A190405 -id:A190405 - cp's OEIS frontend

A190404 Decimal expansion of (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers); based on row 1 of the natural number array, A000027.

8, 2, 0, 8, 1, 6, 2, 8, 0, 3, 2, 7, 5, 7, 6, 9, 3, 3, 1, 4, 6, 9, 2, 1, 3, 8, 5, 1, 1, 2, 7, 1, 4, 7, 1, 7, 1, 1, 3, 0, 3, 0, 7, 6, 8, 9, 7, 8, 3, 6, 9, 8, 7, 3, 9, 0, 2, 3, 2, 5, 8, 1, 1, 1, 9, 0, 0, 7, 2, 3, 0, 1, 8, 6, 6, 6, 7, 5, 8, 8, 7, 8, 0, 0, 1, 8, 2, 0, 8, 5, 8, 1, 1, 6, 7, 9, 5, 6, 6, 5, 4, 3, 0, 4, 4, 8, 6, 7, 6, 5, 8, 1, 7, 1, 8, 0, 9, 7, 3, 0

Offset: 0

Clark Kimberling, May 10 2011

Suppose that F={f(i,j): i>=1, j>=1} is an array of positive integers such that every positive integer occurs exactly once in F.
Let G=G(F) denote the array defined by g(i,j)=(1/2)^f(i,j);
R(i)=Sum_{j>=1} g(i,j); i-th row sum of G;
C(j)=Sum_{i>=1} g(i,j); j-th column sum of G;
U(j)=Sum_{i>=1} g(i,i+j-1); j-th upper diagonal sum of G;
L(i)=Sum_{j>=1} g(i+j,j); i-th lower diagonal sum of G;
R(odds)=Sum_{i>=1} R(2i-1); sum, odd numbered rows of G;
R(evens)=Sum_{i>=1} R(2i); sum, even numbered rows of G;
C(odds)=Sum_{j>=1} R(2j-1); sum, odd numbered cols of G;
C(evens)=Sum_{j>=1} R(2j); sum, even numbered cols of G;
UT=Sum_{j>=1} U(j); sum, upper triangular subarray of G;
LT=Sum_{i>=1} L(i); sum, lower triangular subarray of G.
...
Note that R(odds)+R(evens)=C(odds)+C(evens)=UT+LT=1.
...
For the natural number array F=A000027:
R(1)=0.820816280327576933146921385113... (A190404)
R(2)=0.160408140163788466573460692556...
R(3)=0.0177040700818942332867303462782...
R(4)=0.00103953504094711664336517313909...
R(5)=0.0000314862704735583216825865695447...
...
R(odds)=0.838551840434481240061632331355800... (A190408)
R(evens)=0.161448159565518759938367668644199...(A190409)
...
C(1)=0.64163256065515386629... (A190405)
C(2)=0.28326512131030773259...
C(3)=0.066530242620615465175...
C(4)=0.0080604852412309303507...
C(5)=0.00049597048246186070148...
...
C(odds)=0.7086590131172367153696485920526...(A190410)
C(evens)=0.29134098688276328463035140794... (A190411)
...
D(1)=0.53137210011527713548... (A190406)
D(2)=0.25391006493009715683...
D(3)=0.062744200230554270960...
D(4)=0.0078201298601943136650...
D(5)=0.00048840046110854191952...
...
E(1)=0.12695503246504857842... (A190407)
E(2)=0.015686050057638567740...
E(3)=0.00097751623252428920813...
E(4)=0.000030525028819283869970...
E(5)=0.00000047686626214460406264...
...
UT=0.8563503956097795739814618239914245448... (A190412)
LT=0.1436496043902204260185381760085754551... (A190415)

			0.820816280327576933146921385113...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A190405-A190412, A190415, A299998.

f[i_, j_] := i + (j + i - 2)(j + i - 1)/2;
TableForm[Table[f[i,j],{i,1,10},{j,1,10}]] (* A000027 *)
r[i_] := Sum[2^-f[i, j], {j,1,400}];    (* C(row i) *)
c[j_] := Sum[2^-f[i,j], {i,1,400}];     (* C(col j) *)
d[h_] := Sum[2^-f[i,i+h-1], {i,1,200}]; (* C(udiag h) *)
e[h_] := Sum[2^-f[i+h,i], {i,1,200}];   (* C(ldiag h) *)
RealDigits[r[1], 10, 120, -1]  (* A190404 *)
N[r[1], 30]
N[r[2], 30]
N[r[3], 30]
N[r[4], 30]
N[r[5], 30]
N[r[6], 30]
RealDigits[c[1], 10, 120, -1] (* A190405 *)
N[c[1], 20]
N[c[2], 20]
N[c[3], 20]
N[c[4], 20]
N[c[5], 20]
N[c[6], 20]
RealDigits[d[1], 10, 20, -1] (* A190406 *)
N[d[1], 20]
N[d[2], 20]
N[d[3], 20]
N[d[4], 20]
N[d[5], 20]
N[d[6], 20]
RealDigits[e[1], 10, 20, -1] (* A190407 *)
N[e[1], 20]
N[e[2], 20]
N[e[3], 20]
N[e[4], 20]
N[e[5], 20]
N[e[6], 20]

def A190404(b):  # Generate the constant with b bits of precision
    return N(sum([(1/2)^(1+j*(j+1)/2) for j in range(1,b)])+1/2,b)
A190404(409) # Danny Rorabaugh, Mar 25 2015

A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers).
A190405: Sum_{k>=1} (1/2)^T(k), where T=A000217.
A190406: Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).
A190407: Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).
Equals Product_{k>=1} 1 - 1/(2^(2*k + 1) - 1). - Antonio Graciá Llorente, Oct 01 2024
Equals A299998/2. - Hugo Pfoertner, Oct 01 2024

A299998 Decimal expansion of Sum_{k>=0} 1/2^(k*(k+1)/2).

Original entry on oeis.org

1, 6, 4, 1, 6, 3, 2, 5, 6, 0, 6, 5, 5, 1, 5, 3, 8, 6, 6, 2, 9, 3, 8, 4, 2, 7, 7, 0, 2, 2, 5, 4, 2, 9, 4, 3, 4, 2, 2, 6, 0, 6, 1, 5, 3, 7, 9, 5, 6, 7, 3, 9, 7, 4, 7, 8, 0, 4, 6, 5, 1, 6, 2, 2, 3, 8, 0, 1, 4, 4, 6, 0, 3, 7, 3, 3, 3, 5, 1, 7, 7, 5, 6, 0, 0, 3, 6, 4, 1, 7, 1, 6, 2, 3, 3, 5, 9

Offset: 1

M. F. Hasler, Feb 27 2018

The binary expansion is the characteristic function A010054 of the triangular numbers A000217.

			1.641632560655153866293842770225429434226061537956739747804651622380144603733...

Cf. A000217, A010054, A190405, A333797.

RealDigits[QPochhammer[1/4, 1/4]/QPochhammer[2, 1/4], 10, 100][[1]] (* Amiram Eldar, Aug 13 2020 *)

default(realprecision,200);suminf(k=0,2^(-k*(k+1)/2))

Equals 1 + A190405 (as a decimal number).

Equals Product{k>=1} (1 - 1/2^(2*k))/(1 - 1/2^(2*k-1)). - Amiram Eldar, Aug 13 2020

A076131 a(n) = 2^n*a(n-1) + 1, a(0) = 0.

Original entry on oeis.org

0, 1, 5, 41, 657, 21025, 1345601, 172236929, 44092653825, 22575438758401, 23117249288602625, 47344126543058176001, 193921542320366288900097, 1588605274688440638669594625, 26027708820495411423962638336001, 852875962629993641540407732994080769

Offset: 0

Kyle Hunter (hunterk(AT)raytheon.com), Oct 31 2002

Base-2 expansion is same as base 10 expansion of A076127.

Vincenzo Librandi, Table of n, a(n) for n = 0..80

Cf. A000217, A073587, A076127, A190405.

a[0] = 0; a[n_] := 2^n a[n - 1] + 1; Table[ a[n], {n, 0, 13}]

a(n)=if(n<0,0,subst(Polrev(Vec(sum(k=1,n,x^(k*(k+1)/2)))),x,2))

PARI
```
a(n)=if(n<1,0,1+a(n-1)*2^n)
```

a(n) = floor(c*2^((n+1)*(n+2)/2)) where c = sum(k>=1, 1/2^A000217(k))=0.6416325... - Benoit Cloitre, Nov 01 2002

Edited and extended by Robert G. Wilson v, Oct 31 2002

Formula corrected by Vaclav Kotesovec, Aug 11 2012

A190407 Decimal expansion of Sum_{k>=1} (1/2)^A058331(k); based on a diagonal of the natural number array, A000027.

Original entry on oeis.org

1, 2, 6, 9, 5, 5, 0, 3, 2, 4, 6, 5, 0, 4, 8, 5, 7, 8, 4, 1, 6, 2, 5, 0, 5, 4, 3, 6, 3, 5, 7, 2, 5, 6, 7, 8, 8, 0, 6, 9, 6, 2, 1, 6, 8, 1, 9, 0, 1, 4, 6, 8, 0, 0, 2, 3, 1, 5, 0, 6, 1, 7, 8, 4, 9, 2, 5, 0, 9, 9, 2, 2, 7, 6, 2, 2, 7, 3, 0, 7, 5, 3, 8, 2, 1, 6, 5, 1, 3, 8, 3, 2, 3, 0, 9, 6, 1, 4, 3, 1, 3, 9, 1, 4, 3, 1, 4, 5, 8, 3, 2, 5, 4, 2, 1, 3, 0, 3, 3, 2

Offset: 0

Clark Kimberling, May 10 2011

See A190404.

			0.12695503246504857842...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A058331, A190404, A190405, A190406.

(* See also A190404 *)
RealDigits[(EllipticTheta[3, 0, 1/4]-1)/4, 10, 120] // First (* Jean-François Alcover, Feb 13 2013 *)

th3(x)=1 + 2*suminf(n=1,x^n^2)
(th3(1/4)-1)/4 \\ Charles R Greathouse IV, Jun 06 2016

def A190407(b): # Generate the constant with b bits of precision
    return N(sum([(1/2)^(2*k^2+1) for k in range(1,b)]),b)
A190407(415) # Danny Rorabaugh, Mar 26 2015

Equals Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).

A319015 Decimal expansion of Sum_{k>=0} 1/2^(k^2).

Original entry on oeis.org

1, 5, 6, 4, 4, 6, 8, 4, 1, 3, 6, 0, 5, 9, 3, 8, 5, 7, 9, 3, 3, 4, 7, 2, 9, 2, 7, 4, 2, 7, 2, 4, 7, 5, 6, 6, 2, 3, 0, 6, 2, 5, 8, 2, 6, 9, 9, 7, 0, 4, 3, 9, 0, 4, 6, 4, 4, 4, 5, 0, 5, 5, 9, 6, 0, 2, 8, 4, 8, 0, 1, 3, 3, 1, 7, 9, 5, 7, 8, 4, 0, 6, 6, 5, 9, 1, 3, 0, 6, 4, 0, 1, 6, 2, 4, 6, 9, 1, 4, 8, 4, 4, 7, 4, 0, 2, 4, 7, 1, 6

Offset: 1

Ilya Gutkovskiy, Sep 07 2018

The binary expansion is the characteristic function of the squares (A010052).

This constant is transcendental (Nesterenko, 1996). - Amiram Eldar, Apr 30 2020

			1.5644684136059385793347... = (1.1001000010000001000000001...)_2.
                               | |  |    |      |        |
                               0 1  4    9     16       25

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, On the binary expansions of algebraic numbers, Journal de théorie des nombres de Bordeaux, Vol. 16, No. 3 (2004), pp. 487-518. See p. 490.
Yu. V. Nesterenko, Modular functions and transcendence questions (in Russian), Sbornik: Mathematics, Vol. 187 (1996), pp. 65-96, English translation, ibid., pp. 1319-1348.
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Index entries for transcendental numbers.

Cf. A000290, A002416, A008836, A010052, A190405, A299998.

RealDigits[(1 + EllipticTheta[3, 0, 1/2])/2, 10, 110] [[1]]

suminf(k=0, 1/2^(k^2)) \\ Michel Marcus, Sep 08 2018

Equals (1 + theta_3(1/2))/2, where theta_3 is the Jacobi theta function.
Equals 1 + Sum_{k>=1} lambda(k)/(2^k - 1), where lambda is the Liouville function (A008836). - Amiram Eldar, Apr 30 2020
Equals 1 + Sum_{k>=1} floor(sqrt(k))/2^(k+1) (Shamos, 2011, p. 4). - Amiram Eldar, Mar 12 2024

A319016 Decimal expansion of Sum_{k>=0} 1/2^(k*(k+1)).

Original entry on oeis.org

1, 2, 6, 5, 8, 7, 0, 0, 9, 5, 2, 3, 0, 8, 6, 6, 3, 6, 8, 4, 1, 8, 9, 2, 1, 3, 1, 4, 5, 4, 3, 5, 4, 3, 4, 2, 7, 4, 6, 4, 2, 6, 5, 4, 4, 6, 3, 9, 9, 6, 3, 8, 7, 1, 6, 8, 2, 0, 0, 5, 3, 3, 4, 1, 8, 1, 4, 8, 9, 3, 4, 9, 2, 5, 1, 1, 2, 7, 4, 8, 9, 4, 4, 3, 7, 0, 6, 4, 5, 9, 7, 4, 8, 3, 5, 3, 0, 5, 6, 7, 3, 9, 0, 8, 4, 2, 7, 1, 1, 4

Offset: 1

Ilya Gutkovskiy, Sep 07 2018

The binary expansion is the characteristic function of the oblong numbers (A005369).

The Engel expansion of this constant are the powers of 4 (A000302). - Amiram Eldar, Dec 07 2020

			1.2658700952308663684189... = (1.010001000001000000010000000001...)_2.
                               |  |   |     |       |         |
                               0  2   6    12      20        30

Cf. A000302, A002378, A005369, A053763, A190405, A299998, A319015.

RealDigits[EllipticTheta[2, 0, 1/2]/2^(3/4), 10, 110] [[1]]

suminf(k=0, 1/2^(k*(k+1))) \\ Michel Marcus, Sep 08 2018

Equals theta_2(1/2)/2^(3/4), where theta_2 is the Jacobi theta function.

Equals Product_{k>=1} (1 - 1/4^k)^((-1)^k). - Antonio Graciá Llorente, Oct 01 2024

A332678 Decimal expansion of (1/2) * (1 + 2/1 + 4/(21) + 8/(42*1) + ... ).

Original entry on oeis.org

3, 1, 4, 1, 6, 3, 2, 5, 6, 0, 6, 5, 5, 1, 5, 3, 8, 6, 6, 2, 9, 3, 8, 4, 2, 7, 7, 0, 2, 2, 5, 4, 2, 9, 4, 3, 4, 2, 2, 6, 0, 6, 1, 5, 3, 7, 9, 5, 6, 7, 3, 9, 7, 4, 7, 8, 0, 4, 6, 5, 1, 6, 2, 2, 3, 8, 0, 1, 4, 4, 6, 0, 3, 7, 3, 3, 3, 5, 1, 7, 7, 5, 6, 0, 0, 3, 6, 4, 1, 7, 1, 6, 2, 3, 3, 5, 9, 1, 3, 3, 0, 8, 6

Offset: 1

Drew Edgette, Feb 19 2020

An approximation to Pi.

			3.1416325606551538662938427702254294342260615379567...

Cf. A000796 (Pi), A013705.

c:= sum(2^(j*(3-j)/2-1), j=0..infinity):
evalf(c, 125);  # Alois P. Heinz, Mar 03 2020

suminf(k=0, 2^(k-binomial(k,2)-1)) \\ Andrew Howroyd, Feb 21 2020

Equals (1/2)*Sum_{k>=0} 2^(k-binomial(k,2)). - Andrew Howroyd, Feb 21 2020

Equals A190405 +2.5 = A299998 +1.5. All digits the same but the first one or two. - R. J. Mathar, Mar 10 2020

cp's OEIS Frontend

A190404 Decimal expansion of (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers); based on row 1 of the natural number array, A000027.

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A299998 Decimal expansion of Sum_{k>=0} 1/2^(k*(k+1)/2).

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A076131 a(n) = 2^n*a(n-1) + 1, a(0) = 0.

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A190407 Decimal expansion of Sum_{k>=1} (1/2)^A058331(k); based on a diagonal of the natural number array, A000027.

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A319015 Decimal expansion of Sum_{k>=0} 1/2^(k^2).

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A319016 Decimal expansion of Sum_{k>=0} 1/2^(k*(k+1)).

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A332678 Decimal expansion of (1/2) * (1 + 2/1 + 4/(2*1) + 8/(4*2*1) + ... ).

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A332678 Decimal expansion of (1/2) * (1 + 2/1 + 4/(21) + 8/(42*1) + ... ).