A026049 -id:A026049 - cp's OEIS frontend

A026051 a(n) = (d(n)-r(n))/5, where d = A026049 and r is the periodic sequence with fundamental period (4,1,4,0,1).

23, 31, 40, 53, 68, 86, 109, 135, 167, 203, 244, 292, 345, 406, 473, 547, 630, 720, 820, 928, 1045, 1173, 1310, 1459, 1618, 1788, 1971, 2165, 2373, 2593, 2826, 3074, 3335, 3612, 3903, 4209, 4532, 4870, 5226, 5598, 5987, 6395, 6820, 7265, 7728, 8210, 8713, 9235, 9779, 10343, 10928, 11536

Offset: 7

Clark Kimberling

nonn

A152890 [From Richard Choulet, Dec 14 2008]

a(n)=(n + 7)*(2*n^2 + 13*n + 102)/30 - 1/5*(2 + ( - 1/5*5^(1/2) + 1)*cos(2*n*Pi/5) + (2/5*2^(1/2)*(5 - 5^(1/2))^(1/2))*sin(2*n*Pi/5) + (1/5*5^(1/2) + 1)*cos(4*n*Pi/5) + ( - 2/5*2^(1/2)*(5 + 5^(1/2))^(1/2))*sin(4*n*Pi/5)) [From Richard Choulet, Dec 14 2008]

A026050 a(n) = (d(n)-r(n))/2, where d = A026049 and r is the periodic sequence with fundamental period (1,0,0,1).

Original entry on oeis.org

59, 78, 102, 132, 170, 217, 273, 339, 417, 508, 612, 730, 864, 1015, 1183, 1369, 1575, 1802, 2050, 2320, 2614, 2933, 3277, 3647, 4045, 4472, 4928, 5414, 5932, 6483, 7067, 7685, 8339, 9030, 9758, 10524, 11330, 12177, 13065, 13995, 14969, 15988, 17052, 18162, 19320, 20527, 21783, 23089

Offset: 7

Clark Kimberling

nonn

Conjecture: G.f. x^7*( 59-158*x+203*x^2-202*x^3+145*x^4-45*x^5 ) / ( (x^2+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

A026051 a(n) = (d(n)-r(n))/5, where d = A026049 and r is the periodic sequence with fundamental period (4,1,4,0,1).

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A026050 a(n) = (d(n)-r(n))/2, where d = A026049 and r is the periodic sequence with fundamental period (1,0,0,1).

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

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