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%I #7 Apr 11 2021 23:52:12
%S 0,0,1,1,2,6,6,14,19,36,67,103,194,315,560,971,1651,2895,4933,8581,
%T 14798,25515,44165,76067,131563,227034,392032,677152,1168742,2018769,
%U 3485255,6018422,10392472,17943750,30985861,53501944,92385050,159523542,275451221,475633952
%N a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A277116).
%C The nonnegative integers are displayed as in A277116.
%C Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = 4, f(6) = 2 and f(s) = 1 elsewhere, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see third formula).
%H Steffen Eger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Eger/eger6.html"> Restricted Weighted Integer Compositions and Extended Binomial Coefficients</a>, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,4,2,1).
%H <a href="/index/Ca#calculatordisplay">Index entries for sequences related to calculator display</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: x^2*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + 4*x^3 + 2*x^4 + x^5)/(1 - x^2 - x^3 - x^4 - 4*x^5 - 2*x^6 - x^7).
%F a(n) = a(n-2) + a(n-3) + a(n-4) + 4*a(n-5) + 2*a(n-6) + a(n-7) for n > 13.
%F a(n) = b(n) - b(n-6), where b(n) = [x^n] Sum_{k=max(1,ceiling(n/7))..floor(n/2)} P(x)^k with P(x) = x^2 + x^3 + x^4 + 4*x^5 + 2*x^6 + x^7.
%e a(5) = 6 since 2, 3, 5, 9, 17 and 71 are displayed by 5 segments.
%e __ __ __ __ __ __
%e __| __| |__ |__| | | | |
%e |__ __| __| | | | | |
%e (2) (3) (5) (9) (17) (71)
%t P[x_]:=x^2+x^3+x^4+4x^5+2x^6+x^7; b[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; a[n_]:=b[n]-b[n-6]; Array[a, 40, 0]
%Y Cf. A002426, A004526, A216261, A277116, A331529, A331530, A343314.
%K nonn,base,easy
%O 0,5
%A _Stefano Spezia_, Apr 11 2021
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