A140966 -id:A140966 - cp's OEIS frontend

A321373 Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.

2, 2, -1, 2, 0, 3, 2, 1, 4, 1, 2, 2, 5, 4, 7, 2, 3, 6, 7, 12, 9, 2, 4, 7, 10, 17, 20, 23, 2, 5, 8, 13, 22, 31, 44, 41, 2, 6, 9, 16, 27, 42, 65, 84, 87, 2, 7, 10, 19, 32, 53, 86, 127, 172, 169, 2, 8, 11, 22, 37, 64, 107, 170, 257, 340, 343

Offset: 0

Paul Curtz, Nov 08 2018

Array:
2, -1,  3,  1,  7,  9,  23,  41,  87, ... = (-1)^n*A140966(n)
2,  0,  4,  4, 12, 20,  44,  84, 172, ... = abs(A084247(n+1))
2,  1,  5,  7, 17, 31,  65, 127, 257, ... = A014551(n)
2,  2,  6, 10, 22, 42,  86, 170, 342, ... = A078008(n+2) = A014113(n+1)
2,  3,  7, 13, 27, 53, 107, 213, 427, ... = A048573(n)
2,  4,  8, 16, 32, 64, 128, 256, 512, ... = A000079(n+1)
2,  5,  9, 19, 37, 75, 149, 299, 597, ... = A062092(n)
2,  6, 10, 22, 42, 86, 170, 342, 682, ... = A078008(n+3) = A014113(n+2).
T(n+1,k) = (-1)^k*A140966(k) + (n+1)*A001045(k).
Every row T(n+1,k) has the signature (1,2).
T(0,k) = 2, -2, 2, -2, ... = (-1)^n*2.
T(n+1,k) - T(0,k) = (n+1)*A001045(n).
5*A001045(n) is not in the OEIS.

			Triangle a(n):
  2;
  2, -1;
  2,  0,  3;
  2,  1,  4,  1;
  2,  2,  5,  4,  7;
  2,  3,  6,  7, 12,  9;
  2,  4,  7, 10, 17, 20, 23;
  etc.
Row sums: 2, 1, 5, 8, 20, 39, 83, 166, 338, 677, 1361, 2724, ... = b(n+2).
With b(0) = 2 and b(1) = 0, b(n) = b(n-1) + 2*b(n-2)  + n - 4, n > 1.
b(n) = A001045(n) - A097065(n-1).
b(n) = b(n-2) + A000225(n-2).

Cf. A000079, A001045, A014113, A014551, A048573, A062092, A078008, A084247, A092297, A097073, A140360, A140966.

Cf. A000225, A097065.

T[_, 0] = 2;
T[0, k_] := (2^k + 5(-1)^k)/3;
T[n_ /; n>0, k_ /; k>0] := T[n, k] = T[n-1, k] + (2^k + (-1)^(k+1))/3;
T[, ] = 0;
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 7, 13, 27, 53, 107, 213, 427, 853, 1707, 3413, 6827, 13653, 27307, 54613, 109227, 218453, 436907, 873813, 1747627, 3495253, 6990507, 13981013, 27962027, 55924053, 111848107, 223696213, 447392427, 894784853, 1789569707, 3579139413, 7158278827, 14316557653

Offset: 0

Michael Somos, Jun 17 1999

Number of positive integers requiring exactly n signed bits in the modified non-adjacent form representation. - Ralf Stephan, Aug 02 2003
The n-th entry (n>1) of the sequence is equal to the 1,1-entry of the n-th power of the unnormalized 4 X 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini, Oct 27 2004
Pisano period lengths:  1, 1, 6, 2, 2, 6, 6, 2, 18, 2, 10, 6, 12, 6, 6, 2, 8, 18, 18, 2, ... - R. J. Mathar, Aug 10 2012
For n >= 1, a(n) is the number of ways to tile a strip of length n+2 with blue squares and blue and red dominos, with the restriction that the first two tiles must be the same color. - Guanji Chen and Greg Dresden, Jul 15 2024

			G.f. = 2 + 3*x + 7*x^2 + 13*x^3 + 27*x^4 + 53*x^5 + 107*x^6 + 213*x^7 + 427*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wieb Bosma, Signed bits and fast exponentiation, Journal de théorie des nombres de Bordeaux, Vol. 13, No. 1 (2001), pp. 27-41.
Karl Dilcher and Larry Ericksen, Continued fractions and Stern polynomials, Ramanujan Journal 45.3 (2018): 659-681. See Table 2.
Karl Dilcher and Hayley Tomkins, Square classes and divisibility properties of Stern polynomials, Integers, Vol. 18 (2018), Article #A29.
Petro Kosobutskyy, The Collatz problem as a reverse n->0 problem on a graph tree formed from theta*2^n Jacobsthal-type numbers, arXiv:2306.14635 [math.GM], 2023.
Petro Kosobutskyy and Dariia Rebot, Collatz conjecture 3n+/-1 as a Newton binomial problem, Comp. Des. Sys. Theor. Prac., Lviv Nat'l Polytech. Univ. (Ukraine 2023) Vol. 5, No. 1, 137-145. See p. 140.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Sam Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, Vol. 117, No. 7 (2010), pp. 581-598.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A084214 (first differences).
Cf. A001045, A014551, A024493, A062510.
Cf. A007283, A007679, A078008, A081254.
Cf. A083322, A083944, A083581, A084170.
Cf. A102713, A127978, A130755, A139763.
Cf. A140360, A140966, A166249, A168642.
Cf. A280560, A290604, A340627.
Cf. A112387, A135318.

[(5*2^n+(-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

LinearRecurrence[{1,2},{2,3},40] (* Harvey P. Dale, Dec 11 2017 *)

{a(n) = if( n<0, 0, (5*2^n + (-1)^n) / 3)};

{a(n) = if (n<0 ,0, if( n<2, n+2, a(n-1) + 2*a(n-2)))};

[(5*2^n+(-1)^n)/3 for n in range(35)] # G. C. Greubel, Apr 10 2019

G.f.: (2 + x) / (1 - x - 2*x^2).
a(n) = (5*2^n + (-1)^n) / 3.
a(n) = 2^(n+1) - A001045(n).
a(n) = A084170(n)+1 = abs(A083581(n)-3) = A081254(n+1) - A081254(n) = A084214(n+2)/2.
a(n) = 2*A001045(n+1) + A001045(n) (note that 2 is the limit of A001045(n+1)/A001045(n)). - Paul Barry, Sep 14 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=-charpoly(A,-1). - Milan Janjic, Jan 27 2010
Equivalently, with different offset, a(n) = b(n+1) with b(0)=1 and b(n) = Sum_{i=0..n-1} (-1)^i (1 + (-1)^i b(i)). - Olivier Gérard, Jul 30 2012
a(n) = A000975(n-2)*10 + 5 + 2*(-1)^(n-2), a(0)=2, a(1)=3. - Yuchun Ji, Mar 18 2019
a(n+1) = Sum_{i=0..n} a(i) + 1 + (1-(-1)^n)/2, a(0)=2. - Yuchun Ji, Apr 10 2019
a(n) = 2^n + J(n+1) = J(n+2) + J(n+1) - J(n), where J is A001045. - Yuchun Ji, Apr 10 2019
a(n) = A001045(n+2) + A078008(n) = A062510(n+1) - A078008(n+1) = (A001045(n+2) + A062510(n+1))/2 = A014551(n) + 2*A001045(n). - Paul Curtz, Jul 14 2021
From Thomas Scheuerle, Jul 14 2021: (Start)
a(n) = A083322(n) + A024493(n).
a(n) = A127978(n) - A102713(n).
a(n) = A130755(n) - A166249(n).
a(n) = A007679(n) + A139763(n).
a(n) = A168642(n) XOR A007283(n).
a(n) = A290604(n) + A083944(n). (End)
From Paul Curtz, Jul 21 2021: (Start)
a(n) = 5*A001045(n) - A280560(n+1) = abs(A140360(n+1)) - A280560(n+1).
a(n) = 2^n + A001045(n+1) = A001045(n+3) - A000079(n).
a(n) = A001045(n+4) - A340627(n). (End)
a(n) = A001045(n+5) - A005010(n).
a(n+1) + a(n) = a(n+2) - a(n) = 5*2^n. - Michael Somos, Feb 22 2023
a(n) = A135318(2*n) + A135318(2*n+1) = A112387(2*n) + A112387(2*n+1). - Paul Curtz, Jun 26 2024
E.g.f.: (cosh(x) + 5*cosh(2*x) - sinh(x) + 5*sinh(2*x))/3. - Stefano Spezia, May 18 2025

Formula of Milan Janjic moved here from wrong sequence by Paul D. Hanna, May 29 2010

A163834 a(n) = (4^n + 5)/3.

Original entry on oeis.org

2, 3, 7, 23, 87, 343, 1367, 5463, 21847, 87383, 349527, 1398103, 5592407, 22369623, 89478487, 357913943, 1431655767, 5726623063, 22906492247, 91625968983, 366503875927, 1466015503703, 5864062014807, 23456248059223, 93824992236887, 375299968947543

Offset: 0

Juri-Stepan Gerasimov, Aug 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000027, A135351, A140966, A153643.

Table[(4^n + 5)/3, {n, 0, 50}] (* G. C. Greubel, Aug 05 2017 *)
LinearRecurrence[{5,-4},{2,3},30] (* Harvey P. Dale, Jun 14 2023 *)

x='x+O('x^50); concat([0], Vec((2-7*x)/((4*x-1)*(x-1)))) \\ G. C. Greubel, Aug 05 2017

a(n) = (4^n + 5)/3 = A135351(2*n+1) = A140966(2*n) = A153643(2*n).
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: (2-7*x)/((4*x-1)*(x-1)).
a(n+1) - a(n) = A000302(n).
E.g.f.: (1/3)*(5*exp(x) + exp(4*x)). - G. C. Greubel, Aug 05 2017

Offset set to 0 by R. J. Mathar, Aug 06 2009

A083582 a(n) = (82^n-5(-1)^n)/3.

Original entry on oeis.org

1, 7, 9, 23, 41, 87, 169, 343, 681, 1367, 2729, 5463, 10921, 21847, 43689, 87383, 174761, 349527, 699049, 1398103, 2796201, 5592407, 11184809, 22369623, 44739241, 89478487, 178956969, 357913943, 715827881, 1431655767, 2863311529

Offset: 0

Paul Barry, May 01 2003

Binomial transform of A083581.

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

Cf. A140966.

[(8*2^n-5*(-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

BB := n->if n=1 then 3; > elif n=2 then 1; > else 2*BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 2 to 32 do L:=[op(L),BB(k)]: od: L; # Zerinvary Lajos, Mar 19 2007

f[n_]:=2/(n+1);x=6;Table[x=f[x];Denominator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{1,7},40] (* Harvey P. Dale, May 28 2017 *)

a(n)=(8*2^n-5*(-1)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (8*2^n-5(-1)^n)/3.
G.f.: (1+6*x)/((1-2*x)*(1+x)).
E.g.f.: (8*exp(2*x)-5*exp(-x))/3.
a(n) = 6*A001045(n) + A001045(n+1). - Creighton Dement, Mar 25 2005
a(n) = 2^(n+2)th coefficient of - eta(z)^3 eta(z^5) eta(z^10)^2 /eta(z^2)^2. - Kok Seng Chua (chuaks(AT)ihpc.a-star.edu.sg), Aug 30 2005
a(n) = a(n-1)+2*a(n-2). a(n)+a(n+1) = 8*A000079 = a(n+2)-a(n). - Paul Curtz, Jul 27 2008

A154570 The main diagonal of the successive differences of A154127.

Original entry on oeis.org

1, 3, -4, 2, -6, -2, -14, -18, -46, -82, -174, -338, -686, -1362, -2734, -5458, -10926, -21842, -43694, -87378, -174766, -349522, -699054, -1398098, -2796206, -5592402, -11184814, -22369618, -44739246, -89478482, -178956974, -357913938, -715827886

Offset: 0

Paul Curtz, Jan 12 2009

sign

Index entries for linear recurrences with constant coefficients, signature (1, 2).

Cf. A020988, A047849, A078008, A083582, A140966, A154127.

f[n_]:=2/(n+1);x=6;Table[x=f[x];Numerator[x],{n,0,5!}](* Absolute Values *) (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)

a(n) = a(n-1) + 2*a(n-2), n>0.
a(n+2) = 2*(-1)^(n+1)*A140966(n).
a(n+5) = -2*A083582(n).
a(2n+1) = 3 - A078008(2n) = 3 - A047849(n).
a(2n+2) = -4 - A078008(2n+1) = -4 - A020988(n).
G.f.: (1+2*x-9*x^2)/((1+x)*(1-2*x)). - R. J. Mathar, Feb 25 2009

Edited and extended by R. J. Mathar, Feb 25 2009

A171160 a(n) = a(n-1) + 2*a(n-2) with a(0)=3, a(1)=4.

Original entry on oeis.org

3, 4, 10, 18, 38, 74, 150, 298, 598, 1194, 2390, 4778, 9558, 19114, 38230, 76458, 152918, 305834, 611670, 1223338, 2446678, 4893354, 9786710, 19573418, 39146838, 78293674, 156587350, 313174698, 626349398, 1252698794, 2505397590, 5010795178, 10021590358

Offset: 0

Paul Curtz, Dec 04 2009

J. Mulder, Table of n, a(n) for n = 0..2999
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A078008, A175805.
Cf. A062510, A083582, (-1)^n*A140966.
Cf. A092297, A154879.
Cf. A062092, A083595.

f[n_]:=2/(n+1);x=5;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{3,4},40] (* Harvey P. Dale, Sep 04 2013 *)

Vec(-(x+3)/((x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Feb 10 2015

a(n) = (1/3)*(2*(-1)^n + 7*2^n), with n>=0. - Paolo P. Lava, Dec 14 2009
G.f.: -(x+3) / ((x+1)*(2*x-1)). - Colin Barker, Feb 10 2015
From Paul Curtz, Jun 03 2022: (Start)
a(n) = A078008(n) + A078008(n+1) + A078008(n+2).
a(n) = 2^(n+1) + A078008(n).
a(n) = A001045(n+3) - A001045(n).
(a(n) + a(n+1) = a(n+2) - a(n) = A005009(n).)
a(n) + a(n+3) = A175805(n).
a(n) = A062510(n) + A083582(n-1) with A083582(-1) = 3.
a(n) = A092297(n) + A154879(n). (End)
a(n) = 2*A062092(n-1), for n>0; 2*a(n) = A083595(n+1). - Paul Curtz, Jun 08 2022

Edited by N. J. A. Sloane, Dec 05 2009
More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010
More terms from Max Alekseyev, Apr 24 2010

A156550 a(n) = 5(-1)^nA078008(n).

Original entry on oeis.org

5, 0, 10, -10, 30, -50, 110, -210, 430, -850, 1710, -3410, 6830, -13650, 27310, -54610, 109230, -218450, 436910, -873810, 1747630, -3495250, 6990510, -13981010, 27962030, -55924050, 111848110, -223696210, 447392430, -894784850, 1789569710, -3579139410

Offset: 0

Paul Curtz, Feb 09 2009

Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A001045, A078008, A140966.

LinearRecurrence[{-1, 2}, {5, 0}, 32] (* Robert P. P. McKone, Jan 31 2021 *)

a(n) = A140966(n)+A140966(n+2).
a(n) = 3*A140966(n)-A140966(n+1).
a(n+1) = 10*(-1)^n*A001045(n).
G.f.: 5*(1+x)/(1+x-2*x^2). - R. J. Mathar, Feb 23 2009
a(n) = (5*(2+(-2)^n))/3. - Colin Barker, Jun 10 2012
a(n) = -a(n-1) + 2*a(n-2) for n > 1. - Klaus Purath, Jan 30 2021
E.g.f.: 5*exp(-2*x)*(1 + 2*exp(3*x))/3. - Stefano Spezia, Jan 30 2021

Edited and extended by R. J. Mathar Feb 23 2009

A171501 Inverse binomial transform of A084640.

Original entry on oeis.org

0, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881

Offset: 0

Paul Curtz, Dec 10 2009

a(n) and differences are
0,     1,   3,  -1,    7,   -9,
1,     2,  -4,   8,  -16,   32,  =(-1)^(n+1) * A171449(n),
1,    -6,  12, -24,   48,  -96,
-7,   18, -36,  72, -144,  288,
25,  -54, 108, -216, 432, -864,
Vertical: 1) 0 followed with A168589(n).
          2) (-1 followed with A008776(n) ) signed. See A025192(n).
Main diagonal: see A167747(1+n). - Paul Curtz, Jun 16 2011

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

I:=[0, 1, 3]; [n le 3 select I[n] else -Self(n-1) + 2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 18 2012

CoefficientList[Series[x*(1 + 4*x)/((1 + 2*x)*(1 - x)), {x, 0, 30}], x]
LinearRecurrence[{-1,2},{0,1,3},40] (* Harvey P. Dale, Jan 14 2020 *)

a(n) = A140966(n), n>0.
G.f.: x*(1+4*x) / ( (1+2*x)*(1-x) ). - R. J. Mathar, Jun 14 2011
a(1+n)= (-1)^(1+n) * A001045(1+n) + 2. - Paul Curtz, Jun 16 2011

Mathematica program by Olivier Gérard, Jul 06 2011

A247525 a(n) = 5 * a(n-1) - 2 * a(n-1)^2 / a(n-2), with a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 2, 6, -6, -42, 378, 8694, -356454, -31011498, 5240943162, 1797643504566, -1224195226609446, -1673474874775112682, 4566912933261282509178, 24949045354406386347639414, -272468524315472145302570040294, -5952619850720119958425247670303018

Offset: 0

Michael Somos, Sep 18 2014

sign

G. C. Greubel, Table of n, a(n) for n = 0..83

Cf. A140966.

I:=[1, 2]; [n le 2 select I[n] else 5*Self(n-1) - 2*Self(n-1)^2/Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 05 2018

RecurrenceTable[{a[n] == 5*a[n - 1] - 2*a[n - 1]^2/a[n - 2], a[0] == 1, a[1] == 2}, a, {n, 0, 50}] (* G. C. Greubel, Aug 05 2018 *)

{a(n) = if( n<0, 1 / prod(k=1, -n, (5 + (-2)^-k) / 3), prod(k=0, n-1, (5 + (-2)^k) / 3))};

0 = a(n)*(-5*a(n+1) + a(n+2)) + a(n+1)*(+2*a(n+1)) for all n in Z.

a(n+1) = a(n) * A140966(n) for all n in Z.

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

cp's OEIS Frontend

A321373 Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.

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A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

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PARI

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A163834 a(n) = (4^n + 5)/3.

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A083582 a(n) = (8*2^n-5*(-1)^n)/3.

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A154570 The main diagonal of the successive differences of A154127.

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A171160 a(n) = a(n-1) + 2*a(n-2) with a(0)=3, a(1)=4.

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A156550 a(n) = 5*(-1)^n*A078008(n).

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A083582 a(n) = (82^n-5(-1)^n)/3.

A156550 a(n) = 5(-1)^nA078008(n).