A090905
Left side of irregular triangle of natural numbers in which every row product is a multiple of the previous.
Original entry on oeis.org
1, 2, 3, 5, 9, 15, 27, 47, 87, 167, 327, 635, 1263, 2519, 5007, 10007, 19947, 39875, 79739, 159399, 318779, 637503, 1274999, 2549979, 5099903, 10199787, 20399535, 40799063, 81598083, 163196135, 326392259, 652784499, 1305568943, 2611137839
Offset: 1
The triangle goes as follows:
(1)
(2),
(3,4),
(5,6,7,8),
(9,10,11,12,13,14),
(15,16,17,18,19,20,21,22,23,24,25,26)...
-
a = {{1, 1}}; Do[k = Last@ a[[i - 1]]; While[!Divisible[Pochhammer[Total@ a[[i - 1]], k], Pochhammer @@ a[[i - 1]]], k++]; AppendTo[a, {Total@a[[i - 1]], k}], {i, 2, 17}]; a (* Michael De Vlieger, Dec 15 2016 *)
A090906
Row lengths of the irregular triangle defined in A090905.
Original entry on oeis.org
1, 1, 2, 4, 6, 12, 20, 40, 80, 160, 308, 628, 1256, 2488, 5000, 9940, 19928, 39864, 79660, 159380, 318724, 637496, 1274980, 2549924, 5099884, 10199748, 20399528, 40799020, 81598052, 163196124, 326392240, 652784444, 1305568896, 2611137796
Offset: 1
-
a = {{1, 1}}; Do[k = Last@ a[[i - 1]]; While[! Divisible[Pochhammer[Total@ a[[i - 1]], k], Pochhammer @@ a[[i - 1]]], k++]; AppendTo[a, {Total@a[[i - 1]], k}], {i, 2, 17}]; Last /@ a (* Michael De Vlieger, Dec 15 2016 *)
More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 05 2004
A090907
Ratio of products of successive rows of the irregular triangle defined in A090905.
Original entry on oeis.org
2, 6, 140, 1287, 2139552000, 2949442889323392, 322686644032484531917367528014184448000000
Offset: 1
a(1)=(2!/1!)*(0!/1!)
a(2)=(4!/2!)*(1!/2!)
a(3)=(8!/4!)*(2!/4!)
a(4)=(14!/8!)*(4!/8!)
a(5)=(26!/14!)*(8!/14!)
a(6)=(46!/26!)*(14!/26!)
For n>=6 we have a(n)= ((2*A006992(n))!/(2*A006992(n-1))!)*((2*A006992(n-2))!/(2*A006992(n-1))!), verified for 4<n<21
Edited by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 05 2004
A093910
Product of n-th row of irregular triangle defined in A093911.
Original entry on oeis.org
1, 6, 120, 5040, 5765760, 19275223968000, 13644281345408020027550269440000, 4402827357584746886229433170489943024971625310770489684257669120000000000
Offset: 1
a(3) = 120 because a(1) is the product of 1 successive numbers starting with 1 = 1, and a(2) is the product of 2 successive numbers (2,3) = 6 and a(3) is the product of 3 successive numbers (4,5,6) = 120. All the products have the property that a(n) = 0 (mod a(n - 1)). Thus a(3) = 120.
-
a = {{1, 1}, {2, 2}}; Do[k = Last@ a[[i - 1]]; While[! Divisible[Pochhammer[Total@ a[[i - 1]], k], Pochhammer @@ a[[i - 1]]], k++]; AppendTo[a, {Total@ a[[i - 1]], k}], {i, 3, 8}]; Pochhammer @@ # & /@ a (* Michael De Vlieger, Dec 15 2016 *)
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