A026057 -id:A026057 - cp's OEIS frontend

A026058 a(n) = (d(n)-r(n))/2, where d = A026057 and r is the periodic sequence with fundamental period (0,0,1,0).

13, 25, 41, 63, 90, 123, 162, 209, 263, 325, 395, 475, 564, 663, 772, 893, 1025, 1169, 1325, 1495, 1678, 1875, 2086, 2313, 2555, 2813, 3087, 3379, 3688, 4015, 4360, 4725, 5109, 5513, 5937, 6383, 6850, 7339, 7850, 8385, 8943, 9525, 10131, 10763, 11420, 12103, 12812, 13549, 14313, 15105, 15925

Offset: 4

Clark Kimberling

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

Cf. A026057.

a(n) = - 0.125 - 0.125*( - 1)^n + 0.25*cos(n*Pi/2) + (n + 4)*(n^2 + 20*n + 39)/12. [Richard Choulet, Dec 13 2008]

G.f.: x^4*( 13-14*x+5*x^2+2*x^3-14*x^4+15*x^5-5*x^6 ) / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Jun 22 2013

A026059 a(n) = (d(n)-r(n))/5, where d = A026057 and r is the periodic sequence with fundamental period (1,0,3,1,0).

Original entry on oeis.org

5, 10, 16, 25, 36, 49, 65, 83, 105, 130, 158, 190, 225, 265, 309, 357, 410, 467, 530, 598, 671, 750, 834, 925, 1022, 1125, 1235, 1351, 1475, 1606, 1744, 1890, 2043, 2205, 2375, 2553, 2740, 2935, 3140, 3354, 3577, 3810, 4052, 4305, 4568, 4841, 5125, 5419, 5725, 6042, 6370, 6710, 7061, 7425, 7801, 8189, 8590

Offset: 4

Clark Kimberling

nonn

A094415 Triangle T read by rows: dot product * .

Original entry on oeis.org

1, 4, 5, 10, 13, 13, 20, 26, 28, 26, 35, 45, 50, 50, 45, 56, 71, 80, 83, 80, 71, 84, 105, 119, 126, 126, 119, 105, 120, 148, 168, 180, 184, 180, 168, 148, 165, 201, 228, 246, 255, 255, 246, 228, 201, 220, 265, 300, 325, 340, 345, 340, 325, 300, 265, 286, 341

Offset: 0

Ralf Stephan, May 02 2004

			Triangle begins as:
   1;
   4,  5;
  10, 13, 13;
  20, 26, 28, 26;
  35, 45, 50, 50, 45;
  56, 71, 80, 83, 80, 71;

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

Columns 0-6 are A000292, A008778, A026054, A026057, A026060, A026063, A026066.
Half-diagonal is A050410.
Row sums are A000537.
See also A094414, A088003.

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

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

seq(seq( (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 , k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

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

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

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

T(n, k) = n*(n^2 + 3*n*(1+k) + 2 - 3*k^2)/6 for n >= 0, 0 <= k <= n.

cp's OEIS Frontend

A026058 a(n) = (d(n)-r(n))/2, where d = A026057 and r is the periodic sequence with fundamental period (0,0,1,0).

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A026059 a(n) = (d(n)-r(n))/5, where d = A026057 and r is the periodic sequence with fundamental period (1,0,3,1,0).

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A094415 Triangle T read by rows: dot product * .

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