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

%I #28 Jan 25 2020 01:29:14

%S 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,

%T 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,

%U 100,182,238,224,148,65,17,2,1,10,47,138,282,420,462,372

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

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

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

%C Triangle of expansion of (1+x+2*x^2)*(1+x)^n. - _Philippe Deléham_, Mar 10 2013

%H T. D. Noe, <a href="/A187801/b187801.txt">Rows n = 2..50 of triangle, flattened</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/TriangleGraph.html">MathWorld: Triangle Graph</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/StarGraph.html">MathWorld: Star Graph</a>

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

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

%F Row sums: A000079(n+2) = (4, 8, 16, 32, 64, ...). - _Philippe Deléham_, Mar 10 2013

%e Triangle begins:

%e 1,1,2;

%e 1,2,3,2;

%e 1,3,5,5,2;

%e 1,4,8,10,7,2;

%e 1,5,12,18,17,9,2;

%e 1,6,17,30,35,26,11,2;

%e 1,7,23,47,65,61,37,13,2;

%e 1,8,30,70,112,126,98,50,15,2;

%e 1,9,38,100,182,238,224,148,65,17,2;

%e 1,10,47,138,282,420,462,372,213,82,19,2;

%e 1,11,57,185,420,702,882,834,585,295,101,21,2;

%e 1,12,68,242,605,1122,1584,1716,1419,880,396,122,23,2;

%e 1,13,80,310,847,1727,2706,3300,3135,2299,1276,518,145,25,2;

%e From _Philippe Deléham_, Mar 10 2013: (Start)

%e Row 2: 1+x+2*x^2

%e Row 3: (1+x+2*x^2)*(1+x) = 1+2*x+3*x^2+2*x^3

%e Row 4: (1+x+2*x^2)*(1+x)^2 = 1+3*x+5*x^2+5*x^3+2*x^4

%e 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

%e (End)

%t 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 *)

%Y Cf. A187801, A095660, A107232, A029635, A095660.

%K nonn,tabf

%O 2,3

%A _Jakub Jaroslaw Ciaston_, Jan 06 2013