A059673 -id:A059673 - cp's OEIS frontend

A164874 Triangle read by rows: T(1,1)=2; T(n,k)=2*T(n-1,k)+1, 1<=k

2, 5, 6, 11, 13, 14, 23, 27, 29, 30, 47, 55, 59, 61, 62, 95, 111, 119, 123, 125, 126, 191, 223, 239, 247, 251, 253, 254, 383, 447, 479, 495, 503, 507, 509, 510, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 2046

Offset: 1

Reinhard Zumkeller, Aug 29 2009

All terms contain exactly 1 zero in binary representation.

			Initial rows:
   1:                             2
   2:                        5        6
   3:                  11        13        14
   4:             23        27       29        30
   5:        47        55        59        61        62
   6:    95       111       119      123       125       126
also in binary representation:
                                 10
                            101       110
                      1011      1101      1110
                 10111     11011     11101     11110
           101111    110111    111011    111101    111110
      1011111   1101111   1110111   1111011   1111101   1111110 .

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A030130, A023416, A059673, A055010, A086224, A036563, A000918.

a164874 n k = a164874_tabl !! (n-1) !! (k-1)
a164874_row n = a164874_tabl !! (n-1)
a164874_tabl = map reverse $ iterate f [2] where
   f xs@(x:_) = (2 * x + 2) : map ((+ 1) . (* 2)) xs
-- Reinhard Zumkeller, Mar 31 2015

A164874row[n_] := 2^(n + 1) - 1 - BitShiftRight[2^n, Range[n]];
Array[A164874row, 10] (* Paolo Xausa, Jun 13 2025 *)

from math import isqrt
def A164874(n): return (1<<(a:=(isqrt(n<<3)+1>>1)+1))-(1<<(a*(a-1)>>1)-n)-1 # Chai Wah Wu, May 21 2025

T(n,k) = 2^(n+1) - 2^(n-k) - 1, 1 <= k <= n.
T(n,k) = A030130(n*(n-1)/2 + k + 1);
A023416(T(n,k)) = 1, 1<=k<=n;
A059673(n) = sum of n-th row;
T(n,1) = A055010(n);
T(n,2) = A086224(n-2) for n > 1;
T(n,n-1) = A036563(n+1) for n > 1;
T(n,n) = A000918(n+1).

A059937 Sum of binary numbers with n 1's and two (possibly leading) 0's.

Original entry on oeis.org

0, 7, 45, 186, 630, 1905, 5355, 14308, 36828, 92115, 225225, 540606, 1277874, 2981797, 6881175, 15728520, 35651448, 80215911, 179306325, 398458690, 880803630, 1937768217, 4244635395, 9261022956, 20132658900, 43620761275

Offset: 0

Henry Bottomley, Feb 13 2001

			a(2) = 45 since binary sum of 1100 + 1010 + 1001 + 0110 + 0101 + 0011 is 12 + 10 + 9 + 6 + 5 + 3 = 45.

Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A059672, A059673, A059938.

concat(0, Vec(x*(12*x^2-18*x+7)/((x-1)^3*(2*x-1)^3) + O(x^100))) \\ Colin Barker, Sep 13 2014

a(n) = (2^(n+2) - 1)*n*(n+1)/2 = A059672(n) + A059938(n) = a(n-1)*2*(n+1)/(n-1) + n(n+1)/2.

G.f.: x*(12*x^2-18*x+7) / ((x-1)^3*(2*x-1)^3). - Colin Barker, Sep 13 2014

A059938 Sum of binary numbers with n 1's and two (non-leading) 0's.

Original entry on oeis.org

0, 4, 31, 141, 506, 1590, 4593, 12523, 32740, 82908, 204755, 495561, 1179582, 2768818, 6422437, 14745495, 33554312, 75759480, 169869159, 378535765, 838860610, 1849687854, 4060086041, 8875147011, 19327352556, 41943039700

Offset: 0

Henry Bottomley, Feb 13 2001

			a(2) = 1100_2 + 1010_2 + 1001_2 = 12 + 10 + 9 = 31.

Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A059672, A059673, A059937.

concat(0, Vec(x*(8*x^3-6*x^2-5*x+4)/((x-1)^3*(2*x-1)^3) + O(x^100))) \\ Colin Barker, Sep 14 2014

a(n) = n^2*2^(n+1) - n*(n-1)/2 = A059937(n) - A059672(n) = A059937(n-1) + 2^(n+1)*n*(n+1)/2.

G.f.: x*(8*x^3-6*x^2-5*x+4) / ((x-1)^3*(2*x-1)^3). - Colin Barker, Sep 14 2014

cp's OEIS Frontend

A164874 Triangle read by rows: T(1,1)=2; T(n,k)=2*T(n-1,k)+1, 1<=k

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A059937 Sum of binary numbers with n 1's and two (possibly leading) 0's.

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A059938 Sum of binary numbers with n 1's and two (non-leading) 0's.

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