A187801 Pascal's triangle construction method applied to {1,1,2} as an initial term.

1, 1, 2, 1, 2, 3, 2, 1, 3, 5, 5, 2, 1, 4, 8, 10, 7, 2, 1, 5, 12, 18, 17, 9, 2, 1, 6, 17, 30, 35, 26, 11, 2, 1, 7, 23, 47, 65, 61, 37, 13, 2, 1, 8, 30, 70, 112, 126, 98, 50, 15, 2, 1, 9, 38, 100, 182, 238, 224, 148, 65, 17, 2, 1, 10, 47, 138, 282, 420, 462, 372

Offset

2,3

Comments

A neighborhood decomposition of triangle graph applied to each node gives three identical sequences (independent of start point) {1,3}.

For star graph (depend of start point) generated sequences are: one time {1,3} and three times {1,1,2}.

Triangle of expansion of (1+x+2*x^2)*(1+x)^n. - Philippe Deléham, Mar 10 2013

Links

T. D. Noe, Rows n = 2..50 of triangle, flattened

Eric W. Weisstein, MathWorld: Triangle Graph

Eric W. Weisstein, MathWorld: Star Graph

Formula

For the selection of the initial term: neighborhood decomposition of graph.

For sequence: Pascal's triangle construction method applied to selected initial term.

Row sums: A000079(n+2) = (4, 8, 16, 32, 64, ...). - Philippe Deléham, Mar 10 2013

Example

Triangle begins:
1,1,2;
1,2,3,2;
1,3,5,5,2;
1,4,8,10,7,2;
1,5,12,18,17,9,2;
1,6,17,30,35,26,11,2;
1,7,23,47,65,61,37,13,2;
1,8,30,70,112,126,98,50,15,2;
1,9,38,100,182,238,224,148,65,17,2;
1,10,47,138,282,420,462,372,213,82,19,2;
1,11,57,185,420,702,882,834,585,295,101,21,2;
1,12,68,242,605,1122,1584,1716,1419,880,396,122,23,2;
1,13,80,310,847,1727,2706,3300,3135,2299,1276,518,145,25,2;
From Philippe Deléham, Mar 10 2013: (Start)
Row 2: 1+x+2*x^2
Row 3: (1+x+2*x^2)*(1+x) = 1+2*x+3*x^2+2*x^3
Row 4: (1+x+2*x^2)*(1+x)^2 = 1+3*x+5*x^2+5*x^3+2*x^4
Row 5: (1+x+2*x^2)*(1+x)^3 = 1+4*x+8*x^2+10*x^3+7*x^4+2*x^5
(End)

Mathematica

c = {1, 1, 2}; Join[{c}, t = Table[c = Append[c, 0]; c = c + RotateRight[c], {9}]]; Flatten[t] (* T. D. Noe, Mar 11 2013 *)

Crossrefs

Cf. A187801, A095660, A107232, A029635, A095660.

Sequence in context: A109814 A133088 A059982 * A134388 A055095 A366770

Adjacent sequences: A187798 A187799 A187800 * A187802 A187803 A187804

Keyword

nonn,tabf

Author

Jakub Jaroslaw Ciaston, Jan 06 2013

Status

approved