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A115216
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"Correlation triangle" for 2^n.
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3
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1, 2, 2, 4, 5, 4, 8, 10, 10, 8, 16, 20, 21, 20, 16, 32, 40, 42, 42, 40, 32, 64, 80, 84, 85, 84, 80, 64, 128, 160, 168, 170, 170, 168, 160, 128, 256, 320, 336, 340, 341, 340, 336, 320, 256, 512, 640, 672, 680, 682, 682, 680, 672, 640, 512, 1024, 1280, 1344, 1360, 1364
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row sums are A102301. T(2n,n) gives A002450(n+1). Diagonal sums are A115217. Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the sequence 2^n.
When formated as a square array, this is the self-fussion matrix (as in Example and Mathematica sections) of the sequence (2^n); for interlacing zeros of associated characteristic polynomials, see A202868. [from Clark Kimberling, Dec 26 2011]
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FORMULA
| Number triangle T(n, k)=sum{j=0..n, [j<=k]*2^(k-j)[j<=n-k]*2^(n-k-j)}.
G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x^2*y)) (Christian G. Bower (bowerc(AT)usa.net), Jan 17 2006)
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EXAMPLE
| Triangle begins
1,
2, 2,
4, 5, 4,
8, 10, 10, 8,
16, 20, 21, 20, 16,
32, 40, 42, 42, 40, 32,
Northwest corner of square matrix:
1....2....4....8....16
2....5....10...20...40
4....10...21...42...85
8....20...41...85...170
16...40...84...170..341
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MATHEMATICA
| (* A115216 as a square matrix *)
s[k_] := 2^(k - 1);
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* -1+2^n *)
Table[m[n, n], {n, 1, 12}] (* A002450 *)
(* Clark Kimberling, Dec 26 2011 *)
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CROSSREFS
| Cf. A003983, A202678.
Sequence in context: A128900 A136099 A072454 * A122543 A118003 A159296
Adjacent sequences: A115213 A115214 A115215 * A115217 A115218 A115219
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Jan 16 2006
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