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 A102363 Triangle read by rows, constructed by a Pascal-like rule with left edge = 2^k, right edge = 2^(k+1)-1 (k >= 0). 6
 1, 2, 3, 4, 5, 7, 8, 9, 12, 15, 16, 17, 21, 27, 31, 32, 33, 38, 48, 58, 63, 64, 65, 71, 86, 106, 121, 127, 128, 129, 136, 157, 192, 227, 248, 255, 256, 257, 265, 293, 349, 419, 475, 503, 511, 512, 513, 522, 558, 642, 768, 894, 978, 1014, 1023, 1024, 1025, 1035, 1080, 1200, 1410, 1662, 1872, 1992, 2037, 2047 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS First column right of center divided by 3 equals powers of 4. Right of left edge, sums of rows are divisible by 3. Apparently the number of terms per row plus the number of numbers in natural order skipped per row equals a power of 2. - David Williams, Jun 27 2009 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA G.f.: Sum_{n>=0} x^n * (1+x)^tr(n) = Sum_{n>=0} a(n)*x^n, where tr(n) = A002024(n+1) = floor(sqrt(2*n+1) + 1/2). - Paul D. Hanna, Feb 19 2016 G.f.: Sum_{n>=1} x^(n*(n-1)/2) * (1-x^n)/(1-x) * (1+x)^n = Sum_{n>=0} a(n)*x^n. - Paul D. Hanna, Feb 20 2016 a(n) = A007318(n-1) + a(n-1). - Jon Maiga, Dec 22 2018 EXAMPLE This triangle begins:                             1                          2     3                       4     5     7                    8     9    12    15                16    17    21    27    31             32    33    38    48    58    63          64    65    71    86   106   121   127      128   129   136   157   192   227   248   255   256   257   265   293   349   419   475   503   511 G.f. of this sequence in flattened form: A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 9*x^7 + 12*x^8 + 15*x^9 + 16*x^10 + 17*x^11 + 21*x^12 + 27*x^13 + 31*x^14 + 32*x^15 + ... such that A(x) = (1+x) + x*(1+x)^2 + x^2*(1+x)^2 + x^3*(1+x)^3 + x^4*(1+x)^3 + x^5*(1+x)^3 + x^6*(1+x)^4 + x^7*(1+x)^4 + x^8*(1+x)^4 + x^9*(1+x)^4 + x^10*(1+x)^5 + x^11*(1+x)^5 + x^12*(1+x)^5 + x^13*(1+x)^5 + x^14*(1+x)^5 + x^15*(1+x)^6 + ... MAPLE T:=proc(n, k) if k=0 then 2^n elif k=n then 2^(n+1)-1 else T(n-1, k)+T(n-1, k-1) fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 26 2005 MATHEMATICA t[n_, 0] := 2^n; t[n_, n_] := 2^(n+1)-1; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 15 2013 *) PROG (PARI) /* Print in flattened form: Sum_{n>=0} x^n*(1+x)^tr(n) */ {tr(n) = ceil( (sqrt(8*n+9)-1)/2 )} {a(n) = polcoeff( sum(m=0, n, x^m * (1+x +x*O(x^n))^tr(m) ), n)} for(n=0, 78, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 19 2016 CROSSREFS Cf. A000079, A053220 (row sums), A265939 (central terms). Sequence in context: A278181 A232566 A192649 * A201816 A155900 A274949 Adjacent sequences:  A102360 A102361 A102362 * A102364 A102365 A102366 KEYWORD nonn,tabl,easy AUTHOR David Williams, Mar 15 2005, Oct 05 2007 EXTENSIONS More terms from Emeric Deutsch, Mar 26 2005 STATUS approved

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Last modified January 17 17:07 EST 2019. Contains 319235 sequences. (Running on oeis4.)