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

Author

David Williams, Mar 15 2005, Oct 05 2007

Keywords

• nonn
• tabl
• easy

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

Examples

			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 + ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A000079, A053220 (row sums), A265939 (central terms).

Programs

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 *)

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

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

Extensions

More terms from Emeric Deutsch, Mar 26 2005