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a(n) is the number of integers that can be represented in a 7-segment display by using only n segments (version A010371).
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%I #14 Jan 11 2022 22:05:08

%S 0,0,1,1,3,6,11,14,23,39,71,118,195,317,537,906,1533,2550,4261,7119,

%T 11973,20073,33650,56277,94286,157960,264843,443656,743269,1244915,

%U 2085970,3494922,5855965,9810370,16436113,27536138,46135634,77295509,129501787,216963199,363500178

%N a(n) is the number of integers that can be represented in a 7-segment display by using only n segments (version A010371).

%C The integers are displayed as in A010371, where a 7 is depicted by 4 segments. The negative integers are depicted by using 1 segment more for the minus sign.

%C Since the integer 0 is depicted by 6 segments, in order to avoid considering -0 in the case n = 7, a(7) is obtained by decreasing of a unit the result of the sum A331530(7) + A331530(6) = 7 + 8 = 15, i.e., a(7) = 15 - 1 = 14.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,2,3,3,1).

%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 a(7) = 14, otherwise a(n) = A331530(n) + A331530(n-1).

%F G.f.: x^2*(1 + x + 2*x^2 + 5*x^3 + 6*x^4 + 3*x^5 -2x^8- 3*x^9 - 3*x^10 - x^11)/(1 - x^2 -2 x^4 - 3*x^5 - 3*x^6 - x^7).

%F a(n) = a(n-2) + 2*a(n-4) + 3*a(n-5) + 3*a(n-6) + a(n-7) for n > 13.

%e a(7) = 14 since -111, -71, -41, -17, -14, -9, -6, 8, 12, 13, 15, 21, 31 and 51 are displayed by 7 segments.

%e __ __

%e __ | | | __ | | | __ |__| | __ | | | __ | |__|

%e | | | | | | | | | | |

%e (-111) (-71) (-41) (-17) (-14)

%e __ __ __ __ __ __ __

%e __ |__| __ |__ |__| | __| | __| | |__ __| |

%e __| |__| |__| | |__ | __| | __| |__ |

%e (-9) (-6) (8) (12) (13) (15) (21)

%e __ __

%e __| | |__ |

%e __| | __| |

%e (31) (51)

%t P[x_]:=x^2+2x^4+3x^5+3x^6+x^7; c[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; b[n_]:=c[n]-c[n-6]; (* A331530 *)

%t a[n_]:=If[n!=7,b[n]+b[n-1],14]; Array[a, 41, 0]

%Y Cf. A010371, A331530.

%K nonn,base,easy

%O 0,5

%A _Stefano Spezia_, Dec 31 2021