Counting from 1 to n without any consecutive numbers

Mathematica, 58 bytes, polynomial(n) time

Abs[Sum[(k-1)Hypergeometric2F1[k,k-#,2,2](#-k)!,{k,#}]-1]&

How it works

Rather than iterating over permutations with brute force, we use the inclusion–exclusion principle to count them combinatorially.

Let S be the set of all permutations of [1, …, n] with σ1 = 1, σn = n, and let Si be the set of permutations σ ∈ S such that |σi − σi + 1| = 1. Then the count we are looking for is

|S| − |S1 ∪ ⋯ ∪ Sn − 1| = ∑2 ≤ kn + 1; 1 ≤ i2 < ⋯ < ik − 1 < n (−1)k − 2|Si2 ∩ ⋯ ∩ Sik − 1|.

Now, |Si2 ∩ ⋯ ∩ Sik − 1| only depends on k and on the number j of runs of consecutive indices in [i1, i2, …, ik − 1, ik] where for convenience we fix i1 = 0 and ik = n. Specifically,

|Si2 ∩ ⋯ ∩ Sik − 1| = 2j − 2(nk)!, for 2 ≤ jkn,
|Si2 ∩ ⋯ ∩ Sik − 1| = 1, for j = 1, k = n + 1.

The number of such index sets [i1, i2, …, ik − 1, ik] with j runs is

(k − 1Cj − 1)(nkCj − 2), for 2 ≤ jkn,
1, for j = 1, k = n + 1.

The result is then

(−1)n − 1 + ∑2 ≤ kn2 ≤ jk (−1)k − 2(k − 1Cj − 1)(nkCj − 2)2j − 2(nk)!

The inner sum over j can be written using the hypergeometric 2F1 function:

(−1)n − 1 + ∑2 ≤ kn (−1)k(k − 1)2F1(2 − k, kn; 2; 2)(nk)!

to which we apply a Pfaff transformation that lets us golf away the powers of −1 using an absolute value:

(−1)n − 1 + ∑2 ≤ kn (−1)n(k − 1)2F1(k, kn; 2; 2)(nk)!
= |−1 + ∑1 ≤ kn (k − 1)2F1(k, kn; 2; 2)(nk)!|.

Demo

In[1]:= Table[Abs[Sum[(k-1)Hypergeometric2F1[k,k-#,2,2](#-k)!,{k,#}]-1]&[n],{n,50}]

Out[1]= {1, 0, 0, 0, 0, 2, 10, 68, 500, 4174, 38774, 397584, 4462848, 

>    54455754, 717909202, 10171232060, 154142811052, 2488421201446, 

>    42636471916622, 772807552752712, 14774586965277816, 297138592463202402, 

>    6271277634164008170, 138596853553771517492, 3200958202120445923684, 

>    77114612783976599209598, 1934583996316791634828454, 

>    50460687385591722097602304, 1366482059862153751146376304, 

>    38366771565392871446940748410, 1115482364570332601576605376898, 

>    33544252621178275692411892779180, 1042188051349139920383738392594332, 

>    33419576037745472521641814354312790, 

>    1105004411146009553865786545464526206, 

>    37639281863619947475378460886135133496, 

>    1319658179153254337635342434408766065896, 

>    47585390139805782930448514259179162696722, 

>    1763380871412273296449902785237054760438426, 

>    67106516021125545469475040472412706780911268, 

>    2620784212531087457316728120883870079549134420, 

>    104969402113244439880057492782663678669089779118, 

>    4309132147486627708154774750891684285077633835734, 

>    181199144276064794296827392186304334716629346180848, 

>    7800407552443042507640613928796820288452902805286368, 

>    343589595090843265591418718266306051705639884996218154, 

>    15477521503994968035062094274002250590013877419466108978, 

>    712669883315580566495978374316773450341097231239406211100, 

>    33527174671849317156037438120623503416356879769273672584588, 

>    1610762789255012501855846297689494046193178343355755998487686}

Jelly, 17 16 bytes

ṖḊŒ!ð1;;⁹IỊṀðÐḟL

A monadic link.

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How?

ṖḊŒ!ð1;;⁹IỊṀðÐḟL - Link: number n
Ṗ                - pop (implicit range build) -> [1,n-1]
 Ḋ               - dequeue -> [2,n-1]
  Œ!             - all permutations of [2,n-1]
    ð       ðÐḟ  - filter discard those entries for which this is truthy:
     1;          -   1 concatenated with the entry
       ;⁹        -   ...concatenated with right (n)
         I       -   incremental differences
          Ị      -   is insignificant (absolute value <=1)
           Ṁ     -   maximum
               L - length (the number of valid arrangements)

MATL, 16 bytes

qtq:Y@0&Yc!d|qAs

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For inputs exceeding 12 it runs out of memory.

Explanation

q      % Implicitly input n. Push n-1
tq     % Duplicate and subtract 1: pushes n-2
:      % Range [1 2 ... n-2]
Y@     % Matrix with all permutations, each in a row
0      % Push 0
&Yc    % Append n-1 and predend 0 to each row
!      % Tranpose
d      % Consecutive differences along each column
|      % Absolute value
q      % Subtract 1
A      % All: true if all values in each column are non-zero
s      % Sum. Implicitly display