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%I #23 Jan 14 2023 16:51:48
%S 1,4,7,12,23,48,103,220,463,960,1967,3996,8071,16240,32599,65340,
%T 130847,261888,523999,1048252,2096791,4193904,8388167,16776732,
%U 33553903,67108288,134217103,268434780,536870183,1073741040,2147482807,4294966396,8589933631,17179868160,34359737279,68719475580
%N a(n) = 2^(n+1) - (n-1)^2.
%C Sequences of the form (k-1)^m - m^(k+1):
%C k\m | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
%C -----------------------------------------------------------------------
%C 0 | 1 | -2 | -1 | -4 | -3 | -6 | -5 |
%C 1 | 1 | -1 | -4 | -9 | -16 | -25 | -36 |
%C 2 | 1 | 0 | -7 | -26 | -63 | -124 | -215 |
%C 3 | 1 | 1 | -12 | -73 | -240 | -593 | -1232 |
%C 4 | 1 | 2 | -23 | -216 | -943 | -2882 | -7047 |
%C 5 | 1 | 3 | -43 | -665 | -3840 | -14601 | -42560 |
%C 6 | 1 | 4 | -103 | -2062 | -15759 | -75000 | -264311 |
%C 7 | 1 | 5 | -220 | -6345 | -64240 | -382849 | -1632960 |
%C 8 | 1 | 6 | -463 | -19340 | -259743 | -1936318 | -9960047 |
%C 9 | 1 | 7 | -960 | -58537 | -1044480 | -9732857 | -60204032 |
%C 10 | 1 | 8 | -1967 | -176418 | -4187743 | -48769076 | -362265615 |
%C 11 | 1 | 9 | -3996 | -530441 | -16767216 | -244040625 | -2175782336 |
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4). - _Colin Barker_, Jun 12 2014
%F G.f.: (6*x^3-4*x^2-x+1) / ((x-1)^3*(2*x-1)). - _Colin Barker_, Jun 12 2014
%e 1 = 2^(0+1) - (0-1)^2, 4 = 2^(1+1) - (1-1)^2, 7 = 2^(2+1) - (2-1)^2.
%p A243860:=n->2^(n + 1) - (n - 1)^2; seq(A243860(n), n=0..30); # _Wesley Ivan Hurt_, Jun 12 2014
%t Table[2^(n + 1) - (n - 1)^2, {n, 0, 30}] (* _Wesley Ivan Hurt_, Jun 12 2014 *)
%t LinearRecurrence[{5,-9,7,-2},{1,4,7,12},40] (* _Harvey P. Dale_, Nov 29 2015 *)
%o (Magma) [2^(n+1) - (n-1)^2: n in [0..35]];
%o (PARI) Vec((6*x^3-4*x^2-x+1)/((x-1)^3*(2*x-1)) + O(x^100)) \\ _Colin Barker_, Jun 12 2014
%Y Sequences of the form (k-1)^m - m^(k+1): A000012 (m = 0), A023444 (m = 1), (-1)*(this sequence) for m = 2, A114285 (k = 0),(A000007-A000290) for k = 1, A024001 (k = 2), A024014 (k = 3), A024028 (k = 4), A024042 (k = 5), A024056 (k = 6), A024070 (k = 7), A024084 (k = 8), A024098 (k = 9), A024112 (k = 10), A024126 (k = 11).
%K nonn,easy
%O 0,2
%A _Juri-Stepan Gerasimov_, Jun 12 2014
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