A154570 -id:A154570 - cp's OEIS frontend

A154589 a(n)=A154570(n)+A154570(n+1).

4, -1, -2, -4, -8, -16, -32, -64, -128, -256, -512, -1024, -2048, -4096, -8192, -16384, -32768, -65536

Offset: 0

Paul Curtz, Jan 12 2009

G.f.: (4-9x)/(1-2x). a(n+1)= -A000079(n). R. J. Mathar, Feb 25 2009

E.g.f.: 5 + x/(G(0)+1), where G(k) = 2*k - 1 - x/(1 + x/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Dec 27 2013

Edited by N. J. A. Sloane, Jan 14 2009

A156591 First differences of A154570.

Original entry on oeis.org

2, -7, 6, -8, 4, -12, -4, -28, -36, -92, -164, -348, -676, -1372, -2724, -5468, -10916, -21852, -43684, -87388, -174756, -349532, -699044, -1398108, -2796196, -5592412, -11184804, -22369628, -44739236, -89478492, -178956964, -357913948, -715827876

Offset: 0

Paul Curtz, Feb 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 2).

LinearRecurrence[{1,2},{2,-7,6},40] (* Harvey P. Dale, Feb 01 2025 *)

a(n)+a(n+1) = -A000079(n-1), n>0.
a(n+1) = 4*(-1)^(n+1)- A154879(n).
G.f.: (1-3*x)*(2-3*x)/((1+x)*(1-2*x)). a(n) = -2^n/6+20*(-1)^n/3, n>0. - R. J. Mathar, Feb 25 2009

Edited by R. J. Mathar, Feb 25 2009

A157823 a(n) = A156591(n) + A156591(n+1).

Original entry on oeis.org

-5, -1, -2, -4, -8, -16, -32, -64, -128, -256, -512, -1024, -2048, -4096, -8192, -16384, -32768, -65536, -131072, -262144, -524288, -1048576, -2097152, -4194304, -8388608, -16777216, -33554432, -67108864, -134217728, -268435456, -536870912, -1073741824

Offset: 0

Paul Curtz, Mar 07 2009

A156591 = 2,-7,6,-8,4,-12,... a(n) is companion to A154589 = 4,-1,-2,-4,-8,.For this kind ,companion of sequence b(n) is first differences a(n), second differences being b(n). Well known case: A131577 and A011782. a(n)+b(n)=A000079 or -A000079. a(n)=A154570(n+2)-A154570(n) ,A154570 = 1,3,-4,2,-6,-2,-14,. See sequence(s) identical to its p-th differences (A130785,A130781,A024495,A000749,A138112(linked to Fibonacci),A139761).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Vec(-(9*x-5)/(2*x-1) + O(x^100)) \\ Colin Barker, Feb 03 2015

a(n) = 2*a(n-1) for n>1. G.f.: -(9*x-5) / (2*x-1). - Colin Barker, Feb 03 2015

Edited by Charles R Greathouse IV, Oct 11 2009

A268741 a(n) = 2*a(n - 2) - a(n - 1) for n>1, a(0) = 4, a(1) = 5.

Original entry on oeis.org

4, 5, 3, 7, -1, 15, -17, 47, -81, 175, -337, 687, -1361, 2735, -5457, 10927, -21841, 43695, -87377, 174767, -349521, 699055, -1398097, 2796207, -5592401, 11184815, -22369617, 44739247, -89478481, 178956975, -357913937, 715827887, -1431655761, 2863311535

Offset: 0

Ilya Gutkovskiy, Feb 12 2016

In general, the ordinary generating function for the recurrence relation b(n) = 2*b(n - 2) - b(n - 1) with n>1 and b(0)=k, b(1)=m, is (k + (k + m)*x)/(1 + x - 2*x^2). This recurrence gives the closed form a(n) = ((-2)^n*(k - m) + 2*k + m).

			a(0) = (5 + 3)/2 = 4  because a(1) = 5, a(2) = 3;
a(1) = (3 + 7)/2 = 5  because a(2) = 3, a(3) = 7;
a(2) = (7 - 1)/2 = 3  because a(3) = 7, a(4) = -1, etc.

Ilya Gutkovskiy, Extended graphical example
Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A084247, A140683, A140966, A154570, A171382.

Cf. A001045, A010709, A077925.

[(13-(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Feb 13 2016

Table[(13 - (-2)^n)/3, {n, 0, 33}]
LinearRecurrence[{-1, 2}, {4, 5}, 34]
RecurrenceTable[{a[1] == 4, a[2] == 5, a[n] == 2*a[n-2] - a[n-1]}, a, {n, 50}] (* Vincenzo Librandi, Feb 13 2016 *)

Vec((4 + 9*x)/(1 + x - 2*x^2) + O(x^40)) \\ Michel Marcus, Feb 25 2016

G.f.: (4 + 9*x)/(1 + x - 2*x^2).
a(n) = (13 - (-2)^n)/3.
a(n) = A084247(n) + 3.
a(n) = (-1)^n*A154570(n+1) + 1.
a(n) = (-1)^(n-1)*A171382(n-1) + 2.
Limit_{n -> oo} a(n)/a(n + 1) = -1/2.
a(n) = 4 - (-1)^n *A001045(n). - Paul Curtz, Feb 26 2024

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A154589 a(n)=A154570(n)+A154570(n+1).

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A156591 First differences of A154570.

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A157823 a(n) = A156591(n) + A156591(n+1).

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A268741 a(n) = 2*a(n - 2) - a(n - 1) for n>1, a(0) = 4, a(1) = 5.

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