A026043 -id:A026043 - cp's OEIS frontend

A026045 a(n) = (d(n)-r(n))/5, where d = A026043 and r is the periodic sequence with fundamental period (0,2,3,0,0).

9, 13, 19, 28, 39, 53, 70, 91, 117, 147, 182, 222, 268, 321, 380, 446, 519, 600, 690, 788, 895, 1011, 1137, 1274, 1421, 1579, 1748, 1929, 2123, 2329, 2548, 2780, 3026, 3287, 3562, 3852, 4157, 4478, 4816, 5170, 5541, 5929, 6335, 6760, 7203, 7665, 8146, 8647, 9169, 9711, 10274, 10858, 11464

Offset: 1

Clark Kimberling

nonn

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

A152857 [From Richard Choulet, Dec 14 2008]

a(n)=(n + 5)*(2*n^2 + 11*n + 54)/30 - (1 + ( - 1/2 - 1/10*5^(1/2))*cos(2*n*Pi/5) + (1/10*(3*(5 - 5^(1/2))^(1/2) + 2*(5 + 5^(1/2))^(1/2))*2^(1/2))*sin(2*n*Pi/5) + (1/10*5^(1/2) - 1/2)*cos(4*n*Pi/5) + (1/10*(2*(5 - 5^(1/2))^(1/2) - 3*(5 + 5^(1/2))^(1/2))*2^(1/2))*sin(4*n*Pi/5))/5 [From Richard Choulet, Dec 14 2008]

G.f. x*( 9-14*x+7*x^2+x^3-x^4-8*x^5-6*x^7+14*x^6 ) / ( (x^4+x^3+x^2+x+1)*(x-1)^4 ). - R. J. Mathar, Jun 22 2013

A026044 a(n) = (d(n)-r(n))/2, where d = A026043 and r is the periodic sequence with fundamental period (1,1,0,0).

Original entry on oeis.org

22, 33, 49, 70, 97, 132, 176, 229, 292, 367, 455, 556, 671, 802, 950, 1115, 1298, 1501, 1725, 1970, 2237, 2528, 2844, 3185, 3552, 3947, 4371, 4824, 5307, 5822, 6370, 6951, 7566, 8217, 8905, 9630, 10393, 11196, 12040, 12925, 13852, 14823, 15839, 16900, 18007, 19162, 20366, 21619, 22922, 24277

Offset: 5

Clark Kimberling

nonn

Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-7,4,-1).

LinearRecurrence[{4,-7,8,-7,4,-1},{22,33,49,70,97,132},60] (* Harvey P. Dale, Apr 27 2019 *)

a(n)=(n + 5)*(2*n^2 + 11*n + 54)/12 - 1/4 - (1/4)*cos(Pi*n/2) - (1/4)*sin(Pi*n/2) [From Richard Choulet, Dec 13 2008]

G.f. x^5*( 22-55*x+71*x^2-71*x^3+50*x^4-15*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

A026045 a(n) = (d(n)-r(n))/5, where d = A026043 and r is the periodic sequence with fundamental period (0,2,3,0,0).

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A026044 a(n) = (d(n)-r(n))/2, where d = A026043 and r is the periodic sequence with fundamental period (1,1,0,0).

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

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