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 A352692 #70 Apr 26 2022 20:56:35
%S A352692 4,-3,5,-1,9,7,25,39,89,167,345,679,1369,2727,5465,10919,21849,43687,
%T A352692 87385,174759,349529,699047,1398105,2796199,5592409,11184807,22369625,
%U A352692 44739239,89478489,178956967,357913945,715827879,1431655769,2863311527,5726623065,11453246119,22906492249
%N A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.
%C A352692 Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
%C A352692 4, -3, 5, -1, 9, 7, 25, ...
%C A352692 -7, 8, -6, 10, -2, 18, 14, 50, ...
%C A352692 15, -14, 16, -12, 20, -4, 36, 28, 100, ...
%C A352692 -29, 30, -28, 32, -24, 40, -8, 72, 56, 200, ...
%C A352692 59, -58, 60, -56, 64, -48, 80, -16, 144, 112, 400, ...
%C A352692 ...
%C A352692 The diagonals are given by D(n,n+k) = a(k)*2^n.
%C A352692 D(n,1) = -(-1)^n* A340627(n).
%C A352692 a(n) - a(n) = 0, 0, 0, 0, 0, ... (trivially)
%C A352692 a(n+1) + a(n) = 1, 2, 4, 8, 16, ... = 2^n (by definition)
%C A352692 a(n+2) - a(n) = 1, 2, 4, 8, 16, ... = 2^n
%C A352692 a(n+3) + a(n) = 3, 6, 12, 24, 48, ... = 2^n*3
%C A352692 a(n+4) - a(n) = 5, 10, 20, 40, 80, ... = 2^n*5
%C A352692 a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
%C A352692 (...)
%C A352692 This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
%C A352692 Sums of antidiagonals are A045883(n).
%C A352692 Main diagonal: A192382(n).
%C A352692 First upper diagonal: A054881(n+1).
%C A352692 First subdiagonal: A003683(n+1).
%C A352692 Second subdiagonal: A246036(n).
%C A352692 Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
%C A352692 4, 3, 5, 1, 9, -7, 25, -39, ... = c(n)
%C A352692 -1, 2, -4, 8, -16, 32, -64, 128, ... = -A122803(n)
%C A352692 3, -6, 12, -24, 48, -96, 192, -384, ... =
%C A352692 -9, 18, -36, 72, -144, 288, -576, 1152, ...
%C A352692 27, -54, 108, -216, 432, -864, 1728, -3456, ...
%C A352692 ...
%C A352692 The first subdiagonal is -A000400(n). The second is A169604(n).
%H A352692 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%F A352692 abs(a(n)) = A115335(n-1) for n >= 1.
%F A352692 a(3*n) - (-1)^n*4 = A132805(n).
%F A352692 a(3*n+1) + (-1)^n*4 = A082311(n).
%F A352692 a(3*n+2) - (-1)^n*4 = A082365(n).
%F A352692 From _Thomas Scheuerle_, Mar 29 2022: (Start)
%F A352692 G.f.: (-4 + 7*x)/(-1 + x + 2*x^2).
%F A352692 Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(m + 2*n-k) = a(m)*2^n.
%F A352692 Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(1 + n-k) = -(-1)^n*A340627(n).
%F A352692 a(n) = (11*(-1)^n + 2^n)/3.
%F A352692 a(n + 2*m) = a(n) + A002450(m)*2^n.
%F A352692 a(2*n) = A192382(n+1) + (-1)^n*a(n).
%F A352692 a(n) = ( A045883(n) - Sum_{k=0..n-1}(-1)^k*a(k) )/n, for n > 0. (End)
%F A352692 a(n) = A001045(n) + 4*(-1)^n.
%F A352692 a(n+1) = 2*a(n) -11*(-1)^n.
%F A352692 a(n+2) = a(n) + 2^n.
%F A352692 a(n+4) = a(n) + A020714(n).
%F A352692 a(n+6) = a(n) + A175805(n).
%F A352692 a(2*n) = A163868(n).
%F A352692 a(2*n+1) = (2^(2*n+1) - 11)/3.
%p A352692 a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # _Peter Luschny_, Mar 29 2022
%p A352692 A352691 := proc(n)
%p A352692 (11*(-1)^n + 2^n)/3
%p A352692 end proc: # _R. J. Mathar_, Apr 26 2022
%t A352692 LinearRecurrence[{1, 2}, {4, -3}, 40] (* _Amiram Eldar_, Mar 29 2022 *)
%o A352692 (PARI) a(n) = (11*(-1)^n + 2^n)/3; \\ _Thomas Scheuerle_, Mar 29 2022
%Y A352692 If a(0) = k then A001045 (k=0), A078008 (k=1), A140966 (k=2), A154879 (k=3), this sequence (k=4).
%Y A352692 Essentially the same as A115335.
%Y A352692 Cf. A000079, A002450, A340627.
%Y A352692 Cf. A020714, A175805.
%Y A352692 Cf. A045883, A192382, A003683, A246036.
%Y A352692 Cf. A054881, A000400, A169604, A122803.
%Y A352692 Cf. A132805, A082311, A082365.
%Y A352692 Cf. A024495, A132804, A163868.
%K A352692 sign,easy
%O A352692 0,1
%A A352692 _Paul Curtz_, Mar 29 2022
%E A352692 Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022
%E A352692 Edited by _M. F. Hasler_, Apr 26 2022.