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a(n) = (d(n)-r(n))/2, where d = A026057 and r is the periodic sequence with fundamental period (0,0,1,0).
+20
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
FORMULA
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
a(n) = (d(n)-r(n))/5, where d = A026057 and r is the periodic sequence with fundamental period (1,0,3,1,0).
+20
0
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
Triangle T read by rows: dot product <r,r-1,...,1> * <s+1,s+2,...,r,1,2,...,s>.
+10
10
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
FORMULA
T(n, k) = n*(n^2 + 3*n*(1+k) + 2 - 3*k^2)/6 for n >= 0, 0 <= k <= n.
EXAMPLE
Triangle begins as:
1;
4, 5;
10, 13, 13;
20, 26, 28, 26;
35, 45, 50, 50, 45;
56, 71, 80, 83, 80, 71;
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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 (Magma) [(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 (Sage) [[(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 (GAP) 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
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