A080277 -id:A080277 - cp's OEIS frontend

A080333 Partial sums of A080278.

1, 2, 6, 7, 8, 12, 13, 14, 27, 28, 29, 33, 34, 35, 39, 40, 41, 54, 55, 56, 60, 61, 62, 66, 67, 68, 108, 109, 110, 114, 115, 116, 120, 121, 122, 135, 136, 137, 141, 142, 143, 147, 148, 149, 162, 163, 164, 168, 169, 170, 174, 175, 176, 216, 217, 218, 222, 223, 224, 228, 229

Offset: 1

N. J. A. Sloane, Mar 19 2003

nonn

Klaus Brockhaus, Illustration of A080278 and A080333
B. Dearden, J. Iiams, and J. Metzger, A Function Related to the Rumor Sequence Conjecture , J. Int. Seq. 14 (2011) # 11.2.3.

Cf. A080277, A080278, A333979.

a(n) = fromdigits(Vec(Pol(digits(3*n,3))'),3); \\ Kevin Ryde, Apr 29 2021

a(n) = Sum_{k=0..log_3(n)} 3^k*floor(n/3^k).
a(3^k) = (k+1)*3^k.
a(n) is conjectured to be asymptotic to n*log(n)/log(3). - Klaus Brockhaus, Mar 23 2003 [This follows from the asymptotics of A333979. - Pontus von Brömssen, Sep 06 2020]
a(n) = n + 3*a(floor(n/3)), a(0)=0. - Vladeta Jovovic, Aug 06 2003
G.f.: (1/(1 - x))*Sum_{k>=0} 3^k*x^(3^k)/(1 - x^(3^k)). - Ilya Gutkovskiy, Mar 15 2018
a(n) = A333979(3*n,3). - Pontus von Brömssen, Sep 06 2020

A118822 Numerators of the convergents of the 2-adic continued fraction of zero given by A118821.

Original entry on oeis.org

2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1, -2, 1, 0, 1, 2, -1, 0, -1

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118822(k)/A118823(k) are:
  at k = 4*n: -1/A080277(n);
  at k = 4*n+1: -2/(2*A080277(n)-1);
  at k = 4*n+2: -1/(A080277(n)-1);
  at k = 4*n-1: 0/(-1)^n.
Convergents begin:
  2/1, -1/-1, 0/-1, -1/1, -2/1, 1/0, 0/1, 1/-4,
  2/-7, -1/3, 0/-1, -1/5, -2/9, 1/-4, 0/1, 1/-12,
  2/-23, -1/11, 0/-1, -1/13, -2/25, 1/-12, 0/1, 1/-16,
  2/-31, -1/15, 0/-1, -1/17, -2/33, 1/-16, 0/1, 1/-32, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A118821 (partial quotients), A118823 (denominators).

A118822:=n->sqrt((n+1)^2 mod 8)*(-1)^floor((n+2)/4); seq(A118822(n), n=1..100); # Wesley Ivan Hurt, Jan 01 2014

Table[Sqrt[Mod[(n+1)^2, 8]](-1)^Floor[(n+2)/4], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2014 *)

{a(n)=local(p=+2,q=-1,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[1,1]}
for(n=0,80,print1(a(n),", "))

{a(n) = [2,-1,0,-1,-2,1,0,1][(n-1)%8+1];} \\ Joerg Arndt, Jan 02 2014

Period 8 sequence: [2,-1,0,-1,-2,1,0,1].
G.f.: -x*(x-1)*(x^2+x+2) / ( 1+x^4 ).
a(n) = sqrt((n+1)^2 mod 8)(-1)^floor((n+2)/4). - Wesley Ivan Hurt, Jan 01 2014

A118825 Numerators of the convergents of the 2-adic continued fraction of zero given by A118824.

Original entry on oeis.org

-2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118825(k)/A118826(k) are:
  at k = 4*n: 1/A080277(n);
  at k = 4*n+1: 2/(2*A080277(n)-1);
  at k = 4*n+2: 1/(A080277(n)-1);
  at k = 4*n-1: 0/(-1)^n.
Convergents begin:
  -2/1, -1/1, 0/-1, -1/-1, 2/1, 1/0, 0/1, 1/4,
  -2/-7, -1/-3, 0/-1, -1/-5, 2/9, 1/4, 0/1, 1/12,
  -2/-23, -1/-11, 0/-1, -1/-13, 2/25, 1/12, 0/1, 1/16,
  -2/-31, -1/-15, 0/-1, -1/-17, 2/33, 1/16, 0/1, 1/32, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A118824 (partial quotients), A118826 (denominators), A118822, A230075 (start with a(5)).

A118825:=n->sqrt((n+1)^2 mod 8))*(-1)^floor((n+3)/4); seq(A118825(n), n=1..100); # Wesley Ivan Hurt, Jan 04 2014

Table[Sqrt[Mod[(n+1)^2, 8]](-1)^Floor[(n+3)/4], {n, 100}] (* Wesley Ivan Hurt, Jan 04 2014 *)
PadRight[{},120,{-2,-1,0,-1,2,1,0,1}] (* Harvey P. Dale, May 26 2020 *)

{a(n)=local(p=-2,q=+1,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[1,1]}

Period 8 sequence: [ -2,-1,0,-1,2,1,0,1].
G.f.: -x*(1+x)*(x^2-x+2) / ( 1+x^4 ).
a(n) = sqrt((n+1)^2 mod 8)*(-1)^floor((n+3)/4). - Wesley Ivan Hurt, Jan 04 2014

A118831 Numerators of the convergents of the 2-adic continued fraction of zero given by A118830.

Original entry on oeis.org

-1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118831(k)/A118832(k) are:
at k = 4*n: 1/(2*A080277(n));
at k = 4*n+1: 1/(2*A080277(n)-1);
at k = 4*n+2: 1/(2*A080277(n)-2);
at k = 4*n-1: 0.
Convergents begin:
-1/1, -1/2, 0/-1, -1/-2, 1/1, 1/0, 0/1, 1/8,
-1/-7, -1/-6, 0/-1, -1/-10, 1/9, 1/8, 0/1, 1/24,
-1/-23, -1/-22, 0/-1, -1/-26, 1/25, 1/24, 0/1, 1/32,
-1/-31, -1/-30, 0/-1, -1/-34, 1/33, 1/32, 0/1, 1/64, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A118830 (partial quotients), A118832 (denominators).

{a(n)=local(p=-1,q=+2,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[1,1]}

Period 8 sequence: [ -1,-1,0,-1,1,1,0,1].
G.f.: -x*(1+x+x^3)/(1+x^4). [corrected by R. J. Mathar, Jul 22 2009]
a(n) = -a(n-4). - R. J. Mathar, Jul 22 2009

A240400 Numbers n having a partition into distinct parts of form 3^k-2^k.

Original entry on oeis.org

0, 1, 5, 6, 19, 20, 24, 25, 65, 66, 70, 71, 84, 85, 89, 90, 211, 212, 216, 217, 230, 231, 235, 236, 276, 277, 281, 282, 295, 296, 300, 301, 665, 666, 670, 671, 684, 685, 689, 690, 730, 731, 735, 736, 749, 750, 754, 755, 876, 877, 881, 882, 895, 896, 900, 901

Offset: 1

Jon Perry, Apr 04 2014

nonn

Numbers n such that there are partitions into distinct parts from A001047. - Joerg Arndt, Apr 06 2014

Based on casting binary numbers as ternary numbers. - Jon Perry, Apr 12 2014

			25 = 19 + 5 + 1 so 25 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A241783 (complement).

Cf. A001047, A005187, A080277, A241759.

a240400 n = a240400_list !! (n-1)
a240400_list = filter ((> 0) . a241759) [0..]
-- Reinhard Zumkeller, Apr 28 2014

function trimArray(arr) {
var c, i, j;
c = new Array();
for (j = 0; j < arr.length; j++) c[j] = arr[j];
c.sort(function(a, b) {return a - b;});
i = -1;
while(i++ < c.length - 1)
if (c[i] == c[i + 1]) c.splice(i--, 1);
return c;
}
a = new Array();
for (i = 0; i < 10; i++)
a[i] = Math.pow(3, i) - Math.pow(2, i);
b = new Array();
bc = 0;
for (j = 0; j < 130; j++) {
c = 0;
s = j.toString(2);
sl = s.length;
for (k = 0; k < sl; k++) if (s.charAt(k) == 1) c += a[k];
b[bc++] = c;
}
b = trimArray(b);
document.write(b);

max = 1000; nmax = FindRoot[3^n - 2^n == max, {n, 1}][[1, 2]] // Ceiling; partitions = Select[Table[{3^n - 2^n}, {n, 1, nmax}], (First[#] <= max)&] //. {a___, b_List, c___, d_List, e___} /; Total[b] + Total[d] <= max && FreeQ[p = {a, b, c, d, e}, (j = Join[b, d] // Sort)] && j == Union[j] :> Union[Append[p, j]]; Join[{0}, Total /@ partitions // Sort] (* Jean-François Alcover, Apr 16 2014 *)

PARI
```
a(n)=if(n<2,n%2,n+3*a(floor(n/2)))
```

A241759(a(n)) > 0. - Reinhard Zumkeller, Apr 28 2014

Recursive formula: For n >= 1, a(1)=1 then a(n) = n + 3*a(floor(n/2)). Sum: a(n) = Sum_{k=0..floor(log_2(n))} 3^k*floor(n/2^k). - Benoit Cloitre, Apr 06 2019

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 1, 5, 0, 0, 0, 0, 0, 1, 2, 5, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13

Offset: 0

Pontus von Brömssen, Sep 04 2020

			Array begins:
  n\k|  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
  ---|---------------------------------------------
   0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   2 |  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   3 |  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0
   4 |  4  1  1  0  0  0  0  0  0  0  0  0  0  0  0
   5 |  4  1  1  1  0  0  0  0  0  0  0  0  0  0  0
   6 |  5  2  1  1  1  0  0  0  0  0  0  0  0  0  0
   7 |  5  2  1  1  1  1  0  0  0  0  0  0  0  0  0
   8 | 12  2  2  1  1  1  1  0  0  0  0  0  0  0  0
   9 | 12  6  2  1  1  1  1  1  0  0  0  0  0  0  0
  10 | 13  6  2  2  1  1  1  1  1  0  0  0  0  0  0
  11 | 13  6  2  2  1  1  1  1  1  1  0  0  0  0  0
  12 | 16  7  3  2  2  1  1  1  1  1  1  0  0  0  0
  13 | 16  7  3  2  2  1  1  1  1  1  1  1  0  0  0
  14 | 17  7  3  2  2  2  1  1  1  1  1  1  1  0  0
  15 | 17  8  3  3  2  2  1  1  1  1  1  1  1  1  0
  16 | 32  8  8  3  2  2  2  1  1  1  1  1  1  1  1
64 = 2*3^3 + 1*3^2 + 0*3^1 + 1*3^0, so T(64,3) = 2*3*3^2 + 1*2*3^1 + 0*1*3^0 = 60. Alternatively, using the formula T(n,k) = floor(n/k) + k*T(floor(n/k),k), we get T(64,3) = 21 + 3*T(21,3) = 21 + 3*(7 + 3*T(7,3)) = 42 + 9*(2 + 3*T(2,3)) = 60.

Pontus von Brömssen, Antidiagonals n = 0..99, flattened
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, Journal of Graph Algorithms and Applications 18(2) (2014), 177-209.

Cf. A136013 (column k=2), A080277 (every second term of column k=2), A080333 (every third term of column k=3).

Cf. A007824, A055129, A286561.

import sympy
def A333979(n,k):
  d=sympy.ntheory.factor_.digits(n,k)
  return sum(j*d[-j-1]*k**(j-1) for j in range(1,len(d)-1))

# Second program (faster)
def A333979(n,k):
  return n//k+k*A333979(n//k,k) if n>=k else 0

T(n,k) = floor(n/k) + k*T(floor(n/k),k). Proof: With n = Sum_{j>=0} d_j*k^j we have floor(n/k) + k*T(floor(n/k),k) = Sum_{j>=1} (d_j*k^(j-1) + k*d_j*(j-1)*k^(j-2)) = Sum_{j>=1} d_j*j*k^(j-1) = T(n,k).
T(n,k) = T(n-1,k) + A055129(A286561(n,k),k). Proof: Let n = Sum_{j>=0} d_j*k^j and pick v so that d_j = 0 for j < v and d_v > 0 (so v = A286561(n,k)). Then n - 1 = sum_{j>=0} e_j*k^j, where e_j = k - 1 for j < v, e_v = d_v - 1, and e_j = d_j for j > v. We get T(n,k) - T(n-1,k) = Sum_{j>=1} j*(d_j-e_j)*k^(j-1) = v*k^(v-1) - (k-1)*Sum_{1<=jA055129(A286561(n,k),k).
For fixed k, T(n,k) ~ n*log(n)/(k*log(k)). (The proof for k = 2 by Bannister et al. (p. 182) can be adapted to general k.)
T(n,k) = Sum_{j>=0} k^j*floor(n/k**(j+1)).

A118823 Denominators of the convergents of the 2-adic continued fraction of zero given by A118821.

Original entry on oeis.org

1, -1, -1, 1, 1, 0, 1, -4, -7, 3, -1, 5, 9, -4, 1, -12, -23, 11, -1, 13, 25, -12, 1, -16, -31, 15, -1, 17, 33, -16, 1, -32, -63, 31, -1, 33, 65, -32, 1, -36, -71, 35, -1, 37, 73, -36, 1, -44, -87, 43, -1, 45, 89, -44, 1, -48, -95, 47, -1, 49, 97, -48, 1, -80, -159, 79, -1, 81, 161, -80, 1, -84, -167, 83, -1, 85, 169, -84, 1, -92

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118822(k)/A118823(k) are:
at k = 4*n: -1/A080277(n);
at k = 4*n+1: -2/(2*A080277(n)-1);
at k = 4*n+2: -1/(A080277(n)-1);
at k = 4*n-1: 0/(-1)^n.
Convergents begin:
2/1, -1/-1, 0/-1, -1/1, -2/1, 1/0, 0/1, 1/-4,
2/-7, -1/3, 0/-1, -1/5, -2/9, 1/-4, 0/1, 1/-12,
2/-23, -1/11, 0/-1, -1/13, -2/25, 1/-12, 0/1, 1/-16,
2/-31, -1/15, 0/-1, -1/17, -2/33, 1/-16, 0/1, 1/-32, ...

Cf. A080277; A118821 (partial quotients), A118822 (numerators).

{a(n)=local(p=+2,q=-1,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[2,1]}

a(4*n) = -(-1)^n*A080277(n); a(4*n+1) = -(-1)^n*(2*A080277(n)-1); a(4*n+2) = (-1)^n*(A080277(n)-1); a(4*n-1) = (-1)^n.

A118826 Denominators of the convergents of the 2-adic continued fraction of zero given by A118824.

Original entry on oeis.org

1, 1, -1, -1, 1, 0, 1, 4, -7, -3, -1, -5, 9, 4, 1, 12, -23, -11, -1, -13, 25, 12, 1, 16, -31, -15, -1, -17, 33, 16, 1, 32, -63, -31, -1, -33, 65, 32, 1, 36, -71, -35, -1, -37, 73, 36, 1, 44, -87, -43, -1, -45, 89, 44, 1, 48, -95, -47, -1, -49, 97, 48, 1, 80, -159, -79, -1, -81, 161, 80, 1, 84, -167, -83, -1, -85, 169, 84, 1, 92

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118825(k)/A118826(k) are:
at k = 4*n: 1/A080277(n);
at k = 4*n+1: 2/(2*A080277(n)-1);
at k = 4*n+2: 1/(A080277(n)-1);
at k = 4*n-1: 0.
Convergents begin:
-2/1, -1/1, 0/-1, -1/-1, 2/1, 1/0, 0/1, 1/4,
-2/-7, -1/-3, 0/-1, -1/-5, 2/9, 1/4, 0/1, 1/12,
-2/-23, -1/-11, 0/-1, -1/-13, 2/25, 1/12, 0/1, 1/16,
-2/-31, -1/-15, 0/-1, -1/-17, 2/33, 1/16, 0/1, 1/32, ...

Cf. A006519, A080277; A118824 (partial quotients), A118825 (numerators).

{a(n)=local(p=-2,q=+1,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[2,1]}

a(4*n) = (-1)^n*A080277(n); a(4*n+1) = -(-1)^n*(2*A080277(n)-1); a(4*n+2) = -(-1)^n*(A080277(n)-1); a(4*n-1) = (-1)^n.

A118829 Denominators of the convergents of the 2-adic continued fraction of zero given by A118827.

Original entry on oeis.org

1, -2, -1, 2, 1, 0, 1, -8, -7, 6, -1, 10, 9, -8, 1, -24, -23, 22, -1, 26, 25, -24, 1, -32, -31, 30, -1, 34, 33, -32, 1, -64, -63, 62, -1, 66, 65, -64, 1, -72, -71, 70, -1, 74, 73, -72, 1, -88, -87, 86, -1, 90, 89, -88, 1, -96, -95, 94, -1, 98, 97, -96, 1, -160, -159, 158, -1, 162, 161, -160, 1, -168, -167, 166, -1, 170, 169

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118828(k)/A118829(k) are:
at k = 4*n: -1/(2*A080277(n));
at k = 4*n+1: -1/(2*A080277(n)-1);
at k = 4*n+2: -1/(2*A080277(n)-2);
at k = 4*n-1: 0.
Convergents begin:
1/1, -1/-2, 0/-1, -1/2, -1/1, 1/0, 0/1, 1/-8,
1/-7, -1/6, 0/-1, -1/10, -1/9, 1/-8, 0/1, 1/-24,
1/-23, -1/22, 0/-1, -1/26, -1/25, 1/-24, 0/1, 1/-32,
1/-31, -1/30, 0/-1, -1/34, -1/33, 1/-32, 0/1, 1/-64, ...

Cf. A118827 (partial quotients), A118829 (denominators).

{a(n)=local(p=+1,q=-2,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[1,1]}

a(4*n) = -(-1)^n*2*A080277(n); a(4*n+1) = -(-1)^n*(2*A080277(n)-1); a(4*n+2) = (-1)^n*(2*A080277(n)-2); a(4*n-1) = (-1)^n.

A118832 Denominators of the convergents of the 2-adic continued fraction of zero given by A118830.

Original entry on oeis.org

1, 2, -1, -2, 1, 0, 1, 8, -7, -6, -1, -10, 9, 8, 1, 24, -23, -22, -1, -26, 25, 24, 1, 32, -31, -30, -1, -34, 33, 32, 1, 64, -63, -62, -1, -66, 65, 64, 1, 72, -71, -70, -1, -74, 73, 72, 1, 88, -87, -86, -1, -90, 89, 88, 1, 96, -95, -94, -1, -98, 97, 96, 1, 160, -159, -158, -1, -162, 161, 160, 1, 168, -167, -166, -1, -170, 169, 168

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118831(k)/A118832(k) are:
at k = 4*n: 1/(2*A080277(n));
at k = 4*n+1: 1/(2*A080277(n)-1);
at k = 4*n+2: 1/(2*A080277(n)-2);
at k = 4*n-1: 0.
Convergents begin:
-1/1, -1/2, 0/-1, -1/-2, 1/1, 1/0, 0/1, 1/8,
-1/-7, -1/-6, 0/-1, -1/-10, 1/9, 1/8, 0/1, 1/24,
-1/-23, -1/-22, 0/-1, -1/-26, 1/25, 1/24, 0/1, 1/32,
-1/-31, -1/-30, 0/-1, -1/-34, 1/33, 1/32, 0/1, 1/64, ...

Cf. A080277; A118830 (partial quotients), A118831 (numerators).

{a(n)=local(p=-1,q=+2,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[2,1]}

a(4*n) = (-1)^n*2*A080277(n); a(4*n+1) = -(-1)^n*(2*A080277(n)-1); a(4*n+2) = -(-1)^n*(2*A080277(n)-2); a(4*n-1) = (-1)^n.

cp's OEIS Frontend

A080333 Partial sums of A080278.

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A118822 Numerators of the convergents of the 2-adic continued fraction of zero given by A118821.

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A118825 Numerators of the convergents of the 2-adic continued fraction of zero given by A118824.

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A118831 Numerators of the convergents of the 2-adic continued fraction of zero given by A118830.

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A240400 Numbers n having a partition into distinct parts of form 3^k-2^k.

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A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_j*k^j, then T(n,k) = Sum_{j>=1} d_j*j*k^(j-1).

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A118823 Denominators of the convergents of the 2-adic continued fraction of zero given by A118821.

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A118826 Denominators of the convergents of the 2-adic continued fraction of zero given by A118824.

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).