A065057 -id:A065057 - cp's OEIS frontend

A253768 Partial sums of A098550.

1, 3, 6, 10, 19, 27, 42, 56, 61, 67, 92, 104, 139, 155, 162, 172, 193, 213, 240, 262, 301, 312, 325, 358, 384, 429, 457, 508, 540, 557, 575, 660, 684, 739, 773, 838, 874, 965, 995, 1044, 1082, 1145, 1164, 1206, 1301, 1345, 1402, 1442, 1511, 1561, 1584, 1632, 1747, 1799, 1874, 1920, 2001, 2057, 2144, 2206

Offset: 1

Omar E. Pol, Jan 27 2015

nonn

For the partial sums of EKG sequence see A065057.

David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015.

Cf. A065057, A098550, A250299, A253765, A253766.

f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
Nest[f, {1, 2, 3}, 68] // Accumulate (* Jean-François Alcover, Oct 04 2018, after Robert G. Wilson v *)

A065078 Triangle read by rows: a(n,m) = T[n,m,m] where T[i,j,k] is the 3-dimensional pyramid defined by T[n,m,0]=1 and T[i,j,k]=0 if j>i or k>j and T[i,j,k]=T[i-1,j,k]+T[i,j-1,k]+T[i,j,k-1].

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 10, 18, 1, 4, 22, 79, 162, 1, 5, 40, 220, 831, 1851, 1, 6, 65, 492, 2681, 10488, 24661, 1, 7, 98, 962, 6883, 37367, 149743, 365613, 1, 8, 140, 1715, 15318, 105731, 573051, 2336243, 5863881, 1, 9, 192, 2856, 30840, 258604, 1742770

Offset: 0

Wouter Meeussen, Nov 09 2001

Number of paths to T[n,m,m] counted from the bottom plane (or T[n,m,0]).

			[3,2,2] can be reached from 3*[1,1,0] + 3*[2,1,0] + 2*[2,2,0] + 1*[3,1,0] + 1*[3,2,0], so a(3,2) = 3 + 3 + 2 + 1 + 1 =10.
Triangle begins
1;
1, 1;
1, 2, 3;
1, 3, 10, 18;
1, 4, 22, 79, 162;

Wouter Meeussen, Further comments on this sequence

Last number in each row is A065058.

Cf. A005789, A065057.

T[0, 0, 0] := 1; T[x_, y_, z_] := 0 /; (x< y || y< z); T[u_, v_, 0] := 1; T[, 0, 0] := 1; T[x, y_, z_] := (T[x, y, z]= T[x-1, y, z]+T[x, y-1, z] +T[x, y, z-1]) /; (y<=x ||z<=y); Table[T[x, y, y], {x, 0, 10}, {y, 0, x}]

A256528 Partial sums of A254077.

Original entry on oeis.org

1, 3, 6, 10, 16, 24, 33, 43, 55, 60, 74, 89, 96, 114, 135, 151, 178, 198, 231, 255, 266, 292, 314, 327, 355, 394, 426, 456, 500, 525, 559, 594, 611, 651, 702, 738, 806, 848, 900, 945, 983, 1031, 1050, 1096, 1153, 1176, 1230, 1299, 1349, 1412, 1467, 1516, 1576, 1632, 1697, 1755, 1825, 1854, 1916, 2003, 2034, 2100, 2193, 2257

Offset: 1

Omar E. Pol, May 08 2015

nonn

First differs from A256529 at a(29).

Cf. A065057, A098550, A253768, A254077, A255582, A256529, A256974.

A256529 Partial sums of A255582.

Original entry on oeis.org

1, 3, 6, 10, 16, 24, 33, 43, 55, 60, 74, 89, 96, 114, 135, 151, 178, 198, 231, 255, 266, 292, 314, 327, 355, 394, 426, 456, 490, 515, 532, 567, 618, 658, 700, 738, 774, 793, 837, 894, 940, 985, 1008, 1056, 1125, 1175, 1229, 1281, 1339, 1395, 1424, 1473, 1560, 1623, 1683, 1760, 1822, 1877, 1908, 1973, 2066, 2136, 2202, 2266

Offset: 1

Omar E. Pol, May 08 2015

nonn

First differs from A256528 at a(29).

Cf. A065057, A098550, A253768, A254077, A255582, A256528, A256974.

cp's OEIS Frontend

A253768 Partial sums of A098550.

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A065078 Triangle read by rows: a(n,m) = T[n,m,m] where T[i,j,k] is the 3-dimensional pyramid defined by T[n,m,0]=1 and T[i,j,k]=0 if j>i or k>j and T[i,j,k]=T[i-1,j,k]+T[i,j-1,k]+T[i,j,k-1].

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A256528 Partial sums of A254077.

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A256529 Partial sums of A255582.

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