A001045 -id:A001045 - cp's OEIS frontend

A211892 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n^2) * x^n/n ), where Jacobsthal(n) = A001045(n).

1, 3, 12, 198, 16962, 6762210, 11473594848, 80455865485692, 2306084412391039038, 268657100633050977422322, 126765866001055606588876061400, 241678197713843578271875740922972788, 1858396158245858742065123341776166504084452

Offset: 0

Paul D. Hanna, Apr 24 2012

nonn

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 198*x^3 + 16962*x^4 + 6762210*x^5 +...
such that
log(A(x))/3 = x + 5*x^2/2 + 171*x^3/3 + 21845*x^4/4 + 11184811*x^5/5 + 22906492245*x^6/6 + 187649984473771*x^7/7 +...+ Jacobsthal(n^2)*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,...].

Cf. A211893, A211894, A211895, A211896, A207972, A001045, A155200.

Cf. A231279 (Jacobsthal(n^2)).

{Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))

G.f.: (1+x) * exp( Sum_{n>=1} 2^(n^2) * x^n/n ).

a(n) = A155200(n) + A155200(n-1).

A211894 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^2 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 18, 57, 195, 684, 2460, 8970, 33102, 123204, 461868, 1741410, 6597750, 25099584, 95822928, 366943881, 1408947675, 5422742910, 20915079258, 80820382425, 312839889219, 1212812010804, 4708415402772, 18302630040504, 71230126892088, 277514015733168

Offset: 0

Paul D. Hanna, Apr 25 2012

nonn

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 18*x^3 + 57*x^4 + 195*x^5 + 684*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^2*x^3/3 + 5^2*x^4/4 + 11^2*x^5/5 + 21^2*x^6/6 + 43^2*x^7/7 +...+ Jacobsthal(n)^2*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Cf. A211893, A211895, A211896, A054888, A207969, A001045 (Jacobsthal).

CoefficientList[Series[(1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 18 2020 *)

{Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^2*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff(((1+2*x)^2/((1-x)*(1-4*x) +x*O(x^n)))^(1/3),n)}

G.f.: (1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^2 / 3 * x^n/n ).
a(n) ~ 3^(1/3) * 2^(2*n) / (n^(2/3) * Gamma(1/3)). - Vaclav Kotesovec, Oct 18 2020

A211896 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^4 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 90, 723, 10689, 130428, 1862580, 25594611, 368313993, 5289203262, 77279744418, 1134460916361, 16798605635235, 249994099311288, 3740771822960664, 56208829313956998, 847934859174601650, 12834366187138678836, 194855374723972622988, 2966358133685609559042

Offset: 0

Paul D. Hanna, Apr 25 2012

nonn

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 90*x^3 + 723*x^4 + 10689*x^5 + 130428*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^4*x^3/3 + 5^4*x^4/4 + 11^4*x^5/5 + 21^4*x^6/6 + 43^4*x^7/7 +...+ Jacobsthal(n)^4*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Cf. A211893, A211894, A211895, A207969, A001045 (Jacobsthal).

{Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^4*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff(((1+2*x)^4*(1+8*x)^4/((1-x)*(1-4*x)^6*(1-16*x))+x*O(x^n))^(1/27),n)}

G.f.: ( (1+2*x)^4*(1+8*x)^4 / ((1-x)*(1-4*x)^6*(1-16*x)) )^(1/27).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^4 / 27 * x^n/n ).
a(n) ~ 3^(5/27) * 2^(4*n) / (5^(1/27) * Gamma(1/27) * n^(26/27)). - Vaclav Kotesovec, Oct 18 2020

A286567 Smallest prime factor of the n-th Jacobsthal number: a(n) = A020639(A001045(n)), with a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 3, 5, 11, 3, 43, 5, 3, 11, 683, 3, 2731, 43, 3, 5, 43691, 3, 174763, 5, 3, 23, 2796203, 3, 11, 2731, 3, 5, 59, 3, 715827883, 5, 3, 43691, 11, 3, 1777, 174763, 3, 5, 83, 3, 2932031007403, 5, 3, 47, 283, 3, 43, 11, 3, 5, 107, 3, 11, 5, 3, 59, 2833, 3, 768614336404564651, 715827883, 3, 5, 11, 3, 7327657, 5, 3, 11, 56409643, 3, 1753, 223, 3, 5, 43, 3

Offset: 1

Antti Karttunen & Hans Havermann, May 21 2017

nonn

Hans Havermann, Table of n, a(n) for n = 1..1122

Cf. A001045, A020639, A278165.

Rest[FactorInteger[#][[1,1]]&/@LinearRecurrence[{1,2},{0,1},100]] (* Harvey P. Dale, Sep 23 2018 *)

(define (A286567 n) (A020639 (A001045 n)))

a(n) = A020639(A001045(n)).

A366769 Number of distinct prime divisors of A001045(n) (Jacobsthal numbers).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 4, 3, 3, 1, 6, 3, 2, 3, 5, 2, 6, 1, 4, 4, 2, 4, 8, 2, 2, 3, 6, 2, 6, 1, 6, 5, 3, 2, 9, 2, 6, 5, 6, 2, 6, 4, 7, 4, 5, 3, 11, 1, 2, 5, 6, 5, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 5, 7, 1, 8, 6, 4, 5, 12, 3, 4

Offset: 1

Sean A. Irvine, Oct 21 2023

nonn

			a(12) = 4 because Jacobsthal(12) = 1365 has prime factors {3, 5, 7, 13}.

Amiram Eldar, Table of n, a(n) for n = 1..1122

Cf. A001045, A001221, A022307, A086598, A364818, A366770.

a(n) = omega(Jacobsthal(n)) = A001221(A001045(n)).

A366770 Number of prime factors of A001045(n) (Jacobsthal numbers) (counted with multiplicity).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 3, 1, 5, 1, 5, 3, 3, 1, 6, 3, 2, 5, 5, 2, 6, 1, 4, 4, 2, 4, 9, 2, 2, 3, 7, 2, 7, 1, 6, 6, 3, 2, 9, 2, 6, 5, 6, 2, 8, 5, 7, 4, 5, 3, 12, 1, 2, 6, 6, 5, 8, 2, 6, 4, 8, 2, 13, 2, 4, 6, 6, 5, 7, 1, 9, 9, 4, 5, 13, 3, 4

Offset: 1

Sean A. Irvine, Oct 21 2023

nonn

			a(9) = 3 because Jacobsthal(9) = 171 = 3^2 * 19.

Amiram Eldar, Table of n, a(n) for n = 1..1122

Cf. A001045, A001222, A366769, A366771, A038575.

a(n) = bigomega(Jacobsthal(n)) = A001222(A001045(n)).

A073373 Third convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 4, 18, 60, 195, 576, 1644, 4488, 11925, 30860, 78278, 195012, 478599, 1159080, 2774880, 6575280, 15439065, 35955540, 83118970, 190862860, 435601611, 988620624, 2232236628, 5016441240, 11224087965

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 (4,2,-20,-1,40,8,-32,-16).

Fourth (m=3) column of triangle A073370.

Cf. A001045, A073372.

[(1/4374)*(2^(n+4)*(226 +267*n +90*n^2 +9*n^3) +(-1)^n*(758 +555*n +126*n^2 +9*n^3)): n in [0..40]]; // G. C. Greubel, Sep 29 2022

Table[(1/4374)*(2^(n+4)*(226 +267*n +90*n^2 +9*n^3) +(-1)^n*(758 +555*n +126*n^2 +9*n^3)), {n,0,40}] (* G. C. Greubel, Sep 29 2022 *)

def A073373(n): return (1/4374)*(2^(n+4)*(226+267*n+90*n^2+9*n^3) +(-1)^n*(758 +555*n+126*n^2+9*n^3))
[A073373(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

a(n) = Sum_{k=0..n} b(k) * c(n-k), with b(k) = A001045(k+1) and c(k) = A073372(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, 3) * binomial(n-k, k) * 2^k.
a(n) = ((350+177*n+21*n^2)*(n+1)*U(n+1) + 2*(277+132*n+15*n^2)*(n+2)*U(n))/ (2*9^3) with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1-(1+2*x)*x)^4 = 1/ ( (1+x)^4*(1-2*x)^4 ).
E.g.f.: (1/4374)*(32*(113 + 366*x + 234*x^2 + 36*x^3)*exp(2*x) - (-758 + 690*x - 153*x^2 + 9*x^3)*exp(-x)). - G. C. Greubel, Sep 29 2022

A073374 Fourth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 5, 25, 95, 340, 1106, 3430, 10130, 28915, 80035, 216143, 571225, 1482110, 3783640, 9522740, 23665300, 58149845, 141435985, 340854645, 814589475, 1931900376, 4549699950, 10645737330, 24761578470

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 (5,0,-30,15,81,-30,-120,0,80,32).

Fifth (m=4) column of triangle A073370.

Cf. A001045, A073373.

[(2^(n+5)*(4208+5790*n+2565*n^2+450*n^3+27*n^4) + (-1)^n*(22808+18510*n+ 5265*n^2+630*n^3+27*n^4))/157464: n in [0..40]]; // G. C. Greubel, Sep 29 2022

Table[(2^(n+5)*(4208+5790*n+2565*n^2+450*n^3+27*n^4) + (-1)^n*(22808+18510*n+ 5265*n^2+630*n^3+27*n^4))/157464, {n,0,40}] (* G. C. Greubel, Sep 29 2022 *)

def A073374(n): return (2^(n+5)*(4208+5790*n+2565*n^2+450*n^3+27*n^4) + (-1)^n*(22808+18510*n+ 5265*n^2+630*n^3+27*n^4))/157464
[A073374(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073373(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+4, 4) * binomial(n-k, k) * 2^k.
a(n) = (5*(2968 +1974*n +411*n^2 +27*n^3)*(n+1)*U(n+1) + 2*(9412 +6099*n +1248*n^2 +81*n^3)*(n+2)*U(n))/(4!*3^7) with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1-(1+2*x)*x)^5 = 1/((1+x)*(1-2*x))^5.
E.g.f.: (1/157464)*(512*(263 + 1104*x + 1026*x^2 + 306*x^3 + 27*x^4)*exp(2*x) + (22808 - 24432*x + 7344*x^2 - 792*x^3 + 27*x^4)*exp(-x)). - G. C. Greubel, Sep 29 2022

A073375 Fifth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 6, 33, 140, 546, 1932, 6454, 20448, 62271, 183202, 523887, 1461516, 3991400, 10698072, 28203612, 73265056, 187822125, 475788222, 1192287117, 2958453036, 7274927646, 17741533668, 42937126290

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 (6,-3,-40,45,126,-141,-252,180,320,-48,-192,-64).

Sixth (m=5) column of triangle A073370.

Cf. A001045, A073374.

[(n+3)*(n+9)*(2^(6+n)*(9*n^3 +117*n^2 +438*n +400) +(-1)^n*(9*n^3 +207*n^2 + 1518*n +3560))/787320: n in [0..40]]; // G. C. Greubel, Sep 29 2022

Table[(n+3)*(n+9)*(2^(6+n)*(9*n^3 +117*n^2 +438*n +400) +(-1)^n*(9*n^3 +207*n^2 + 1518*n +3560))/787320, {n,0,40}] (* G. C. Greubel, Sep 29 2022 *)

def A073375(n): return (n+3)*(n+9)*(2^(6+n)*(9*n^3 +117*n^2 +438*n +400) +(-1)^n*(9*n^3 +207*n^2 + 1518*n +3560))/787320
[A073375(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073374(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+5, 5) * binomial(n-k, k) * 2^k.
a(n) = (n+3)*(n+9)*((3080 +1086*n +93*n^2)*(n+1)*U(n+1) + 2*(1660 +591*n +51*n^2) *(n+2)*U(n))/(5!*3^7) with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1-(1+2*x)*x)^6 = 1/((1+x)*(1-2*x))^6.
E.g.f.: (1/787320)*(1024*(675 +3470*x +4195*x^2 +1830*x^3 +315*x^4 +18*x^5 )*exp(2*x) + (96120 -115640*x +42440*x^2 -6360*x^3 +405*x^4 -9*x^5)*exp(-x)). - G. C. Greubel, Sep 29 2022

A073376 Sixth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 7, 42, 196, 826, 3150, 11256, 38004, 122787, 381997, 1151458, 3376968, 9671284, 27123292, 74669472, 202181112, 539342181, 1419492627, 3690464106, 9487902396, 24143758254, 60861096714

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 (7,-7,-49,91,161,-357,-363,714,644,-728,-784,224, 448,128).

Seventh (m=6) column of triangle A073370.

Cf. A001045, A073375.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^7 )); // G. C. Greubel, Sep 29 2022

Table[(2^(n+7)*(297760 +500640*n +302778*n^2 +87255*n^3 +12915*n^4 +945*n^5 +27*n^6) +(-1)^n*(4402000 +4038132*n +1453788*n^2 +265545*n^3 +26145*n^4 +1323*n^5 +27*n^6))/(6!*3^10), {n,0,40}] (* G. C. Greubel, Sep 29 2022 *)

def A073376(n): return (2^(n+7)*(297760 +500640*n +302778*n^2 +87255*n^3 +12915*n^4 +945*n^5 +27*n^6) +(-1)^n*(4402000 +4038132*n +1453788*n^2 +265545*n^3 +26145*n^4 +1323*n^5 +27*n^6))/(factorial(6)*3^10)
[A073376(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

a(n) = Sum_{k=0..n} b(k) * c(n-k), with b(k) = A001045(k+1) and c(k) = A073375(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+6, 6) * binomial(n-k, k) * 2^k.
a(n) = ((1+n)*(4884880 +4449396*n +1525272*n^2 +247653*n^3 +19152*n^4 +567*n^5)* U(n+1) + 2*(2+n)*(2321720 +2182242*n +765993*n^2 +126621*n^3 +9927*n^4 +297*n^5 )*U(n))/(6!*3^9) with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1-(1+2*x)*x)^7 = 1/((1+x)*(1-2*x))^7.
E.g.f.: (1/(6!*3^10))*( 4096*(9305 +56535*x +83745*x^2 +47700*x^3 +12060*x^4 +1350*x^5 +54*x^6)*exp(2*x) + (4402000 -5784960*x +2454120*x^2 -457920*x^3 +41130*x^4 -1728*x^5 +27*x^6)*exp(-x)). - G. C. Greubel, Sep 29 2022

cp's OEIS Frontend

A211892 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n^2) * x^n/n ), where Jacobsthal(n) = A001045(n).

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A211894 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^2 * x^n/n ), where Jacobsthal(n) = A001045(n).

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A211896 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^4 * x^n/n ), where Jacobsthal(n) = A001045(n).

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A286567 Smallest prime factor of the n-th Jacobsthal number: a(n) = A020639(A001045(n)), with a(1)=a(2)=1.

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A366769 Number of distinct prime divisors of A001045(n) (Jacobsthal numbers).

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A366770 Number of prime factors of A001045(n) (Jacobsthal numbers) (counted with multiplicity).

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A073373 Third convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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A073374 Fourth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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