A001045 -id:A001045 - cp's OEIS frontend

A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

1, 1, 1, 3, 2, 1, 5, 7, 3, 1, 11, 16, 12, 4, 1, 21, 41, 34, 18, 5, 1, 43, 94, 99, 60, 25, 6, 1, 85, 219, 261, 195, 95, 33, 7, 1, 171, 492, 678, 576, 340, 140, 42, 8, 1, 341, 1101, 1692, 1644, 1106, 546, 196, 52, 9, 1

Offset: 0

Wolfdieter Lang, Aug 02 2002

The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(1+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.
Riordan array (1/(1-x-2*x^2), x/(1-x-2*x^2)). - Paul Barry, Mar 15 2005
Subtriangle (obtained by dropping the first column) of the triangle given by (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013
The number of ternary words of length n having k letters equal 2 and 0,1 avoid runs of odd lengths. - Milan Janjic, Jan 14 2017

			Triangle begins as:
    1;
    1,   1;
    3,   2,   1;
    5,   7,   3,   1;
   11,  16,  12,   4,   1;
   21,  41,  34,  18,   5,   1;
   43,  94,  99,  60,  25,   6,   1;
   85, 219, 261, 195,  95,  33,   7,   1;
  171, 492, 678, 576, 340, 140,  42,   8,   1;
The triangle (0, 1, 2, -2, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  3,  2,  1;
  0,  5,  7,  3,  1;
  0, 11, 16, 12,  4,  1;
  0, 21, 41, 34, 18,  5,  1; - _Philippe Deléham_, Feb 19 2013

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
W. Lang, First 10 rows.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Columns: A001045 (k=0), A073371 (k=1), A073372 (k=2), A073373 (k=3), A073374 (k=4), A073375 (k=5), A073376 (k=6), A073377 (k=7), A073378 (k=8), A073379 (k=9).
Cf. A002605 (row sums), A006130 (diagonal sums), A073399, A073400.
Cf. A000045, A077957.

A073370:= func< n,k | (&+[Binomial(n-j,k)*Binomial(n-k-j,j)*2^j: j in [0..Floor((n-k)/2)]]) >;
[A073370(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 01 2022

# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> (2^n - (-1)^n) / 3); # Peter Luschny, Oct 07 2022

T[n_, k_]:= T[n, k]= Sum[Binomial[n-j,k]*Binomial[n-k-j,j]*2^j, {j,0,Floor[(n- k)/2]}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 01 2022 *)

def A073370(n,k): return binomial(n,k)*sum( 2^j * binomial(2*j,j) * binomial(n-k,2*j)/binomial(n,j) for j in range(1+(n-k)//2))
flatten([[A073370(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 01 2022

T(n, m) = Sum_{k=0..floor((n-m)/2)} binomial(n-k, m)*binomial(n-m-k, k)*2^k, if n > m, else 0.
Sum_{k=0..n} T(n, k) = A002605(n+1).
T(n, m) = (1*(n-m+1)*T(n, m-1) + 2*2*(n+m)*T(n-1, m-1))/((1^2 + 4*2)*m), n >= m >= 1, T(n, 0) = A001045(n+1), n >= 0, else 0.
T(n, m) = (p(m-1, n-m)*1*(n-m+1)*T(n-m+1) + q(m-1, n-m)*2*(n-m+2)*T(n-m))/(m!*9^m), n >= m >= 1, with T(n) = T(n, m=0) = A001045(n+1), else 0; p(k, n) = Sum_{j=0..k} (A(k, j)*n^(k-j) and q(k, n) = Sum_{j=0..k} B(k, j)*n^(k-j), with the number triangles A(k, m) = A073399(k, m) and B(k, m) = A073400(k, m).
G.f.: 1/(1-(1+2*x)*x)^(m+1) = 1/((1+x)*(1-2*x))^(m+1), m >= 0, for column m (without leading zeros).
T(n, 0) = A001045(n), T(1, 1) = 1, T(n, k) = 0 if k>n, T(n, k) = T(n-1, k-1) + 2*T(n-2, k) + T(n-1, k) otherwise. - Paul Barry, Mar 15 2005
G.f.: (1+x)*(1-2*x)/(1-x-2*x^2-x*y) for the triangle including the 1, 0, 0, 0, 0, ... column. - R. J. Mathar, Aug 11 2015
From Peter Bala, Oct 07 2019: (Start)
Recurrence for row polynomials: R(n,x) = (1 + x)*R(n-1,x) + 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 1 + x.
The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 - 2*x/(1 + ... + x/(1 - 2*x/(1)))) (with 2*n partial numerators). Cf. A110441. (End)
From G. C. Greubel, Oct 01 2022: (Start)
T(n, k) = binomial(n,k)*Sum_{j=0..floor((n-k)/2)} 2^j*binomial(2*j, j)*binomial(n-k, 2*j)/binomial(n, j).
T(n, k) = binomial(n, k)*Hypergeometric2F1([(k-n)/2, (k-n+1)/2], [-2*n], -8).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006130(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000045(n+1). (End)

A130249 Maximal index k of a Jacobsthal number such that A001045(k)<=n (the 'lower' Jacobsthal inverse).

Original entry on oeis.org

0, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=1 (see A130250 for another version). a(n)+1 is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

			a(12)=5, since A001045(5)=11<=12, but A001045(6)=21>12.

G. C. Greubel, Table of n, a(n) for n = 0..10000

For partial sums see A130251.
Other related sequences A130250, A130253, A105348, A001045, A130233, A130241.
Cf. A000523, A078008 (runlengths).

[Floor(Log(3*n+1)/Log(2)): n in [0..30]]; // G. C. Greubel, Jan 08 2018

Table[Floor[Log[2, 3*n + 1]], {n, 0, 50}] (* G. C. Greubel, Jan 08 2018 *)

for(n=0, 30, print1(floor(log(3*n+1)/log(2)), ", ")) \\ G. C. Greubel, Jan 08 2018

a(n) = logint(3*n+1, 2); \\ Ruud H.G. van Tol, May 12 2024

def A130249(n): return (3*n+1).bit_length()-1 # Chai Wah Wu, Jun 08 2022

a(n) = floor(log_2(3n+1)).
a(n) = A130250(n+1) - 1 = A130253(n) - 1.
G.f.: 1/(1-x)*(Sum_{k>=1} x^A001045(k)).
a(n) = A000523(3*n+1). - Ruud H.G. van Tol, May 12 2024

A265745 a(n) is the number of Jacobsthal numbers (A001045) needed to sum to n using the greedy algorithm.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Dec 17 2015

Sum of digits in "Jacobsthal greedy base", A265747.
It would be nice to know for sure whether this sequence gives also the least number of Jacobsthal numbers that add to n, i.e., that there cannot be even better nongreedy solutions.
The integer 63=21+21+21 has 3 for its 'non-greedy' solution, and a(63) = 5 for its greedy solution 63=43+11+5+3+1. - Yuriko Suwa, Jul 11 2021
Positions where a(n) is different from A372555(n) are n=63, 84, 148, 169, 191, 212, 234, 255, etc. See A372557. - Antti Karttunen, May 07 2024

			a(0) = 0, because no numbers are needed to form an empty sum, which is zero.
For n=1 we need just A001045(2) = 1, thus a(1) = 1.
For n=2 we need A001045(2) + A001045(2) = 1 + 1, thus a(2) = 2.
For n=4 we need A001045(3) + A001045(2) = 3 + 1, thus a(4) = 2.
For n=6 we form the greedy sum as A001045(4) + A001045(2) = 5 + 1, thus a(6) = 2. Alternatively, we could form the sum as A001045(3) + A001045(3) = 3 + 3, but the number of summands in that case is no less.
For n=7 we need A001045(4) + A001045(2) + A001045(2) = 5 + 1 + 1, thus a(7) = 3.
For n=8 we need A001045(4) + A001045(3) = 5 + 3, thus a(8) = 2.
For n=10 we need A001045(4) + A001045(4) = 5 + 5, thus a(10) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..10923

Cf. A001045, A130249, A197911, A003158, A265746, A265747, A364377, A364379, A364380, A372288, A372555, A372557.
Cf. A054111 (apparently the positions of the first occurrence of each n > 0).
Cf. also A007895, A014420, A053610, A265404, A265743, A265744.

jacob[n_] := (2^n - (-1)^n)/3; maxInd[n_] := Floor[Log2[3*n + 1]]; A265745[n_] := A265745[n] = 1 + A265745[n - jacob[maxInd[n]]]; A265745[0] = 0; Array[A265745, 100, 0] (* Amiram Eldar, Jul 21 2023 *)

A130249(n) = floor(log(3*n + 1)/log(2));
A001045(n) = (2^n - (-1)^n) / 3;
A265745(n) = {if(n == 0, 0, my(d = n - A001045(A130249(n))); if(d == 0, 1, 1 + A265745(d)));} \\ Amiram Eldar, Jul 21 2023

def greedyJ(n): n1 = (3*n+1).bit_length() - 1; return (2**n1 - (-1)**n1)//3
def a(n): return 0 if n == 0 else 1 + a(n - greedyJ(n))
print([a(n) for n in range(107)]) # Michael S. Branicky, Jul 11 2021

a(0) = 0; for n >= 1, a(n) = 1 + a(n - A001045(A130249(n))). [This formula uses a simple greedy algorithm.]

A073371 Convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 2, 7, 16, 41, 94, 219, 492, 1101, 2426, 5311, 11528, 24881, 53398, 114083, 242724, 514581, 1087410, 2291335, 4815680, 10097401, 21126862, 44117867, 91963996, 191384541, 397682154, 825190479, 1710033272, 3539371201, 7317351686

Offset: 0

Wolfdieter Lang, Aug 02 2002

PSumSIGN transform of A045883(n-1). - Michael Somos, Jul 10 2003
Numbers of the form ((6*m+4)*2^m + (-1)^(m-1)*(3*m+4))/27. - Artur Jasinski, Feb 09 2007
With [0, 0, 0] prepended, this is an "autosequence" of the first kind, whose companion is [0, 0, 2, 3, 12, 25, 66, ...], that is A099429. - Jean-François Alcover, Jul 10 2022

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wieb Bosma, Signed bits and fast exponentiation, J. Th. Nombres de Bordeaux, 13 no. 1 (2001), p. 27-41.
OEIS Wiki, Autosequence

Second (m=1) column of triangle A073370.

Cf. A001045, A062111, A099429, A127976.

[((5+3*n)*2^(n+2) + (-1)^n*(7+3*n))/27: n in [0..40]]; // G. C. Greubel, Sep 28 2022

Table[((6n+4)*2^n + (-1)^(n-1)(3n+4))/27, {n, 100}] (* Artur Jasinski, Feb 09 2007 *)

a(n) = if(n<-3, 0, ((5+3*n)*2^(n+2)+(7+3*n)*(-1)^n)/27)

def A073371(n): return ((5+3*n)*2^(n+2) + (-1)^n*(7+3*n))/27
[A073371(n) for n in range(40)] # G. C. Greubel, Sep 28 2022

a(n) = Sum_{k=0..n} b(k) * b(n-k), where b(k) = A001045(k+1).
a(n) = Sum_{k=0..floor(n/2)} (n-k+1) * binomial(n-k, k) * 2^k.
a(n) = ((n+1)*U(n+1) + 4*(n+2)*U(n))/9 with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1 - (1+2*x)*x)^2.
G.f.: 1/((1+x)*(1-2*x))^2.
a(n) = ((5+3*n)*2^(n+2) + (7+3*n)*(-1)^n)/27.
a(n) = ((6*n+4)*2^(n) + (-1)^(n-1)*(3*n+4))/27. - Artur Jasinski, Feb 09 2007
E.g.f.: (1/27)*(4*(5+6*x)*exp(2*x) + (7-3*x)*exp(-x)). - G. C. Greubel, Sep 28 2022

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

A128209 Jacobsthal numbers(A001045) + 1.

Original entry on oeis.org

1, 2, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532, 5726623062

Offset: 0

Paul Barry, Feb 19 2007

Row sums of A128208.
Essentially the same as A052953. - R. J. Mathar, Jun 14 2008
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 1, a(n+1) is the number of different representations of matrix P^(-1)+I+P by sum of permutation matrices. - Vladimir Shevelev, Apr 12 2010
a(n) is the rank of Fibonacci(n+2) in row n of A049456 (regarded as an irregular triangle read by rows). - N. J. A. Sloane, Nov 23 2016

V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A167030, A153643, A049456.

[1+2^n/3-(-1)^n/3: n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

CoefficientList[Series[(1-3*x^2)/(1-2*x-x^2+2*x^3),{x,0,40}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)

a(n)=2^n\/3+1 \\ Charles R Greathouse IV, Jan 31 2012

def A128209(n): return ((1<Chai Wah Wu, Apr 17 2025

a(n) = 1 + 2^n/3 - (-1)^n/3.

G.f.: (1-3*x^2)/(1 - 2*x - x^2 + 2*x^3).

A130253 Number of Jacobsthal numbers (A001045) <=n.

Original entry on oeis.org

1, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Hieronymus Fischer, May 20 2007

Partial sums of the Jacobsthal indicator sequence (A105348).

For n<>1, we have a(A001045(n))=n+1.

			a(9)=5 because there are 5 Jacobsthal numbers <=9 (0,1,1,3 and 5).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

For partial sums see A130252. Other related sequences A001045, A130249, A130250, A130253, A105348. Also A130233, A130235, A130241, A108852, A130245.

[Ceiling(Log(3*n+2)/Log(2)): n in [0..30]]; // G. C. Greubel, Jan 08 2018

Table[1+Floor[Log[2,3n+1]],{n,0,100}] (* Harvey P. Dale, Jul 03 2013 *)

a(n)=logint(3*n+1,2)+1 \\ Charles R Greathouse IV, Oct 03 2016

def A130253(n): return (3*n+1).bit_length() # Chai Wah Wu, Apr 17 2025

a(n) = floor(log_2(3n+1)) + 1 = ceiling(log_2(3n+2)).
a(n) = A130249(n) + 1 = A130250(n+1).
G.f.: 1/(1-x)*(Sum_{k>=0} x^A001045(k)).

A154879 Third differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

3, -2, 4, 0, 8, 8, 24, 40, 88, 168, 344, 680, 1368, 2728, 5464, 10920, 21848, 43688, 87384, 174760, 349528, 699048, 1398104, 2796200, 5592408, 11184808, 22369624, 44739240, 89478488, 178956968, 357913944, 715827880, 1431655768, 2863311528, 5726623064

Offset: 0

Paul Curtz, Jan 16 2009

Second differences of A078008. First differences of the sequence (-1)^(n+1)*A084247(n).

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

Cf. A115341.

[(1/3)*(8*(-1)^n+2^n): n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

Differences[LinearRecurrence[{1,2},{0,1},40],3] (* or *) LinearRecurrence[ {1,2},{3,-2},40] (* Harvey P. Dale, Apr 20 2018 *)

def A154879(n): return ((1<2 else (3,-2,4)[n] # Chai Wah Wu, Apr 18 2025

a(n) + a(n+1) = A000079(n), n > 1.
a(n+3) = 8*A001045(n) = 4*A078008(n+1) = 2*A097073(n+1).
From R. J. Mathar, Jan 23 2009: (Start)
a(n) = a(n-1) + 2*a(n-2).
G.f.: (3-5*x)/((1+x)*(1-2*x)). (End)

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

Typo in A-number in formula corrected by R. J. Mathar, Feb 23 2009

A322942 Jacobsthal triangle, coefficients of orthogonal polynomials J(n, x) where J(n, 0) are the Jacobsthal numbers (A001045 with a(0) = 1). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 5, 3, 1, 5, 12, 10, 4, 1, 11, 27, 28, 16, 5, 1, 21, 62, 75, 52, 23, 6, 1, 43, 137, 193, 159, 85, 31, 7, 1, 85, 304, 480, 456, 290, 128, 40, 8, 1, 171, 663, 1170, 1254, 916, 480, 182, 50, 9, 1, 341, 1442, 2793, 3336, 2750, 1652, 742, 248, 61, 10, 1

Offset: 0

Peter Luschny, Jan 03 2019

The name 'Jacobsthal triangle' used here is not standard.

			The first few polynomials are:
  J(0, x) = 1;
  J(1, x) = x + 1;
  J(2, x) = x^2 + 2*x + 1;
  J(3, x) = x^3 + 3*x^2 +  5*x + 3;
  J(4, x) = x^4 + 4*x^3 + 10*x^2 + 12*x + 5;
  J(5, x) = x^5 + 5*x^4 + 16*x^3 + 28*x^2 + 27*x + 11;
  J(6, x) = x^6 + 6*x^5 + 23*x^4 + 52*x^3 + 75*x^2 + 62*x + 21;
The triangle starts:
  [0]   1;
  [1]   1,   1;
  [2]   1,   2,    1;
  [3]   3,   5,    3,    1;
  [4]   5,  12,   10,    4,   1;
  [5]  11,  27,   28,   16,   5,   1;
  [6]  21,  62,   75,   52,  23,   6,   1;
  [7]  43, 137,  193,  159,  85,  31,   7,  1;
  [8]  85, 304,  480,  456, 290, 128,  40,  8, 1;
  [9] 171, 663, 1170, 1254, 916, 480, 182, 50, 9, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Row sums are A152035, alternating row sums are A000007, values at x=1/2 are A323232, values at x=0 (first column) are A152046.

Cf. A000045, A001045, A006138, A321620.

R:=PowerSeriesRing(Rationals(), 30);
g:= func< n,x | (&+[Binomial(n-k,k)*2^k*(x+1)^(n-2*k): k in [0..Floor(n/2)]]) >;
f:= func< n,x | n le 1 select (x+1)^n else g(n,x) - 2*g(n-2,x) >;
A322942:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[A322942(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 20 2023

J := proc(n) option remember;
`if`(n < 3, [1, x+1, x^2 + 2*x + 1][n+1], (x+1)*J(n-1) + 2*J(n-2));
sort(expand(%)) end: seq(print(J(n)), n=0..11); # Computes the polynomials.
seq(seq(coeff(J(n), x, k), k=0..n), n=0..11);

J[n_, x_]:= J[n, x]= If[n<3, (x+1)^n, (x+1)*J[n-1, x] + 2*J[n-2, x]];
T[n_, k_]:= Coefficient[J[n, x], x, k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Jun 17 2019 *)
(* Second program *)
A322942[n_, k_]:= Coefficient[Series[Boole[n==0] + (I*Sqrt[2])^n*(ChebyshevU[n, (x+1)/(2*Sqrt[2]*I)] + ChebyshevU[n-2, (x+ 1)/(2*Sqrt[2]*I)]), {x,0,50}], x, k];
Table[A322942[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 20 2023 *)

# use[riordan_square from A321620]
riordan_square((2*x^2 - 1)/((x + 1)*(2*x - 1)), 9)

J(n, x) = (x+1)*J(n-1, x) + 2*J(n-2, x) for n >= 3.
T(n, k) = [x^k] J(n, x).
Equals the Riordan square (cf. A321620) generated by (2*x^2-1)/((x + 1)*(2*x - 1)).
Sum_{k=0..n} T(n, k) = A152035(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
From G. C. Greubel, Sep 20 2023: (Start)
T(n, k) = [x^k]( [n=0] + (i*sqrt(2))^n*(ChebyshevU(n, (x+1)/(2*sqrt(2)*i)) + ChebyshevU(n-2, (x+1)/(2*sqrt(2)*i))) ).
G.f.: (1 - 2*t^2)/(1 - (x+1)*t - 2*t^2).
Sum_{k=0..floor(n/2)} T(n-k, k) = (2/3)*[n=0] + A006138(n-1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = 2*[n=0] + Fibonacci(n-2). (End)

A049883 Primes in the Jacobsthal sequence (A001045).

Original entry on oeis.org

3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243

Offset: 1

Judson Neer

nonn

All terms, except a(2) = 5, are of the form (2^p + 1)/3 - the Wagstaff primes A000979 = {3, 11, 43, 683, 2731, 43691, 174763, ...}.
Indices of prime Jacobsthal numbers are listed in A107036 = {3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, ...}.
For n > 1, A107036(n) = A000978(n) (numbers m such that (2^m + 1)/3 is prime). - Alexander Adamchuk, Oct 10 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..23

Cf. A001045, A000978, A000979, A107036.

Select[Table[(2^n + (-1)^(n - 1))/3, {n, 200}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011 *)

A130250 Minimal index k of a Jacobsthal number such that A001045(k) >= n (the 'upper' Jacobsthal inverse).

Original entry on oeis.org

0, 1, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=2 (see A130249 for another version). a(n+1) is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

			a(10)=5 because A001045(5) = 11 >= 10, but A001045(4) = 5 < 10.

G. C. Greubel, Table of n, a(n) for n = 0..5000

For partial sums see A130252.

Cf. A001045, A105348, A130234, A130242, A130249, A130253.

[0] cat [Ceiling(Log(2,3*n-1)): n in [1..120]]; // G. C. Greubel, Mar 18 2023

Table[If[n==0, 0, Ceiling[Log[2, 3*n-1]]], {n,0,120}] (* G. C. Greubel, Mar 18 2023 *)

def A130250(n): return (3*n-2).bit_length() if n else 0 # Chai Wah Wu, Apr 17 2025

def A130250(n): return 0 if (n==0) else ceil(log(3*n-1, 2))
[A130250(n) for n in range(121)] # G. C. Greubel, Mar 18 2023

a(n) = ceiling(log_2(3n-1)) = 1 + floor(log_2(3n-2)) for n >= 1.
a(n) = A130249(n-1) + 1 = A130253(n-1) for n >= 1.
G.f.: (x/(1-x))*Sum_{k>=0} x^A001045(k).

cp's OEIS Frontend

A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

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A130249 Maximal index k of a Jacobsthal number such that A001045(k)<=n (the 'lower' Jacobsthal inverse).

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A265745 a(n) is the number of Jacobsthal numbers (A001045) needed to sum to n using the greedy algorithm.

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A073371 Convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n >= 0, with itself.

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A128209 Jacobsthal numbers(A001045) + 1.

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A130253 Number of Jacobsthal numbers (A001045) <=n.

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