A060867 -id:A060867 - cp's OEIS frontend

A090024 Number of distinct lines through the origin in the n-dimensional lattice of side length 8.

0, 1, 45, 571, 5841, 55651, 515025, 4702531, 42649281, 385447171, 3476958705, 31332052291, 282184860321, 2540643522691, 22870684139985, 205860600134851, 1852867557848961, 16676418630942211, 150090820212050865

Offset: 0

Joshua Zucker, Nov 20 2003

Equivalently, lattice points where the gcd of all the coordinates is 1.

			a(2) = 45 because in 2D the lines have slope 0, 1/8, 3/8, 5/8, 7/8, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/6, 5/6, 1/5, 2/5, 3/5, 4/5, 1/4, 3/4, 1/3, 2/3, 1/2, 1 and their reciprocals.

Gennady Eremin, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (20,-140,430,-579,270).

a(n) = T(n, 5) from A090030. Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023 are for dimension n with side lengths 1, 2, 3, 4, 5, 6, 7 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

Table[9^n - 5^n - 3^n - 2^n + 2, {n, 0, 20}]

[9**n-5**n-3**n-2**n+2 for n in range(30)] # Gennady Eremin, Mar 12 2022

a(n) = 9^n - 5^n - 3^n - 2^n + 2.

G.f.: -x*(291*x^3-189*x^2+25*x+1)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)*(9*x-1)). [Colin Barker, Sep 04 2012]

A090026 Number of distinct lines through the origin in 4-dimensional cube of side length n.

Original entry on oeis.org

0, 15, 65, 225, 529, 1185, 2065, 3745, 5841, 9105, 13025, 19105, 25521, 35361, 45825, 59905, 75425, 96865, 117841, 147505, 177041, 214961, 254401, 306321, 355249, 420929, 485489, 565265, 645377, 748081, 841841, 966881, 1086241, 1230401, 1373185, 1549825

Offset: 0

Joshua Zucker, Nov 25 2003

nonn

Equivalently, number of lattice points where the GCD of all coordinates = 1.

			a(2) = 65 because the 65 points with at least one coordinate=2 all make distinct lines and the remaining 15 points and the origin are on those lines.

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 are for n dimensions with side length 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 are this sequence for 2, 3, 4, 5, 6, 7 dimensions. A090030 is the table for n dimensions, side length k.

aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]];lines[n_, k_] := (k + 1)^n - Sum[Floor[k/i - 1]*aux[n, i], {i, 1, Floor[k/2]}] - 1;Table[lines[4, k], {k, 0, 40}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A090026(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A090026(k1)
        j, k1 = j2, n//j2
    return (n+1)**4-c+15*(j-n-1) # Chai Wah Wu, Mar 30 2021

a(n) = A090030(4, n).

a(n) = (n+1)^4 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

A090027 Number of distinct lines through the origin in 5-dimensional cube of side length n.

Original entry on oeis.org

0, 31, 211, 961, 2851, 7471, 15541, 31471, 55651, 95821, 152041, 239791, 351331, 517831, 723241, 1007041, 1352041, 1821721, 2359051, 3082921, 3904081, 4956901, 6151651, 7677901, 9334261, 11445361, 13746181, 16566691, 19644031, 23432851, 27408331, 32333581

Offset: 0

Joshua Zucker, Nov 25 2003

nonn

Equivalently, number of lattice points where the GCD of all coordinates = 1.

			a(2) = 211 because the 211 points with at least one coordinate=2 all make distinct lines and the remaining 31 points and the origin are on those lines.

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 are for n dimensions with side length 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 are this sequence for 2, 3, 4, 5, 6, 7 dimensions. A090030 is the table for n dimensions, side length k.

aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]];lines[n_, k_] := (k + 1)^n - Sum[Floor[k/i - 1]*aux[n, i], {i, 1, Floor[k/2]}] - 1;Table[lines[5, k], {k, 0, 40}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A090027(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A090027(k1)
        j, k1 = j2, n//j2
    return (n+1)**5-c+31*(j-n-1) # Chai Wah Wu, Mar 30 2021

a(n) = A090030(5, n).

a(n) = (n+1)^5 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

A090028 Number of distinct lines through the origin in 6-dimensional cube of side length n.

Original entry on oeis.org

0, 63, 665, 3969, 14833, 45801, 112825, 257257, 515025, 980217, 1720145, 2934505, 4693473, 7396137, 11112129, 16464385, 23555441, 33430033, 45927505, 62881561, 83865257, 111331241, 144772201, 187839225, 238778281, 303522401, 379323785

Offset: 0

Joshua Zucker, Nov 25 2003

nonn

Equivalently, lattice points where the GCD of all coordinates = 1.

			a(2) = 665 because the 665 points with at least one coordinate=2 all make distinct lines and the remaining 63 points and the origin are on those lines.

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 are for n dimensions with side length 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 are this sequence for 2, 3, 4, 5, 6, 7 dimensions. A090030 is the table for n dimensions, side length k.

aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]];lines[n_, k_] := (k + 1)^n - Sum[Floor[k/i - 1]*aux[n, i], {i, 1, Floor[k/2]}] - 1;Table[lines[6, k], {k, 0, 40}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A090028(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A090028(k1)
        j, k1 = j2, n//j2
    return (n+1)**6-c+63*(j-n-1) # Chai Wah Wu, Mar 30 2021

a(n) = A090030(6, n).

a(n) = (n+1)^6 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

A090029 Number of distinct lines through the origin in 7-dimensional cube of side length n.

Original entry on oeis.org

0, 127, 2059, 16129, 75811, 277495, 804973, 2078455, 4702531, 9905365, 19188793, 35533303, 61846723, 104511583, 168681913, 266042113, 405259513, 607140745, 883046011, 1269174145, 1780715833, 2472697501, 3366818491, 4548464341

Offset: 0

Joshua Zucker, Nov 25 2003

nonn

Equivalently, lattice points where the GCD of all coordinates = 1.

			a(2) = 2059 because the 2059 points with at least one coordinate=2 all make distinct lines and the remaining 127 points and the origin are on those lines.

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 are for n dimensions with side length 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 are this sequence for 2, 3, 4, 5, 6, 7 dimensions. A090030 is the table for n dimensions, side length k.

aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]];lines[n_, k_] := (k + 1)^n - Sum[Floor[k/i - 1]*aux[n, i], {i, 1, Floor[k/2]}] - 1;Table[lines[7, k], {k, 0, 40}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A090029(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A090029(k1)
        j, k1 = j2, n//j2
    return (n+1)**7-c+127*(j-n-1) # Chai Wah Wu, Mar 30 2021

a(n) = A090030(7, n).

a(n) = (n+1)^7 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

A090030 Triangle read by rows: T(n,k) = number of distinct lines through the origin in the n-dimensional cubic lattice of side length k with one corner at the origin.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 5, 7, 0, 0, 1, 9, 19, 15, 0, 0, 1, 13, 49, 65, 31, 0, 0, 1, 21, 91, 225, 211, 63, 0, 0, 1, 25, 175, 529, 961, 665, 127, 0, 0, 1, 37, 253, 1185, 2851, 3969, 2059, 255, 0, 0, 1, 45, 415, 2065, 7471, 14833, 16129, 6305, 511, 0, 0, 1, 57, 571, 3745, 15541, 45801, 75811, 65025, 19171, 1023, 0

Offset: 0

Joshua Zucker, Nov 24 2003

Equivalently, number of lattice points where the GCD of all coordinates = 1.

			T(n,1) = 2^n-1 because there are 2^n-1 lattice points other than the corner, all of which make distinct lines. T(n,2) = 3^n - 2^n because if the given corner is the origin, all the points with coordinates in {0,1} make lines that are redundant with a point containing a coordinate 2.
Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1,  0;
  0, 1,  3,   0;
  0, 1,  5,   7,    0;
  0, 1,  9,  19,   15,    0;
  0, 1, 13,  49,   65,   31,     0;
  0, 1, 21,  91,  225,  211,    63,     0;
  0, 1, 25, 175,  529,  961,   665,   127,    0;
  0, 1, 37, 253, 1185, 2851,  3969,  2059,  255,   0;
  0, 1, 45, 415, 2065, 7471, 14833, 16129, 6305, 511, 0;
  ...

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 give T(n, k) for k = 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 give T(n, k) for n=2, 3, 4, 5, 6, 7 respectively. A090225 counts only points with at least one coordinate = k.
T(n,n) gives A081474.
Cf. A008683.

aux[n_, k_] := If[k==0, 0, (k+1)^n-k^n-Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]]-1}]];lines[n_, k_] := (k+1)^n-Sum[Floor[k/i-1]*aux[n, i], {i, 1, Floor[k/2]}]-1;lines[n, k]

With A(n, k) = A090225(n, k), T(n, k) =(k+1)^n - 1 - the sum for 0 < i < k of Floor[k/i-1]*A(n, i)

T(n,k) = Sum_{i=1..n-k} moebius(i)*((floor((n-k)/i)+1)^k-1). - Vladeta Jovovic, Dec 03 2004

A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 18 2001

			1.1373387363441965966969133683013475838308493098...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Harry J. Smith, Table of n, a(n) for n=1..2000
Steven R. Finch, Digital Search Tree Constants. [Broken link]
Steven R. Finch, Digital Search Tree Constants. [From the Wayback machine]
Eric Weisstein's World of Mathematics, Tree Searching.

Cf. A060867, A065442, A065608, A066766, A067616.

RealDigits[NSum[1/(2^k - 1)^2, {k, 1, Infinity}, PrecisionGoal -> 40, AccuracyGoal -> 40, WorkingPrecision -> 500, NSumTerms -> 50, NSumExtraTerms -> 50]][[1]] (* Peter Bertok (peter(AT)bertok.com), Dec 04 2001 *)
RealDigits[(Log[2] QPolyGamma[0, 1, 1/2] + QPolyGamma[1, 1, 1/2])/Log[2]^2 - 1, 10, 20][[1]] (* Eric W. Weisstein, Jun 02 2025 *)

{ default(realprecision, 2080); x=suminf(k=1, 1/(2^k - 1)^2); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065443.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

Equals Sum_{n>=1} 1/A060867(n).
From Amiram Eldar, Oct 16 2022: (Start)
Equals Sum_{k>=1} k/(2^(k+1)-1).
Equals A066766 - A065442. (End)
Equals Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 evaluated at q = 1/2 (see A065608). - Peter Bala, Oct 16 2022

More terms from Peter Bertok (peter(AT)bertok.com), Dec 04 2001

A090021 Number of distinct lines through the origin in the n-dimensional lattice of side length 5.

Original entry on oeis.org

0, 1, 21, 175, 1185, 7471, 45801, 277495, 1672545, 10056991, 60405081, 362615815, 2176242705, 13059083311, 78359348361, 470170570135, 2821066729665, 16926530042431, 101559568723641, 609358576700455, 3656154951181425

Offset: 0

Joshua Zucker, Nov 19 2003

Equivalently, lattice points where the gcd of all the coordinates is 1.

			a(2) = 21 because in 2D the lines have slope 0, 1/5, 2/5, 3/5, 4/5, 1/4, 3/4, 1/3, 2/3, 1/2, 1 and their reciprocals.

Index entries for linear recurrences with constant coefficients, signature (12,-47,72,-36).

a(n) = T(n, 5) from A090030. Cf. A000225, A001047, A060867, A090020, A090022, A090023, A090024 are for dimension n with side lengths 1, 2, 3, 4, 6, 7, 8 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

Table[6^n - 3^n - 2*2^n + 2, {n, 0, 25}]
LinearRecurrence[{12,-47,72,-36},{0,1,21,175},30] (* Harvey P. Dale, Jul 18 2016 *)

a(n) = 6^n - 3^n - 2*2^n + 2.

G.f.: -x*(30*x^2-9*x-1)/((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). [Colin Barker, Sep 04 2012]

A090023 Number of distinct lines through the origin in the n-dimensional lattice of side length 7.

Original entry on oeis.org

0, 1, 37, 415, 3745, 31471, 257257, 2078455, 16704865, 133935391, 1072633177, 8585561095, 68702163985, 549687102511, 4397773276297, 35183283965335, 281470638631105, 2251782504544831, 18014329402322617, 144114912035163175, 1152920401607386225

Offset: 0

Joshua Zucker, Nov 20 2003

Equivalently, lattice points where the gcd of all the coordinates is 1.

			a(2) = 37 because in 2D the lines have slope 0, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/6, 5/6, 1/5, 2/5, 3/5, 4/5, 1/4, 3/4, 1/3, 2/3, 1/2, 1 and their reciprocals.

Index entries for linear recurrences with constant coefficients, signature (18,-115,330,-424,192).

Equals A090030(n+7,n).

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090024 are for dimension n with side lengths 1, 2, 3, 4, 5, 6, 8 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

Table[8^n - 4^n - 3^n - 2^n + 2, {n, 0, 20}]

[8**n-4**n-3**n-2**n+2 for n in range(25)] # Gennady Eremin, Mar 09 2022

a(n) = 8^n - 4^n - 3^n - 2^n + 2.

G.f.: -x*(200*x^3-136*x^2+19*x+1)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(8*x-1)). - Colin Barker, Sep 04 2012

A007179 Dual pairs of integrals arising from reflection coefficients.

Original entry on oeis.org

0, 1, 1, 4, 6, 16, 28, 64, 120, 256, 496, 1024, 2016, 4096, 8128, 16384, 32640, 65536, 130816, 262144, 523776, 1048576, 2096128, 4194304, 8386560, 16777216, 33550336, 67108864, 134209536, 268435456, 536854528, 1073741824, 2147450880, 4294967296, 8589869056

Offset: 0

N. J. A. Sloane, Simon Plouffe

			From _Gus Wiseman_, Feb 26 2022: (Start)
Also the number of integer compositions of n with at least one odd part. For example, the a(1) = 1 through a(5) = 16 compositions are:
  (1)  (1,1)  (3)      (1,3)      (5)
              (1,2)    (3,1)      (1,4)
              (2,1)    (1,1,2)    (2,3)
              (1,1,1)  (1,2,1)    (3,2)
                       (2,1,1)    (4,1)
                       (1,1,1,1)  (1,1,3)
                                  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
J. Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J. Phys. A 14 (1981), 357-367.
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Column k=2 of A309748.
Odd bisection is A000302.
Even bisection is A006516 = 2^(n-1)*(2^n - 1).
The complement is counted by A077957, internal version A027383.
The internal case is A274230, even bisection A134057.
A000045(n-1) counts compositions without odd parts, non-singleton A077896.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A034871, A097805, and A345197 count compositions by alternating sum.
A052952 (or A074331) counts non-singleton compositions without even parts.
Cf. A000918, A033484, A052955, A060867, A116406, A138364.

[Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011

f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;

LinearRecurrence[{2,2,-4},{0,1,1},30] (* Harvey P. Dale, Nov 30 2015 *)
Table[2^(n-1)-If[EvenQ[n],2^(n/2-1),0],{n,0,15}] (* Gus Wiseman, Feb 26 2022 *)

Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016

From Paul Barry, Apr 28 2004: (Start)
  Binomial transform is (A000244(n)+A001333(n))/2.
  G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).
  a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).
  a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)
G.f.: (1+x*Q(0))*x/(1-x), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
a(n) = A011782(n+2) - A077957(n) - Gus Wiseman, Feb 26 2022

cp's OEIS Frontend

A090024 Number of distinct lines through the origin in the n-dimensional lattice of side length 8.

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A090026 Number of distinct lines through the origin in 4-dimensional cube of side length n.

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A090027 Number of distinct lines through the origin in 5-dimensional cube of side length n.

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A090028 Number of distinct lines through the origin in 6-dimensional cube of side length n.

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A090029 Number of distinct lines through the origin in 7-dimensional cube of side length n.

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A090030 Triangle read by rows: T(n,k) = number of distinct lines through the origin in the n-dimensional cubic lattice of side length k with one corner at the origin.

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A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

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