204, 195, 195, 204, 222, 249, 385, 340, 305, 280, 265, 260, 265, 280, 305, 340, 506, 451, 407, 374, 352, 341, 341, 352, 374, 407, 451

Offset: 0

Ralf Stephan, May 02 2004

Offset for r (the rows) is 1, for s (the columns) it is 0.

			Triangle begins as:
   1;
   5,  4;
  14, 11, 11;
  30, 24, 22, 24;
  55, 45, 40, 40, 45;
  91, 76, 67, 64, 67, 76;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Columns 0-6 are A000330, A006527, A026037, A026040, A026043, A026046, A026049.
Row sums are A000537.
See also A094415, A088003.

Flat(List([0..12], n-> List([0..n-1], k-> n*((n+1)*(2*n+1) -3*k*(n-k))/6 ))); # G. C. Greubel, Oct 30 2019

[n*((n+1)*(2*n+1) -3*k*(n-k))/6: k in [0..n-1], n in [0..12]]; // G. C. Greubel, Oct 30 2019

T:=proc(r,s) if s>=r then 0 else r*(2*r^2+3*r+1-3*r*s+3*s^2)/6 fi end: for r from 1 to 11 do seq(T(r,s),s=0..r-1) od; # yields sequence in triangular form # Emeric Deutsch, Nov 27 2006

Table[n*((n+1)*(2*n+1) -3*k*(n-k))/6, {n,0,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

T(n,k) = n*((n+1)*(2*n+1) -3*k*(n-k))/6;
for(n=0,12, for(k=0,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019

[[n*((n+1)*(2*n+1) -3*k*(n-k))/6 for k in (0..n-1)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

T(r, s) = r*(2*r^2 + 3*r - 3*r*s + 1 + 3*s^2)/6, r >= 1, 0 <= s <= r-1.

More terms from G. C. Greubel, Oct 30 2019

cp's OEIS Frontend

A026041 a(n) = d(n)/2, where d = A026040.

Views

Author

Keywords

Links

Programs

Magma

Mathematica

PARI

Formula

A026042 a(n) = (d(n)-r(n))/5, where d = A026040 and r is the periodic sequence with fundamental period (4,0,4,3,4).

Views

Author

Keywords

Links

Crossrefs

Formula

A094414 Triangle T read by rows: dot product <1,2,...,r> * .

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions