A001045 -id:A001045 - cp's OEIS frontend

A073372 Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 3, 12, 34, 99, 261, 678, 1692, 4149, 9959, 23568, 55014, 127031, 290457, 658602, 1482240, 3314025, 7365915, 16285300, 35832810, 78500811, 171293293, 372412782, 806963364, 1743173469, 3754782351, 8066319768, 17285917742, 36957928479, 78847115649

Offset: 0

Wolfdieter Lang, Aug 02 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-11,-6,12,8).

Third (m=2) column of triangle A073370.

Cf. A001045, A073371.

[(2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162: n in [0..40]]; // G. C. Greubel, Sep 28 2022

CoefficientList[Series[-(-1+x+2x^2)^(-3),{x,0,78}],x] (* or *) Table[(-3*(-1)^n*n^2+3*2^(n+2)*n^2-15*(-1)^n*n+9*2^(n+2)*n-16*(-1)^n+2^(n+4))/162,{n,42}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

def A073372(n): return (2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162
[A073372(n) for n in range(40)] # G. C. Greubel, Sep 28 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073371(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2) * binomial(n-k, k) * 2^k.
a(n) = ((30+9*n)*(n+1)*U(n+1) + 2*(33+9*n)*(n+2)*U(n))/162 with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1 - (1+2*x)*x)^3.
E.g.f.: (1/162)*(32*(4 + 9*x + 3*x^2)*exp(2*x) + (34 - 24*x + 3*x^2)*exp(-x)). - G. C. Greubel, Sep 28 2022

A073399 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n>=0, (generalized (1,2)-Fibonacci). Companion triangle is A073400.

Original entry on oeis.org

1, 9, 30, 63, 531, 1050, 405, 6165, 29610, 44520, 2511, 59454, 502821, 1789614, 2245320, 15309, 517104, 6686631, 41182344, 120133692, 131891760, 92583, 4214349, 76790673, 714174327, 3559509360, 8966770308, 8862693840

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..

The k-th convolution of U0(n) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^k), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073400(k,m).

			k=2: U2(n)=((9*n+30)*(n+1)*U0(n+1)+(9*n+33)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 9,30; 63,531,1050; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073400-A073402, A073372.

Recursion for row polynomials defined in the comments: see A073401.

A073400 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073399.

Original entry on oeis.org

2, 9, 33, 45, 396, 831, 243, 3744, 18297, 28236, 1377, 32481, 273483, 968679, 1210140, 8019, 268029, 3418767, 20681811, 58920534, 62686440, 47385, 2130138, 38186478, 347584284, 1683064737, 4075425738, 3810867480

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073399(k,m).

			k=2: U2(n)=((9*n+30)*(n+1)*U0(n+1)+(9*n+33)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 9,33; 45,396,831; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073399, A073401.

Recursion for row polynomials defined in the comments: see A073401.

A091593 Reversion of Jacobsthal numbers A001045.

Original entry on oeis.org

1, -1, -1, 5, -3, -21, 51, 41, -391, 407, 1927, -6227, -2507, 49347, -71109, -236079, 966129, 9519, -7408497, 13685205, 32079981, -167077221, 60639939, 1209248505, -2761755543, -4457338681, 30629783831, -22124857219, -206064020315, 572040039283, 590258340811

Offset: 0

Paul Barry, Jan 23 2004

Hankel transform is (-2)^C(n+1,2). - Paul Barry, Apr 28 2009

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A001263, A154825.

a := n -> hypergeom([-n,-n-1], [2], -2);
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 22 2014
# Using function CompInv from A357588.
CompInv(25, n -> (2^n - (-1)^n)/3 ); # Peter Luschny, Oct 07 2022

a[n_] := Hypergeometric2F1[-n - 1, -n - 1, 2, -2] + (n + 1)*Hypergeometric2F1[-n, -n, 3, -2]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 03 2016, after Vladimir Kruchinin *)

a(n) := hypergeometric([ -n - 1, -n - 1 ], [ 2 ], -2) + (n + 1) * hypergeometric([ -n, -n ], [ 3 ], -2); /* Vladimir Kruchinin, Oct 12 2011 */

# Algorithm of L. Seidel (1877)
def A091593_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; b = false; h = 1
    for i in range(2*n) :
        if b :
            for k in range(1, h, 1) : D[k] += -2*D[k+1]
            R.append(D[1])
        else :
            for k in range(h, 0, -1) : D[k] += D[k-1]
            h += 1
        b = not b
    return R
A091593_list(30)  # Peter Luschny, Oct 19 2012

G.f.: (-(1+x)+sqrt(1+2*x+9*x^2))/(4*x^2). - Corrected by Seiichi Manyama, May 12 2019
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*C(k)*(-1)^(n-k)*2^k, where C(n) is A000108(n). - Paul Barry, May 16 2005
G.f.: 1/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+ ... (continued fraction). - Paul Barry, Apr 28 2009
a(n) = Sum_{i=0..n} (2^(i)*(-1)^(n-i)*binomial(n+1,i)^2*(n-i+1)/(i+1))/(n+1). - Vladimir Kruchinin, Oct 12 2011
Conjecture: (n+2)*a(n) +(2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
a(n) = (-1)^n*hypergeom([-n/2, (1-n)/2], [2], -8). - Peter Luschny, May 28 2014
R. J. Mathar's conjecture confirmed by Maple using this hypergeom form. - Robert Israel, Sep 22 2014
a(n) = Sum_{k = 0..n} (-2)^k * (1/(n+1))*binomial(n+1, k)*binomial(n+1, k+1) =  Sum_{k = 0..n} (-2)^k * N(n+1,k+1), where N(n,k) = A001263(n,k) are the Narayana numbers. - Peter Bala, Sep 01 2023

A280049 Canonical representation of n as a sum of distinct Jacobsthal numbers J(n) (A001045) (see Comments for details); also binary numbers that end in an even number of zeros.

Original entry on oeis.org

1, 11, 100, 101, 111, 1001, 1011, 1100, 1101, 1111, 10000, 10001, 10011, 10100, 10101, 10111, 11001, 11011, 11100, 11101, 11111, 100001, 100011, 100100, 100101, 100111, 101001, 101011, 101100, 101101, 101111, 110000, 110001, 110011, 110100, 110101, 110111

Offset: 1

N. J. A. Sloane, Dec 31 2016

Every positive integer has a unique expression as a sum of distinct Jacobsthal numbers in which the index of the smallest summand is odd, with J(1) = 1 and J(2) = 1 both allowed. [Carlitz-Scoville-Hoggatt, 1972]. - Based on a comment in A001045 from Ira M. Gessel, Dec 31 2016.

The highest-order bits are on the left. Interpreting these as binary numbers we get A003159.

			9 = 5+3+1 = J(4)+J(3)+J(1) = 1101.

Lars Blomberg, Table of n, a(n) for n = 1..10000
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550.

Cf. A001045, A003159, A007088.

FromDigits[IntegerDigits[#, 2]] & /@ Select[Range[100], EvenQ[IntegerExponent[#, 2]] &] (* Amiram Eldar, Jul 14 2023 *)

lista(kmax) = for(k = 1, kmax, if(!(valuation(k, 2)%2), print1(fromdigits(binary(k), 10), ", "))); \\ Amiram Eldar, Jul 14 2023

from itertools import count, islice
def A280049_gen(): # generator of terms
    return map(lambda n:int(bin(n)[2:]),filter(lambda n:(n&-n).bit_length()&1,count(1)))
A280049_list = list(islice(A280049_gen(),20)) # Chai Wah Wu, Mar 19 2024

def A280049(n):
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return int(bin(m)[2:]) # Chai Wah Wu, Jan 30 2025

a(n) = A007088(A003159(n)). - Amiram Eldar, Jul 14 2023

Corrected a(5), a(16) and more terms from Lars Blomberg, Jan 02 2017

A153643 Jacobsthal numbers A001045 incremented by 2.

Original entry on oeis.org

2, 3, 3, 5, 7, 13, 23, 45, 87, 173, 343, 685, 1367, 2733, 5463, 10925, 21847, 43693, 87383, 174765, 349527, 699053, 1398103, 2796205, 5592407, 11184813, 22369623, 44739245, 89478487, 178956973, 357913943, 715827885, 1431655767, 2863311533, 5726623063

Offset: 0

Paul Curtz, Dec 30 2008

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

Cf. A001045, A007395, A010684, A078008, A128209.

a:=[2,3,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Apr 02 2019

I:=[2,3,3]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) -2*Self(n-3): n in [1..40]]; // G. C. Greubel, Apr 02 2019

LinearRecurrence[{1,2},{0,1}, 40] + 2 (* Harvey P. Dale, May 26 2014 *)
LinearRecurrence[{2,1,-2},{2,3,3}, 40] (* Georg Fischer, Apr 02 2019 *)

my(x='x+O('x^40)); Vec( (2-x-5*x^2)/((1-x^2)*(1-2*x)) ) \\ G. C. Greubel, Apr 02 2019

def A153643(n): return ((1<Chai Wah Wu, Apr 18 2025

((2-x-5*x^2)/((1-x^2)*(1-2*x))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 02 2019

a(n) = 2 + A001045(n) = A001045(n) + A007395(n) = 1 + A128209(n).
a(n) - A010684(n) = A078008(n), first differences of A001045. - Paul Curtz, Jan 17 2009
G.f.: (2 - x - 5*x^2)/((1+x)*(1-x)*(1-2*x)). - R. J. Mathar, Jan 23 2009
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n >= 3. - Andrew Howroyd, Feb 26 2018

Edited and extended by R. J. Mathar, Jan 23 2009

A193449 Products of the Jacobsthal numbers and the integers: a(n) = n * A001045(n+1).

Original entry on oeis.org

0, 1, 6, 15, 44, 105, 258, 595, 1368, 3069, 6830, 15015, 32772, 70993, 152922, 327675, 699056, 1485477, 3145734, 6640975, 13981020, 29360121, 61516466, 128625315, 268435464, 559240525, 1163220318, 2415919095, 5010795188, 10379504289, 21474836490, 44381328715

Offset: 0

Olivier Gérard, Jul 26 2011

a(n) = n * A001045(n+1).

This sequence is the sum of several triangles of integers (see formula section)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Cf. A001045, Equals second column of A124860, equals sum of A193450 or A193451.

[n*(2^(n + 1) + (-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Oct 21 2012

Table[Sum[n Binomial[n, k] HypergeometricPFQ[{1, -k}, {-n}, -1], {k, 0, n}], {n, 0, 35}]
CoefficientList[Series[(x*(1 + 4*x))/(2*x^2 + x - 1)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Oct 21 2012 *)

def A193449(n): return (((1<Chai Wah Wu, Apr 18 2025

G.f.:  x*(1 + 4*x)/( 2*x^2+x-1)^2
a(n) = n*(2^(n + 1) + (-1)^n)/3
a(n)= sum( sum( (-1)^(j+k)*(j+k)*C(n-k+j,j), j=0..k), k=0..n)
a(n)= sum( n*C(n, k)*2F1( (1, -k); -n )(-1), k=0..n)
a(n)= sum( sum( (-1)^j*n*C(n-j,k-j), j=0..k), k=0..n)
a(n)= sum( (1+2*k)*C(n+1, k+1)*2F1( (1, n+2); k+2 )(-1)  - C(n+2, k+2) 2F1( (2, n+3); k+3 )(-1)  - (-1)^(k) * 2^(k-n-2) * (n-3*k+1) , k=0..n) with C(n,k) the binomial coefficient and 2F1( ) the hypergeometric function.

A197911 Representable by A001045 (Jacobsthal sequence). Complement of A003158.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 85, 86, 88

Offset: 0

Philippe Deléham, Oct 19 2011

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060.
The sequence of Jacobsthal numbers A001045 begins [1, 1, 3, 5, 11, 21, ...] with two occurrences of the term 1. Allowing for this, we find that the numbers representable as a sum of distinct Jacobsthal numbers form A050292. - Peter Bala, Feb 02 2013
Partial sums of A056832. - Reinhard Zumkeller, Jul 29 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550 (see page 499).

Cf. A001045, A003158, A010060. A050292.

Cf. A056832.

a197911 n = a197911_list !! n
a197911_list = scanl (+) 0 a056832_list
-- Reinhard Zumkeller, Jul 29 2014

def A197911(n): return n+sum((~(i+1)&i).bit_length()&1 for i in range(n)) # Chai Wah Wu, Jan 09 2023

a(n) = Sum_{k>=0} A030308(n,k)*A001045(k+2).

A293433 a(n) is the number of the proper divisors of n that are Jacobsthal numbers (A001045).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 21, whose proper divisors are [1, 3, 7], both 1 and 3 are in A001045, thus a(21) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..21845

Cf. A001045, A147612, A293431, A293434.

Cf. also A293435.

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A293433(n) = sumdiv(n,d,(dA147612(d));

from sympy import divisors
def A293433(n): return sum(1 for d in divisors(n,generator=True) if d(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

a(n) = Sum_{d|n, dA147612(d).
a(n) = A293431(n) - A147612(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A001045(n) = 1.718591611927... . - Amiram Eldar, Jul 05 2025

A073401 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073402.

Original entry on oeis.org

1, 30, 9, 1050, 531, 63, 44520, 29610, 6165, 405, 2245320, 1789614, 502821, 59454, 2511, 131891760, 120133692, 41182344, 6686631, 517104, 15309, 8862693840, 8966770308, 3559509360, 714174327, 76790673, 4214349, 92583

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073402(k,m).

			k=2: U2(n)=((30+9*n)*(n+1)*U0(n+1)+(33+9*n)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 30,9; 1050,531,63; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073399, A073372, A073402, A073400.

Recursion for row polynomials defined in the comments: p(k, n)= (n+2)*p(k-1, n+1)+4*(n+2*(k+1))*p(k-1, n)+2*(n+3)*q(k-1, n+1); q(k, n)= (n+1)*p(k-1, n+1)+4*(n+2*(k+1))*q(k-1, n), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

cp's OEIS Frontend

A073372 Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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A073399 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n>=0, (generalized (1,2)-Fibonacci). Companion triangle is A073400.

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A073400 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073399.

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A091593 Reversion of Jacobsthal numbers A001045.

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A280049 Canonical representation of n as a sum of distinct Jacobsthal numbers J(n) (A001045) (see Comments for details); also binary numbers that end in an even number of zeros.

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A153643 Jacobsthal numbers A001045 incremented by 2.

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A193449 Products of the Jacobsthal numbers and the integers: a(n) = n * A001045(n+1).

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