A235163 - OEIS

%I #33 Sep 26 2021 08:11:46

%S 9,26,75,217,629,1826,5307,15438,44941,130900,381444,1111926,3242224,

%T 9455987,27583372,80472698,234799873,685149328,1999414181,5835044495,

%U 17029601028,49702671494,145066398937,423412132499,1235854038791,3607255734629,10529101874491

%N Number of positive integers with n digits in which adjacent digits differ by at most 1.

%H Alois P. Heinz, <a href="/A235163/b235163.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,1,6,-1).

%F a(n) = Sum_{r=0..9} u(n,r) where u(n,r) = 0 if r<0 or r>9, u(1,0) = 0, u(1,r) = 1 for 1<=r<=9, and otherwise u(n,r) = u(n-1,r-1) + u(n-1,r) + u(n-1,r+1).

%F G.f.: -x*(3*x^4-18*x^3-9*x^2+28*x-9)/(x^5-6*x^4-x^3+10*x^2-6*x+1). - _Alois P. Heinz_, Jan 12 2014

%e a(2) = 26: 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99.

%p u:= proc(n, r) option remember; `if`(n=1, `if`(r=0, 0, 1),

%p       add(`if`(r+i in [$0..9], u(n-1, r+i), 0), i=-1..1))

%p     end:

%p a:= n-> add(u(n, r), r = 0..9):

%p seq(a(n), n=1..30);  # _Alois P. Heinz_, Jan 12 2014

%t CoefficientList[Series[-x*(3*x^4-18*x^3-9*x^2+28*x-9)/(x^5-6*x^4-x^3+10*x^2-6*x+1),{x,0,30}],x]//Rest (* _Harvey P. Dale_, Aug 13 2019 *)

%o (Python)

%o from functools import cache

%o @cache

%o def u(n, r):

%o     if r < 0 or r > 9: return 0

%o     if n == 1: return (r > 0)

%o     return u(n-1, r-1) + u(n-1, r) + u(n-1, r+1)

%o def a(n): return sum(u(n, r) for r in range(10))

%o print([a(n) for n in range(1, 28)]) # _Michael S. Branicky_, Sep 26 2021

%Y Cf. A032981, A090994, A126364 (allowing leading zeros).

%K nonn,base,easy

%O 1,1

%A _Gerry Leversha_, Jan 04 2014