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a(1) = 6; for n>1, a(n) = 7 * a(n-1) + 7 - n.
7

%I #25 May 29 2023 07:13:06

%S 6,47,333,2334,16340,114381,800667,5604668,39232674,274628715,

%T 1922401001,13456807002,94197649008,659383543049,4615684801335,

%U 32309793609336,226168555265342,1583179886857383,11082259208001669,77575814456011670,543030701192081676

%N a(1) = 6; for n>1, a(n) = 7 * a(n-1) + 7 - n.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-15,7).

%F G.f.: x * (6 - 7 * x)/((1 - x)^2 * (1 - 7 * x)).

%F a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).

%F a(n) = 5 * A014830(n) + n.

%F a(n) = (5*7^(n+1) + 6*n - 35)/36.

%F a(n) = Sum_{k=0..n-1} (7 - n + k)*7^k.

%F E.g.f.: exp(x)*(35*(exp(6*x) - 1) + 6*x)/36. - _Stefano Spezia_, May 29 2023

%t LinearRecurrence[{9, -15, 7}, {6, 47, 333}, 21] (* _Amiram Eldar_, Apr 23 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(x*(6-7*x)/((1-x)^2*(1-7*x)))

%o (PARI) a(n) = (5*7^(n+1)+6*n-35)/36;

%o (PARI) b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);

%o a(n) = b(n, 7);

%Y Cf. A064617, A353094, A353095, A353096, A353097, A353099, A353100.

%Y Cf. A014830.

%K nonn,easy

%O 1,1

%A _Seiichi Manyama_, Apr 23 2022