This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343314 #7 Apr 11 2021 23:52:02
%S A343314 0,0,1,1,2,7,5,16,19,39,77,103,226,334,636,1106,1827,3386,5568,10059,
%T A343314 17281,29890,52771,90283,159191,274976,479035,835476,1447278,2528496,
%U A343314 4386143,7640592,13293308,23106132,40245277,69946521,121762316,211791205,368418674,641125867
%N A343314 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A063720).
%C A343314 The nonnegative integers are displayed as in A063720.
%C A343314 Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = 5 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 A343314 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 A343314 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,5,1,1).
%H A343314 <a href="/index/Ca#calculatordisplay">Index entries for sequences related to calculator display</a>
%H A343314 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A343314 G.f.: x^2*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + 5*x^3 + x^4 + x^5)/(1 - x^2 - x^3 - x^4 - 5*x^5 - x^6 - x^7).
%F A343314 a(n) = a(n-2) + a(n-3) + a(n-4) + 5*a(n-5) + a(n-6) + a(n-7) for n > 13.
%F A343314 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 + 5*x^5 + x^6 + x^7.
%e A343314 a(6) = 5 since 0, 14, 41, 77 and 111 are displayed by 6 segments.
%e A343314 __ __ __
%e A343314 | | | |__| |__| | | | | | |
%e A343314 |__| | | | | | | | | |
%e A343314 (0) (14) (41) (77) (111)
%t A343314 P[x_]:=x^2+x^3+x^4+5x^5+x^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 A343314 Cf. A002426, A004526, A063720, A216261, A331529, A331530, A343315.
%K A343314 nonn,base,easy
%O A343314 0,5
%A A343314 _Stefano Spezia_, Apr 11 2021