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A209274 Table T(n,k) = n*(n+2^k-1)/2, n, k > 0 read by antidiagonals.

%I A209274 #26 Jan 05 2018 14:45:22
%S A209274 1,2,3,4,5,6,8,9,9,10,16,17,15,14,15,32,33,27,22,20,21,64,65,51,38,30,
%T A209274 27,28,128,129,99,70,50,39,35,36,256,257,195,134,90,63,49,44,45,512,
%U A209274 513,387,262,170,111,77,60,54,55,1024,1025,771,518,330,207,133,92,72,65,66
%N A209274 Table T(n,k) = n*(n+2^k-1)/2, n, k > 0  read by antidiagonals.
%C A209274 Column number 1 A000217 n*(n+1)/2,
%C A209274 column number 2 A000096 n*(n+3)/2,
%C A209274 column number 3 A055999 n*(n+7)/2,
%C A209274 column number 4 A056121 n*(n+15)/2,
%C A209274 column number 5 A132758 n*(n+31)/2.
%C A209274 Row number 1 A000079 2^k,
%C A209274 row number 2 A000051 2^k + 1.
%H A209274 Boris Putievskiy, <a href="/A209274/b209274.txt">Rows n = 1..140 of triangle, flattened</a>
%H A209274 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F A209274 a(n) = A002260(n)*(A002260(n)+2^A004736(n)-1)/2.
%F A209274 a(n) = i*(i+2^j-1)/2,
%F A209274 where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
%e A209274 The start of the sequence as table:
%e A209274   1....2...4...8...16...32...64...
%e A209274   3....5...9..17...33...65..129...
%e A209274   6....9..15..27...51...99..195...
%e A209274   10..14..22..38...70..134..262...
%e A209274   15..20..30..50...90..170..330...
%e A209274   21..27..39..63..111..207..399...
%e A209274   28..35..49..77..133..245..469...
%e A209274   . . .
%e A209274 The start of the sequence as triangle array read by rows:
%e A209274   1;
%e A209274   2,3;
%e A209274   4,5,6;
%e A209274   8,9,9,10;
%e A209274   16,17,15,14,15;
%e A209274   32,33,27,22,20,21;
%e A209274   64,65,51,38,30,27,28;
%e A209274   . . .
%e A209274 Row number r contains r numbers.
%t A209274 b[n_] := n - d[n]*(d[n] + 1)/2; c[n_] := (d[n]^2 + 3*d[n] + 4)/2 - n; d[n_] := Floor[(-1 + Sqrt[8*n - 7])/2]; a[n_] := b[n]*(b[n] + 2^c[n] - 1)/2; Table[a[n], {n, 1, 50}] (* _G. C. Greubel_, Jan 04 2018 *)
%o A209274 (Python)
%o A209274 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A209274 i=n-t*(t+1)/2
%o A209274 j=(t*t+3*t+4)/2-n
%o A209274 result = i*(i+2**j-1)/2
%o A209274 (PARI) a(n, k) = n*(n+2^k-1)/2
%o A209274 array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
%o A209274 /* Print initial 7 rows and 8 columns of table as follows */
%o A209274 array(7, 8) \\ _Felix Fröhlich_, Jan 05 2018
%Y A209274 Cf. A000217, A000096, A055999, A056121, A132758, A000079, A000051.
%K A209274 nonn,tabl
%O A209274 1,2
%A A209274 _Boris Putievskiy_, Jan 15 2013