A033156 -id:A033156 - cp's OEIS frontend

A003314 Binary entropy function: a(1)=0; for n > 1, a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }.

0, 2, 5, 8, 12, 16, 20, 24, 29, 34, 39, 44, 49, 54, 59, 64, 70, 76, 82, 88, 94, 100, 106, 112, 118, 124, 130, 136, 142, 148, 154, 160, 167, 174, 181, 188, 195, 202, 209, 216, 223, 230, 237, 244, 251, 258, 265, 272, 279, 286, 293, 300, 307, 314, 321, 328, 335

Offset: 1

N. J. A. Sloane

Morris gives many other interesting properties of this function.

a(n) is a convex function of n. (See the Morris reference.)

			a(6) = 6 + min {1+12, 2+8, 5+5} = 6 + 10 = 16.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.4.9, Eq. (19). p. 374.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 1..32768 (terms 1..1000 from T. D. Noe)
Mareike Fischer, Extremal values of the Sackin balance index for rooted binary trees, arXiv:1801.10418 [q-bio.PE], 2018.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 4.
D. Chistikov, S. Iván, A. Lubiw, and J. Shallit, Fractional coverings, greedy coverings, and rectifier networks, arXiv preprint arXiv:1509.07588 [cs.CC], 2015-2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
D. Knuth, Letter to N. J. A. Sloane, date unknown
R. Morris, Some theorems on sorting, SIAM J. Appl. Math., 17 (1969), 1-6.

Cf. A054248, A070941, A123753.

a003314 n = a003314_list !! (n-1)
a003314_list = 0 : f [0] [2..] where
   f vs (w:ws) = y : f (y:vs) ws where
     y = w + minimum (zipWith (+) vs $ reverse vs)
-- Reinhard Zumkeller, Aug 13 2013

A003314 := proc(n) local i,j; option remember; if n<=2 then n elif n=3 then 5 else j := 10^10; for i from 1 to n-1 do if A003314(i)+A003314(n-i) < j then j := A003314(i)+A003314(n-i); fi; od; n+j; fi; end;

a[1] = 0; a[n_] := If[OddQ[n], n + a[(n-1)/2 + 1] + a[(n-1)/2], 2*(n/2 + a[n/2])];
Table[a[n], {n, 1, 57}] (* Jean-François Alcover, Oct 15 2012 *)
a[n_] := n + n IntegerLength[n, 2] - 2^IntegerLength[n, 2];
Table[a[n], {n, 1, 57}] (* Peter Luschny, Dec 02 2017 *)

PARI
```
a(n)=if(n<2,0,n+a(n\2)+a((n+1)\2))
```

a(n)=local(m);if(n<2,0,m=length(binary(n-1));n*m-2^m+n)

def A003314(n):
    return n*int(math.log(4*n,2))-2**int(math.log(2*n,2)) # Indranil Ghosh, Feb 03 2017

def A003314(n):
    s, i, z = n-1, n-1, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A003314(n) for n in range(1, 58)]) # Peter Luschny, Nov 30 2017

def A003314(n): return n*(1+(m:=(n-1).bit_length()))-(1<Chai Wah Wu, Mar 29 2023

a(1) = 0; a(n) = n + a([n/2]) + a(n-[n/2]). (See the Morris reference.)
a(n) = A001855(n)+n-1. - Michael Somos Feb 07 2004
a(n) = n + a(floor(n/2)) + a(ceiling(n/2)) = n*floor(log_2(4n))-2^floor(log_2(2n)) = A033156(n) - n = n*A070941(n) - A062383(n). - Henry Bottomley, Jul 03 2002
a(1) = 0 and for n>1: a(n) = a(n-1) + A070941(2*n-1). Also a(n) = A123753(n-1) - 1. - Reinhard Zumkeller, Oct 12 2006
a(n) = A123753(n-1) - 1. - Peter Luschny, Nov 30 2017

A123753 Partial sums of A070941.

Original entry on oeis.org

1, 3, 6, 9, 13, 17, 21, 25, 30, 35, 40, 45, 50, 55, 60, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343

Offset: 0

Reinhard Zumkeller, Oct 12 2006

nonn

Peter Luschny, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47.

Cf. A001855, A003314, A033156, A054248, A061168, A083652, A097383, A295508.

A123753 := proc(n) local i, J, z; i := n+1: J := i; i := i-1; z := 1;
while 0 <= i do J := J+i; i := i-z; z := z+z od; J end:
seq(A123753(n), n=0..57); # Peter Luschny, Nov 30 2017
# Alternatively:
a := n -> (n+1)*(1 + ilog2(2*n+3)) - 2^ilog2(2*n+3) + 1:
seq(a(n), n=0..57); # Peter Luschny, Dec 02 2017

a[n_] := (n + 1)(1 + IntegerLength[n + 1, 2]) - 2^IntegerLength[n + 1, 2] + 1;
Table[a[n], {n, 0, 57}] (* Peter Luschny, Dec 02 2017 *)

def A123753(n):
    s, i, z = n+1, n, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A123753(n) for n in range(0, 58)]) # Peter Luschny, Nov 30 2017

def A123753(n): return (n+1)*(1+(m:=n.bit_length()))-(1<Chai Wah Wu, Mar 29 2023

a(n) = A003314(n+1)+1. - Reinhard Zumkeller, Oct 12 2006

Let bil(n) = floor(log_2(n)) + 1 for n>0, bil(0) = 0 and b(n) = n + n*bil(n) - 2^bil(n) + 1 then a(n) = b(n+1). (This suggests that '0' be prepended to this sequence.) - Peter Luschny, Dec 02 2017

A275670 G.f. A(x,y) satisfies: A(x,y) = xy + A(x,xy)^2, with A(0,y) = 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 4, 0, 8, 1, 0, 16, 4, 0, 32, 14, 0, 64, 40, 0, 128, 108, 2, 0, 256, 272, 12, 0, 512, 664, 52, 0, 1024, 1568, 188, 0, 2048, 3632, 608, 1, 0, 4096, 8256, 1816, 12, 0, 8192, 18528, 5128, 76, 0, 16384, 41088, 13856, 360, 0, 32768, 90304, 36176, 1446, 0, 65536, 196864, 91856, 5192, 4, 0, 131072, 426368, 227968, 17192, 42, 0, 262144, 918016, 555040, 53504, 284, 0, 524288, 1966848, 1329696, 158588, 1496, 0, 1048576, 4195328, 3141632, 451824, 6704, 0, 2097152, 8914432, 7334208, 1245936, 26772, 6

Offset: 0

Paul D. Hanna, Aug 04 2016

Compare g.f. to G(x,y) = x*y + G(x*y,y)^2 with G(0,y) = 0, which generates triangle A138157.
Apparently, the g.f. of column n equals y^n*x^A033156(n) * P(n,x)/Q(n,x), where:
Q(n,x) = Product_{k=1..n} (1 - 2*x^k)^floor(n/k),
and P(n,x) is of degree A024916(n) - A033156(n).

			G.f.: A(x,y) = 1 + y*x + 2*y*x^2 + 4*y*x^3 + (y^2 + 8*y)*x^4 + (4*y^2 + 16*y)*x^5 + (14*y^2 + 32*y)*x^6 + (40*y^2 + 64*y)*x^7 + (2*y^3 + 108*y^2 + 128*y)*x^8 + (12*y^3 + 272*y^2 + 256*y)*x^9 + (52*y^3 + 664*y^2 + 512*y)*x^10 + (188*y^3 + 1568*y^2 + 1024*y)*x^11 + (y^4 + 608*y^3 + 3632*y^2 + 2048*y)*x^12 +...
such that A(x,y) = x*y + A(x,x*y)^2, with A(0,y) = 1; further,
A(x,y) = x*y + ( x^2*y + A(x,x^2*y)^2 )^2,
A(x,y) = x*y + ( x^2*y + ( x^3*y + A(x,x^3*y)^2 )^2 )^2, etc.
This table of coefficients in g.f. A(x,y) begins:
1;
0, 1;
0, 2;
0, 4;
0, 8, 1;
0, 16, 4;
0, 32, 14;
0, 64, 40;
0, 128, 108, 2;
0, 256, 272, 12;
0, 512, 664, 52;
0, 1024, 1568, 188;
0, 2048, 3632, 608, 1;
0, 4096, 8256, 1816, 12;
0, 8192, 18528, 5128, 76;
0, 16384, 41088, 13856, 360;
0, 32768, 90304, 36176, 1446;
0, 65536, 196864, 91856, 5192, 4;
0, 131072, 426368, 227968, 17192, 42;
0, 262144, 918016, 555040, 53504, 284;
0, 524288, 1966848, 1329696, 158588, 1496;
0, 1048576, 4195328, 3141632, 451824, 6704;
0, 2097152, 8914432, 7334208, 1245936, 26772, 6;
0, 4194304, 18876416, 16943680, 3342784, 98060, 80;
0, 8388608, 39848960, 38785536, 8761720, 335704, 636;
0, 16777216, 83890176, 88063616, 22508448, 1088496, 3844;
0, 33554432, 176166912, 198506624, 56822624, 3375096, 19492;
0, 67108864, 369106944, 444562432, 141270272, 10080760, 87184, 4;
0, 134217728, 771764224, 989807872, 346507120, 29167000, 354628, 80;
0, 268435456, 1610629120, 2192154880, 839762496, 82113648, 1338376, 812;
0, 536870912, 3355467776, 4831741952, 2013427136, 225746384, 4753320, 5916;
0, 1073741824, 6979354624, 10603063808, 4781027584, 607828752, 16052296, 35000;
0, 2147483648, 14495563776, 23174734336, 11254280416, 1606760304, 51954808, 178904, 1; ...
Row polynomials begin:
n=0: 1;
n=1: y;
n=2: 2*y;
n=3: 4*y;
n=4: 8*y + y^2;
n=5: 16*y + 4*y^2;
n=6: 32*y + 14*y^2;
n=7: 64*y + 40*y^2;
n=8: 128*y + 108*y^2 + 2*y^3;
n=9: 256*y + 272*y^2 + 12*y^3;
n=10: 512*y + 664*y^2 + 52*y^3;
n=11: 1024*y + 1568*y^2 + 188*y^3;
n=12: 2048*y + 3632*y^2 + 608*y^3 + y^4;
n=13: 4096*y + 8256*y^2 + 1816*y^3 + 12*y^4;
n=14: 8192*y + 18528*y^2 + 5128*y^3 + 76*y^4;
n=15: 16384*y + 41088*y^2 + 13856*y^3 + 360*y^4;
n=16: 32768*y + 90304*y^2 + 36176*y^3 + 1446*y^4;
n=17: 65536*y + 196864*y^2 + 91856*y^3 + 5192*y^4 + 4*y^5; ...
the first row in which y^m appears is given by n = A033156(m), where A033156 begins:
[1, 4, 8, 12, 17, 22, 27, 32, 38, 44, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 200, ...].
Generating functions of initial columns.
G.f. of column 0: 1
G.f. of column 1: y*x/(1-2*x).
G.f. of column 2: y^2*x^4/((1-2*x)^2*(1-2*x^2)).
G.f. of column 3: y^3*2*x^8/((1-2*x)^3*(1-2*x^2)*(1-2*x^3)).
G.f. of column 4: y^4*x^12*(1 + 4*x - 10*x^3)/((1-2*x)^4*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)).
G.f. of column 5: y^5*x^17*(4 + 2*x + 8*x^2 - 28*x^4)/((1-2*x)^5*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)*(1-2*x^5)).
G.f. of column 6: y^6*x^22*(6 + 8*x - 20*x^3 - 24*x^4 - 36*x^5 - 56*x^6 + 16*x^7 + 176*x^8 + 224*x^9 - 336*x^11)/((1-2*x)^6*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)).
G.f. of column 7: y^7*x^27*(4 + 24*x + 4*x^2 - 12*x^3 - 72*x^5 - 112*x^6 - 96*x^7 + 112*x^8 - 64*x^9 + 64*x^10 + 496*x^11 + 576*x^12 - 1056*x^14) / ((1-2*x)^7*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)).
G.f. of column 8: y^8*x^32*(1 + 24*x + 36*x^2 - 4*x^3 - 88*x^4 - 202*x^5 - 14*x^6 - 82*x^7 - 168*x^8 + 400*x^9 + 440*x^10 + 892*x^11 + 1292*x^12 - 660*x^13 - 800*x^14 - 688*x^15 - 1776*x^16 - 1136*x^17 - 4504*x^18 - 2672*x^19 + 4672*x^20 + 5664*x^21 + 12672*x^22 - 13728*x^24) / ((1-2*x)^8*(1-2*x^2)^4*(1-2*x^3)^2*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)).
G.f. of column 9: y^9*x^38*(8 + 60*x + 72*x^2 + 16*x^3 - 238*x^4 - 584*x^5 - 232*x^6 + 172*x^7 + 328*x^8 + 52*x^9 + 1012*x^10 + 2636*x^11 + 1464*x^12 + 520*x^13 - 2040*x^14 - 664*x^15 - 2360*x^16 - 8712*x^17 - 13008*x^18 - 3696*x^19 + 12080*x^20 + 15392*x^21 + 1456*x^22 - 11040*x^23 + 18112*x^24 + 37728*x^25 + 47040*x^26 - 34304*x^27 - 78144*x^28 - 73216*x^29 + 91520*x^31) / ((1-2*x)^9*(1-2*x^2)^4*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)).
G.f. of column 10: y^10*x^44*(28 + 96*x + 198*x^2 - 160*x^3 - 864*x^4 - 596*x^5 - 856*x^6 - 384*x^7 + 3652*x^8 + 4752*x^9 + 696*x^10 - 2972*x^11 + 3928*x^12 + 4848*x^13 - 8360*x^14 - 18768*x^15 - 11000*x^16 - 14184*x^17 - 9896*x^18 + 17184*x^19 + 23664*x^20 + 7904*x^21 + 34480*x^22 + 53472*x^23 + 54160*x^24 + 68160*x^25 + 10560*x^26 - 166208*x^27 - 203488*x^28 - 86720*x^29 - 23552*x^30 + 13632*x^31 + 67584*x^32 - 95232*x^33 - 232256*x^34 + 129536*x^35 + 677632*x^36 + 624000*x^37 + 355840*x^38 - 67584*x^39 - 988416*x^40 - 1464320*x^41 + 1244672*x^43) / ((1-2*x)^10*(1-2*x^2)^5*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)^2*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)*(1-2*x^10)).
...
The g.f. of column n, y^n * x^A033156(n) * P(n,x)/Q(n,x), appears to have the following denominator:
Q(n,x) = Product_{k=1..n} (1 - 2*x^k)^floor(n/k), with
P(n,x) being a polynomial of degree A024916(n) - A033156(n),
where A024916(n) = Sum_{k=1..n} k*floor(n/k).
...

Paul D. Hanna, Table of n, a(n) for n = 0..1606, for rows 0..122 of triangle in flattened form.

Cf. A274965 (row sums), A275691 (antidiagonal sums), A033156.

Cf. variant: A138157.

/* Print first N rows of this triangle: */ N=32;
{a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = A^2 + y*x^(n+1-k)); polcoeff(A, n)}
{for(n=0, N, for(k=0,n, if(k==0,print1(polcoeff(a(n)+y*O(y^n),k,y)", "), if(polcoeff(a(n)+y*O(y^n),k,y)==0,break,print1(polcoeff(a(n)+y*O(y^n),k,y),", "))));print(""))}

G.f. A(x,y) satisfies: 1 = ...(((((A(x,y) - x*y)^(1/2) - x^2*y)^(1/2) - x^3*y)^(1/2) - x^4*y)^(1/2) - x^5*y)^(1/2) -...- x^n*y)^(1/2) -..., an infinite series of nested square roots.

A340298 a(n) = a(floor(n/2)) + a(ceiling(n/2)) + n*floor(log_2(n)) for n >= 2, a(n) = n for n <= 1.

Original entry on oeis.org

0, 1, 4, 8, 16, 22, 28, 38, 56, 65, 74, 83, 92, 105, 118, 139, 176, 189, 202, 215, 228, 241, 254, 267, 280, 297, 314, 331, 348, 373, 398, 439, 512, 530, 548, 566, 584, 602, 620, 638, 656, 674, 692, 710, 728, 746, 764, 782, 800, 822, 844, 866, 888, 910, 932, 954

Offset: 0

Alois P. Heinz, Jan 03 2021

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A033156, A340257, A340301.

a:= proc(n) option remember; `if`(n<2, n, (h->
      a(h)+a(n-h)+n*ilog2(n))(iquo(n, 2)))
    end:
seq(a(n), n=0..55);

a(2^n) = A340257(n).

A295513 a(n) = n*bil(n) - 2^bil(n) where bil(0) = 0 and bil(n) = floor(log_2(n)) + 1 for n>0.

Original entry on oeis.org

-1, -1, 0, 2, 4, 7, 10, 13, 16, 20, 24, 28, 32, 36, 40, 44, 48, 53, 58, 63, 68, 73, 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278

Offset: 0

Peter Luschny, Dec 02 2017

sign

Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.

Cf. A001855, A003314, A033156, A054248, A061168, A083652, A097383, A123753, A295508.

A295513 := proc(n) local i, s, z; s := -1; i := n-1; z := 1;
while 0 <= i do s := s+i; i := i-z; z := z+z od; s end:
seq(A295513(n), n=0..57);

a[n_] := n IntegerLength[n, 2] - 2^IntegerLength[n, 2];
Table[a[n], {n, 0, 58}]

def A295513(n): return n*(m:=(n-1).bit_length())-(1<Chai Wah Wu, Mar 29 2023

A001855(n) = a(n) + 1.
A033156(n) = a(n) + 2n.
A003314(n) = a(n) + n.
A083652(n) = a(n+1) + 2.
A061168(n) = a(n+1) - n + 1.
A123753(n) = a(n+1) + n + 2.
A097383(n) = a(n+1) - div(n-1, 2).
A054248(n) = a(n) + n + rem(n, 2).

cp's OEIS Frontend

A003314 Binary entropy function: a(1)=0; for n > 1, a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }.

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A123753 Partial sums of A070941.

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A275670 G.f. A(x,y) satisfies: A(x,y) = x*y + A(x,x*y)^2, with A(0,y) = 1.

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A340298 a(n) = a(floor(n/2)) + a(ceiling(n/2)) + n*floor(log_2(n)) for n >= 2, a(n) = n for n <= 1.

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A295513 a(n) = n*bil(n) - 2^bil(n) where bil(0) = 0 and bil(n) = floor(log_2(n)) + 1 for n>0.

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A275670 G.f. A(x,y) satisfies: A(x,y) = xy + A(x,xy)^2, with A(0,y) = 1.