A083581 -id:A083581 - cp's OEIS frontend

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

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

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

A084244 a(0)=1, a(1)=5, a(n) = -3*a(n-1), n>1.

Original entry on oeis.org

1, 5, -15, 45, -135, 405, -1215, 3645, -10935, 32805, -98415, 295245, -885735, 2657205, -7971615, 23914845, -71744535, 215233605, -645700815, 1937102445, -5811307335, 17433922005, -52301766015, 156905298045, -470715894135, 1412147682405, -4236443047215

Offset: 0

Paul Barry, May 23 2003

Binomial transform is A083581.

Winston de Greef, Table of n, a(n) for n = 0..2081
Index entries for linear recurrences with constant coefficients, signature (-3).

Cf. A083581.

Join[{1},NestList[-3#&,5,30]] (* Harvey P. Dale, Jan 29 2013 *)

a(n)=8*0^n/3 - 5*(-3)^n/3 \\ Winston de Greef, Mar 19 2023

a(n) = 8*0^n/3 - 5*(-3)^n/3.
G.f.: (1 + 8*x)/(1 + 3*x).
E.g.f.: (8 - 5*exp(-3*x))/3.

A084640 Generalized Jacobsthal numbers.

Original entry on oeis.org

0, 1, 5, 11, 25, 51, 105, 211, 425, 851, 1705, 3411, 6825, 13651, 27305, 54611, 109225, 218451, 436905, 873811, 1747625, 3495251, 6990505, 13981011, 27962025, 55924051, 111848105, 223696211, 447392425, 894784851, 1789569705, 3579139411

Offset: 0

Paul Barry, Jun 06 2003

This is the sequence A(0,1;1,2;4) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From _Wolfdieter Lang_, Oct 18 2010]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A000225, A084639.

a084640 n = a084640_list !! n
a084640_list = 0 : 1 : (map (+ 4) $
   zipWith (+) (map (* 2) a084640_list) (tail a084640_list))
-- Reinhard Zumkeller, May 23 2013

[5*2^n/3+(-1)^n/3-2: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

LinearRecurrence[{2,1,-2},{0,1,5},40] (* Harvey P. Dale, Oct 27 2015 *)

x='x+O('x^50); Vec(x*(1+3*x)/((1-x^2)*(1-2*x))) \\ G. C. Greubel, Sep 26 2017

G.f.: x*(1+3*x)/((1-x^2)*(1-2*x)).
a(n) = a(n-1) + 2a(n-2) + 4, a(0)=0, a(1)=1.
a(n) = (5*2^n + (-1)^n - 6)/3.
a(n) = A001045(n+2) + 4*A000975(n-3).
a(n+1) - 2*a(n) = period 2: repeat 1, 3. - Paul Curtz, Apr 03 2008
Contribution from Paul Curtz, Dec 10 2009: (Start)
a(n+2) - a(n) = A020714(n).
Le the array D(n,k) of the first differences be defined via D(0,k) = a(k); D(n+1,k) = D(n,k+1)-D(n,k).
Then D(n,n) = 4*A131577(n); D(1,k) = A084214(k+1); D(2,k) = A115102(k-1) for k>0; D(3,k) = (-1)^(k+1)*A083581(k). (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=0, a(1)=1, a(2)=5. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010

A230448 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A226205(n+1), n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 1, 2, 4, 5, 1, 3, 6, 9, 16, 1, 4, 9, 15, 25, 39, 1, 5, 13, 24, 40, 64, 105, 1, 6, 18, 37, 64, 104, 169, 272, 1, 7, 24, 55, 101, 168, 273, 441, 715, 1, 8, 31, 79, 156, 269, 441, 714, 1156, 1869, 1, 9, 39, 110, 235, 425, 710, 1155, 1870, 3025, 4896

Offset: 0

Johannes W. Meijer, Oct 19 2013

Triangle T(n, k) is related to the Kn1p sums of the ‘Races with Ties’ triangle A035317. See A230447 for the Kn1p sums and A180662 for the definitions of these sums.
The row sums equal ((-1)^n*3*A083581(n) + A022379(2*n+2))/15.
Note that the partial fraction expansion of the G.f. of the terms in the n-th row of the square array Tsq(n, k) = T(n+k, k) is related to A014334, the exponential convolution of the Fibonacci numbers with themselves, and to A000032, the Lucas numbers.

			The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1
1|  1,  0
2|  1,  1,  3
3|  1,  2,  4,   5
4|  1,  3,  6,   9,  16
5|  1,  4,  9,  15,  25,   39
6|  1,  5, 13,  24,  40,   64,  105
7|  1,  6, 18,  37,  64,  104,  169,   272
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
n/k 0   1   2    3    4    5      6     7
------------------------------------------------
0|  1,  0,  3,   5,  16,  39,   105,  272
1|  1,  1,  4,   9,  25,  64,   169,  441
2|  1,  2,  6,  15,  40,  104,  273,  714
3|  1,  3,  9,  24,  64,  168,  441, 1155
4|  1,  4, 13,  37, 101,  269,  710, 1865
5|  1,  5, 18,  55, 156,  425, 1135, 3000
6|  1,  6, 24,  79, 235,  660, 1795, 4795
7|  1,  7, 31, 110, 345, 1005, 2800, 7595

Cf. (Triangle columns) A000012, A001477, A152950, A226205, A007598, A001654, A064831, A080145

T := proc(n, k) option remember: if k=0 then return(1) elif k=n then return(combinat[fibonacci](n+2)*combinat[fibonacci](n-1)) else procname(n-1, k-1) + procname(n-1, k) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End first program.
T := proc(n, k): add(A035317(n+k-p-2, p), p=0..k) end: A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program.

T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = F(n+2) * F(n-1) = A226205(n+1) with F(n) = A000045(n), the Fibonacci numbers, n >= 0 and 0 <= k <= n.
T(n, k) = sum(A035317(n+k-p-2, p), p=0..k), n >= 0 and 0 <= k <= n.
T(n+p+2, p-2) = A080239(n+2*p-1) - sum(A035317(n-k+p-1, k+p-1), k=0..floor(n/2)), n >= 0 and p >= 2.
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
Tsq(n, k) = sum(Tsq(n-1, i), i=0..k), n >= 1 and k >= 0, with Tsq(0, k) = A226205(k+1).
The two G.f.’s given below generate the terms in the n-th row of the square array Tsq(n, k). The remarkable second G.f. is the partial fraction expansion of the first G.f..
G.f.: 1/((1-x)^(n-2)*(1+x)*(x^2-3*x+1)), n >= 0.
G.f.: sum((-1)^(n+k-1)*A014334(k+2)/(2^(k+2)*(x-1)^(n-k-2)), k=0..n-3) + 1/(5*2^(n-2)*(1+x)) + (A000032(n+1) - A000032(n-1)*x)/(5*(x^2-3*x+1)), n >= 0.

cp's OEIS Frontend

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

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

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A084244 a(0)=1, a(1)=5, a(n) = -3*a(n-1), n>1.

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A084640 Generalized Jacobsthal numbers.

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A230448 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A226205(n+1), n >= 0 and 0 <= k <= n.

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