A102699 -id:A102699 - cp's OEIS frontend

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0

Offset: 0

Alois P. Heinz, Dec 03 2012

Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017

			A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,   5,   6,   7, ...
  0,  0,  2,  4,  6,   8,  10,  12, ...
  0,  0,  2,  6, 10,  14,  18,  22, ...
  0,  0,  2,  8, 16,  24,  32,  40, ...
  0,  0,  2, 12, 26,  42,  58,  74, ...
  0,  0,  2, 16, 42,  72, 104, 136, ...
  0,  0,  2, 24, 68, 126, 188, 252, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360.
Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4.
Main diagonal gives: A102699.
Cf. A198632, A188866, A276562, A208727, A208671.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-1, j, k), j=1..k),
      `if`(i>1, b(n-1, i-1, k), 0)+
      `if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017

A153338 Number of zig-zag paths from top to bottom of a 2n-1 by 2n-1 square whose color is not that of the top right corner.

Original entry on oeis.org

0, 2, 18, 116, 650, 3372, 16660, 79592, 371034, 1697660, 7654460, 34106712, 150499908, 658707896, 2863150440, 12371226064, 53178791162, 227561427612, 969890051884, 4119092850680, 17438036501676, 73611934643368, 309935825654168, 1301878616066736

Offset: 1

Joseph Myers, Dec 24 2008

			a(3) = 3*2 ^ (2*3 - 2) - (2*3 - 1) * binomial(2*3 - 2, 3 - 1) = 18. - _Indranil Ghosh_, Feb 19 2017

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Cf. A102699, A153334, A153335, A153336, A153337.

[(n)*2^(2*n-2)-(2*n-1)*Binomial(2*n-2, n-1): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

Table[n 2^(2 n - 2) - (2 n - 1) Binomial[2 n - 2, n - 1], {n, 22}] (* Michael De Vlieger, Sep 17 2015 *)

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A153338(n):
    return str(n*2**(2*n-2)-(2*n-1)*C(2*n-2,n-1)) # Indranil Ghosh, Feb 19 2017

a(n) = n*2^(2*n-2) - (2*n-1)*binomial(2*n-2,n-1).

4^n*(n+1)-C(2*n,n)*(2*n+1) = Sum_{k=1..n} C(2*(n-k),n-k)*C(2*k,k)*k*(H(k)-H(n-k)) for n >= 0; H(n) denote the harmonic numbers. This identity is attributed to Maillard. - Peter Luschny, Sep 17 2015

a(23)-a(24) from Vincenzo Librandi, Sep 18 2015

A153334 Number of zig-zag paths from top to bottom of an n X n square whose color is that of the top right corner.

Original entry on oeis.org

1, 1, 4, 8, 24, 52, 136, 296, 720, 1556, 3624, 7768, 17584, 37416, 83024, 175568, 383904, 807604, 1746280, 3657464, 7839216, 16357496, 34812144, 72407728, 153204064, 317777032, 669108496, 1384524656, 2903267040, 5994736336

Offset: 1

Joseph Myers, Dec 24 2008

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Cf. A102699, A153335, A153336, A153337, A153338.

Table[If[Mod[n,2]==0, (n+1)*2^(n-2)-2(n-1) Binomial[n-2,(n-2)/2], (n+1)*2^(n-2)-(n-1)  Binomial[n-1,(n-1)/2]],{n,1,30}] (* Indranil Ghosh, Feb 19 2017 *)

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A153334(n):
    if n%2==0: return str(int((n+1)*2**(n-2)-2*(n-1)*C(n-2,(n-2)/2)))
    else: return str(int((n+1)*2**(n-2)-(n-1)*C(n-1,(n-1)/2))) # Indranil Ghosh, Feb 19 2017

a(n) = (n+1)2^(n-2) - 2(n-1)binomial(n-2,(n-2)/2) for n even, a(n) = (n+1)2^(n-2) - (n-1)binomial(n-1,(n-1)/2) for n odd.

A153335 Number of zig-zag paths from top to bottom of an n X n square whose color is not that of the top right corner.

Original entry on oeis.org

0, 1, 2, 8, 18, 52, 116, 296, 650, 1556, 3372, 7768, 16660, 37416, 79592, 175568, 371034, 807604, 1697660, 3657464, 7654460, 16357496, 34106712, 72407728, 150499908, 317777032, 658707896, 1384524656, 2863150440, 5994736336

Offset: 1

Joseph Myers, Dec 24 2008

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Cf. A102699, A153334, A153336, A153337, A153338.

Table[If[Mod[n,2]==0, (n+1)*2^(n-2)-2(n-1) Binomial[n-2,(n-2)/2], (n+1)*2^(n-2)-(n) Binomial[n-1,(n-1)/2]],{n,1,30}] (* Indranil Ghosh, Feb 19 2017 *)

a(n) = if (n % 2, (n+1)*2^(n-2) - n*binomial(n-1,(n-1)/2), (n+1)*2^(n-2) - 2*(n-1)*binomial(n-2,(n-2)/2)); \\ Michel Marcus, Feb 19 2017

import math
def C(n, r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A153335(n):
    if n%2==0: return str(int((n+1)*2**(n-2)-2*(n-1)*C(n-2, (n-2)/2)))
    else: return str(int((n+1)*2**(n-2)-(n)*C(n-1, (n-1)/2))) # Indranil Ghosh, Feb 19 2017

a(n) = (n+1)2^(n-2) - 2(n-1)binomial(n-2,(n-2)/2) for n even, a(n) = (n+1)2^(n-2) - (n)binomial(n-1,(n-1)/2) for n odd.

A153336 Number of zig-zag paths from top to bottom of a 2n by 2n square whose color is that of the top right corner.

Original entry on oeis.org

1, 8, 52, 296, 1556, 7768, 37416, 175568, 807604, 3657464, 16357496, 72407728, 317777032, 1384524656, 5994736336, 25816193952, 110652549620, 472302724408, 2008499580504, 8513063608304, 35975584631128, 151621915797840

Offset: 1

Joseph Myers, Dec 24 2008, Dec 31 2008

			a(3) = (2*3 + 1)*2 ^ (2*3 - 2) - 2*(2*3 - 1) * binomial(2*3 - 2, 3 - 1) = 52. - _Indranil Ghosh_, Feb 19 2017

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Cf. A102699, A153334, A153335, A153337, A153338.

Table[(2n+1) 2^(2n-2)-2(2n-1) Binomial[2n-2,n-1],{n,1,22}] (* Indranil Ghosh, Feb 19 2017 *)

a(n) = (2*n+1)*2^(2*n-2) - 2*(2*n-1)*binomial(2*n-2, n-1); \\ Michel Marcus, Feb 19 2017

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A153336(n):
    return str((2*n+1)*2**(2*n-2)-2*(2*n-1)*C(2*n-2,n-1)) # Indranil Ghosh, Feb 19 2017

a(n) = (2n+1)2^(2n-2) - 2(2n-1)binomial(2n-2,n-1).

A153337 Number of zig-zag paths from top to bottom of a 2n-1 by 2n-1 square whose color is that of the top right corner.

Original entry on oeis.org

1, 4, 24, 136, 720, 3624, 17584, 83024, 383904, 1746280, 7839216, 34812144, 153204064, 669108496, 2903267040, 12526343584, 53779871552, 229895033832, 978965187184, 4154438114480, 17575883030496, 74150192517808

Offset: 1

Joseph Myers, Dec 24 2008

			a(3) = 3 * 2 ^ (2*3 - 2) - 2* (3 - 1) * binomial(2*3 - 2, 3 - 1) = 24. - _Indranil Ghosh_, Feb 19 2017

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Cf. A102699, A153334, A153335, A153336, A153338.

Table[(n)2^(2n-2)-2(n-1) Binomial[2n-2,n-1],{n,1,22}] (* Indranil Ghosh, Feb 19 2017 *)

a(n) = n*2^(2*n-2) - 2*(n-1)*binomial(2*n-2,n-1); \\ Michel Marcus, Feb 19 2017

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A153337(n):
    return str(n*2**(2*n-2)-2*(n-1)*C(2*n-2,n-1)) # Indranil Ghosh, Feb 19 2017

a(n) = n*2^(2n-2) - 2(n-1)*binomial(2n-2,n-1).

A152086 a(n) = Sum_{k=1..n-1} k*A110971(n,k).

Original entry on oeis.org

1, 3, 8, 21, 52, 126, 296, 685, 1556, 3498, 7768, 17122, 37416, 81308, 175568, 377469, 807604, 1721970, 3657464, 7746838, 16357496, 34459428, 72407728, 151851986, 317777032, 663908196, 1384524656, 2883208740, 5994736336, 12448784824, 25816193952, 53479331357, 110652549620

Offset: 2

N. J. A. Sloane, Sep 20 2009

nonn

Paolo Xausa, Table of n, a(n) for n = 2..1000
M. A. Michels and U. Knauer, The congruence classes of paths and cycles, Discr. Math., 309 (2009), 5352-5359. See p. 5356.

Cf. A110971, A102699.

Main diagonal of A377000.

A110971[n_] := (n+1)*2^(n-2) - If[OddQ[n], (n-1/2)*Binomial[n-1, (n-1)/2], 2*(n-1)*Binomial[n-2, (n-2)/2]];
Array[A110971, 50, 2] (* Paolo Xausa, Oct 13 2024 *)

from math import comb
def A152086(n): return ((n+1<>1)>>1 if n&1 else (n-1)*comb(n-2,n-2>>1)<<1)) # Chai Wah Wu, Oct 28 2024

a(n) = A102699(n)/2. - Paolo Xausa, Oct 13 2024, from N. J. A. Sloane formula in A102699.

More terms from Paolo Xausa, Oct 13 2024

A377000 Array read by ascending antidiagonals: T(n,k) = number of n-esthetic numbers with k digits.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 8, 6, 1, 6, 9, 12, 13, 8, 1, 7, 11, 16, 21, 21, 12, 1, 8, 13, 20, 29, 36, 34, 16, 1, 9, 15, 24, 37, 52, 63, 55, 24, 1, 10, 17, 28, 45, 68, 94, 108, 89, 32, 1, 11, 19, 32, 53, 84, 126, 169, 189, 144, 48, 1, 12, 21, 36, 61, 100, 158, 232, 305, 324, 233, 64, 1

Offset: 2

Paolo Xausa, Oct 12 2024

A number is n-esthetic if, when written in base n, adjacent digits differ by 1: see De Koninck and Doyon (2009), where T(n,k) is denoted by N_q(r).

			Array begins (cf. De Koninck and Doyon (2009), table on p. 155):
  n\k| 1   2   3   4    5    6    7    8     9    10  ...
  -------------------------------------------------------
   2 | 1,  1,  1,  1,   1,   1,   1,   1,    1,    1, ... = A000012
   3 | 2,  3,  4,  6,   8,  12,  16,  24,   32,   48, ... = A029744 (from n = 2)
   4 | 3,  5,  8, 13,  21,  34,  55,  89,  144,  233, ... = A000045 (from n = 4)
   5 | 4,  7, 12, 21,  36,  63, 108, 189,  324,  567, ... = A228879
   6 | 5,  9, 16, 29,  52,  94, 169, 305,  549,  990, ...
   7 | 6, 11, 20, 37,  68, 126, 232, 430,  792, 1468, ...
   8 | 7, 13, 24, 45,  84, 158, 296, 557, 1045, 1966, ...
   9 | 8, 15, 28, 53, 100, 190, 360, 685, 1300, 2475, ...
  10 | 9, 17, 32, 61, 116, 222, 424, 813, 1556, 2986, ... = A090994
  ...                                               \______ A152086 (main diagonal)

Paolo Xausa, Table of n, a(n) for n = 2..11326 (first 150 antidiagonals, flattened).
Jean-Marie De Koninck and Nicolas Doyon, Esthetic Numbers, Ann. Sci. Math. Québec 33 (2009), No. 2, pp. 155-164.
Giovanni Resta, Esthetic Numbers, Numbers Aplenty, 2013.
Branko J. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33 (arXiv version, arXiv:0704.0750 [math.DG], 2007).

Cf. A000012 (row n = 2), A029744 (row n = 3), A000045 (row n = 4), A228879 (row n = 5), A090994 (row n = 10).
Cf. A102699, A152086 (main diagonal).
Diagonal above the main diagonal appears to be A206603.

A377000[n_, k_] := Round[2^k/(n+1)*Sum[If[m != (n+1)/2, Cos[#]^k*(Cot[#] + Csc[#])^2 & [Pi*m/(n+1)], 0], {m, 1, n, 2}]];
Table[A377000[n-k+1, k], {n, 2, 15}, {k, n-1}]

from itertools import count, islice
from functools import lru_cache
@lru_cache(maxsize=None)
def A377000_N(q,r,i):
    if r==1 and i==0: return 0
    if r==1: return 1
    if q==2: return r+i&1^1
    if i == 0: return A377000_N(q,r-1,1)
    if i == q-1: return A377000_N(q,r-1,q-2)
    return A377000_N(q,r-1,i-1)+A377000_N(q,r-1,i+1)
def A377000_T(n,k): return sum(A377000_N(n,k,i) for i in range(n))
def A377000_gen(): # generator of terms
    for n in count(2):
        for k in range(1,n):
            yield A377000_T(n-k+1,k)
A377000_list = list(islice(A377000_gen(),100)) # Chai Wah Wu, Oct 21 2024

All of the following formulas are taken from De Koninck and Doyon (2009).
T(n,k) = 2^k/(n+1) * Sum_{m=1..n, m odd, m != (n+1)/2} cos(p)^k*(cot(p) + csc(p))^2, where p = Pi*m/(n+1).
T(n,1) = n - 1.
T(2,k) = 1.
T(3,k) = 2^((k+1)/2) if k is odd, 3*2^((k-2)/2) if k is even = A029744(k+1).
T(4,k) = A000045(k+3).
T(5,k) = 4*3^((k-1)/2) if k is odd, 7*3^((k-2)/2) if k is even = A228879(k-1).
Conjectures from Chai Wah Wu, Oct 21 2024: (Start)
Conjecture 1: For even n, T(n,k) is the number of meaningful differential operations of the k-th order on the space R^(n-1).
Conjecture 2: For each n, the row T(n,k) satisfies a linear recurrence. For example:
T(6,k) = T(6,k-1) + 2*T(6,k-2) - T(6,k-3) for k > 3 (A090990).
T(7,k) = 4*T(7,k-2) - 2*T(7,k-4) for k > 4.
T(8,k) = T(8,k-1) + 3*T(8,k-2) - 2*T(8,k-3) - T(8,k-4) for k > 4 (A090992).
T(9,k) = 5*T(9,k-2) - 5*T(9,k-4) for k > 4.
T(10,k) = T(10,k-1) + 4*T(10,k-2) - 3*T(10,k-3) - 3*T(10,k-4) + T(10,k-5) for k > 5.
T(11,k) = 6*T(11,k-2) - 9*T(11,k-4) + 2*T(11,k-6) for k > 6.
T(12,k) = T(12,k-1) + 5*T(12,k-2) - 4*T(12,k-3) - 6*T(12,k-4) + 3*T(12,k-5) + T(12,k-6) for k > 6 (A129638).
...
Note that for even n, Conjecture 1 implies Conjecture 2 due to (Malesevic, 1998).
Conjecture 3: T(n,n-2) = A182555(n-2). (End)

A110971 Triangle T(n,k) (n >= 2, 1 <= k <= n-1) read by rows: row n gives epispectrum of a path P_n (see reference for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 6, 1, 1, 2, 7, 11, 10, 1, 1, 2, 8, 14, 24, 14, 1, 1, 2, 9, 16, 35, 42, 22, 1, 1, 2, 10, 18, 45, 68, 81, 30, 1, 1, 2, 11, 20, 55, 89, 149, 138, 46, 1, 1, 2, 12, 22, 66, 110, 216, 282, 250, 62, 1, 1, 2, 13, 24, 78, 132, 285, 422, 577, 419

Offset: 2

N. J. A. Sloane, Sep 20 2009

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  4,  1;
  1, 2,  6,  6,  1;
  1, 2,  7, 11, 10,   1;
  1, 2,  8, 14, 24,  14,   1;
  1, 2,  9, 16, 35,  42,  22,   1;
  1, 2, 10, 18, 45,  68,  81,  30,   1;
  1, 2, 11, 20, 55,  89, 149, 138,  46,    1;
  1, 2, 12, 22, 66, 110, 216, 282, 250,   62,   1;
  1, 2, 13, 24, 78, 132, 285, 422, 577,  419,  94,   1;
  1, 2, 14, 26, 91, 156, 364, 568, 945, 1070, 732, 126, 1;

Toufik Mansour, Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
M. A. Michels and U. Knauer, The congruence classes of paths and cycles, Discr. Math., 309 (2009), 5352-5359.

Cf. A152086, A102699.

A198335 Triangle read by rows: T(n,k) is 1/2 of the number of walks of length k (1<=k<=n-1) in the path graph on n vertices (n>=2).

Original entry on oeis.org

1, 2, 3, 3, 5, 8, 4, 7, 12, 21, 5, 9, 16, 29, 52, 6, 11, 20, 37, 68, 126, 7, 13, 24, 45, 84, 158, 296, 8, 15, 28, 53, 100, 190, 360, 685, 9, 17, 32, 61, 116, 222, 424, 813, 1556, 10, 19, 36, 69, 132, 254, 488, 941, 1812, 3498, 11, 21, 40, 77, 148, 286, 552, 1069, 2068, 4010, 7768

Offset: 2

Emeric Deutsch, Dec 01 2011

Sum of entries in row n is A144952(n) (n>=2).

T(n,n-1)=(1/2)A102699(n).

			T(3,1)=2 and T(3,2)=3 because in the path a - b - c we have 4 walks of length 1 (ab, bc, ba, cb) and 6 walks of length 2 (aba, abc, bab, bcb, cbc, cba).
Triangle starts:
1;
2,3;
3,5,8;
4,7,12,21;
5,9,16,29,52;

G. Rucker and C. Rucker, Walk counts, labyrinthicity and complexity of acyclic and cyclic graphs and molecules, J. Chem. Inf. Comput. Sci., 40 (2000), 99-106.

Cf. A144952, A102699

with(GraphTheory): T := proc (n, k) local G, A, B: G := PathGraph(n): A := AdjacencyMatrix(G): B := A^k: if k < n then (1/2)*add(add(B[i, j], i = 1 .. n), j = 1 .. n) else 0 end if end proc: for n from 2 to 12 do seq(T(n, k), k = 1 .. n-1) end do; # yields sequence in triangular form

It is known that if A is the adjacency matrix of a graph G, then the (i,j)-entry of the matrix A^k is equal to the number of walks from vertex i to vertex j. Consequently, T(n,k) is 1/2 of the sum of the entries of the matrix A^k (see the Maple program).

Keyword tabl added by Michel Marcus, Apr 09 2013

cp's OEIS Frontend

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A153338 Number of zig-zag paths from top to bottom of a 2n-1 by 2n-1 square whose color is not that of the top right corner.

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A153334 Number of zig-zag paths from top to bottom of an n X n square whose color is that of the top right corner.

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A153335 Number of zig-zag paths from top to bottom of an n X n square whose color is not that of the top right corner.

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A153336 Number of zig-zag paths from top to bottom of a 2n by 2n square whose color is that of the top right corner.

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A153337 Number of zig-zag paths from top to bottom of a 2n-1 by 2n-1 square whose color is that of the top right corner.

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A152086 a(n) = Sum_{k=1..n-1} k*A110971(n,k).

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