A102363 Triangle read by rows, constructed by a Pascal-like rule with left edge = 2^k, right edge = 2^(k+1)-1 (k >= 0).

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

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