A016778 -id:A016778

A016922

a(n) = (6*n+1)^2.

+10
25

1, 49, 169, 361, 625, 961, 1369, 1849, 2401, 3025, 3721, 4489, 5329, 6241, 7225, 8281, 9409, 10609, 11881, 13225, 14641, 16129, 17689, 19321, 21025, 22801, 24649, 26569, 28561, 30625, 32761, 34969, 37249, 39601, 42025, 44521, 47089, 49729, 52441, 55225

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Except for 2, exponents e such that x^e-x+1 is reducible.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: ( 1+46*x+25*x^2 ) / (1-x)^3. - R. J. Mathar, Mar 10 2011

a(n) = A016921(n)^2 = A000290(A016921(n)). - Wesley Ivan Hurt, Dec 06 2013

a(n) = 24*A005449(n)+1. - Jean-Bernard François, Oct 12 2014

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Wesley Ivan Hurt, Oct 13 2014

Sum_{n>=0} 1/a(n) = A086727. - Amiram Eldar, Nov 16 2020

MAPLE

A016922:=n->(6*n+1)^2; seq(A016922(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

MATHEMATICA

Table[(6n+1)^2, {n, 0, 100}] (* or *)

CoefficientList[Series[(1 + 46*x + 25*x^2)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 13 2014 *)

LinearRecurrence[{3, -3, 1}, {1, 49, 169}, 50] (* Harvey P. Dale, Feb 17 2023 *)

PROG

(Magma) [(6*n+1)^2: n in [0..60]]; // Vincenzo Librandi, May 04 2011

(PARI) a(n)=(6*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

CROSSREFS

Cf. A000290, A005449, A086727, A016778 (bisection), A016921.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

A016814

a(n) = (4*n + 1)^2.

+10
23

1, 25, 81, 169, 289, 441, 625, 841, 1089, 1369, 1681, 2025, 2401, 2809, 3249, 3721, 4225, 4761, 5329, 5929, 6561, 7225, 7921, 8649, 9409, 10201, 11025, 11881, 12769, 13689, 14641, 15625, 16641, 17689, 18769, 19881, 21025, 22201, 23409, 24649, 25921, 27225, 28561, 29929

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

A bisection of A016754. Sequence arises from reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..200 from Ivan Panchenko).

Leo Tavares, Illustration: Square trapeziums

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = a(n-1) + 32*n - 8, n > 0. - Vincenzo Librandi, Dec 15 2010

From George F. Johnson, Sep 28 2012: (Start)

G.f.: (1 + 22*x + 9*x^2)/(1 - x)^3.

a(n+1) = a(n) + 16 + 8*sqrt(a(n)).

a(n+1) = 2*a(n) - a(n-1) + 32 = 3*a(n) - 3*a(n-1) + a(n-2).

a(n-1)*a(n+1) = (a(n) - 16)^2 ; a(n+1) - a(n-1) = 16*sqrt(a(n)).

a(n) = A016754(2*n) = (A016813(n))^2. (End)

Sum_{n>=0} 1/a(n) = G/2 + Pi^2/16, where G is the Catalan constant (A006752). - Amiram Eldar, Jun 28 2020

Product_{n>=1} (1 - 1/a(n)) = 2*Gamma(5/4)^2/sqrt(Pi) = 2 * A068467^2 * A087197. - Amiram Eldar, Feb 01 2021

From G. C. Greubel, Dec 28 2022: (Start)

a(2*n) = A017078(n).

a(2*n+1) = A017126(n).

E.g.f.: (1 + 24*x + 16*x^2)*exp(x). (End)

a(n) = A272399(n+1) - A014105(n). - Leo Tavares, Dec 24 2023

MAPLE

A016814:=n->(4*n+1)^2; seq(A016814(k), k=0..100); # Wesley Ivan Hurt, Nov 02 2013

MATHEMATICA

(4*Range[0, 40] +1)^2 (* or *) LinearRecurrence[{3, -3, 1}, {1, 25, 81}, 40] (* Harvey P. Dale, Nov 20 2012 *)

Accumulate[32Range[0, 47] - 8] + 9 (* Alonso del Arte, Aug 19 2017 *)

PROG

(PARI) a(n)=(4*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

(Magma) [(4*n+1)^2: n in [0..40]]; // G. C. Greubel, Dec 28 2022

(SageMath) [(4*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

CROSSREFS

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), this sequence (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

Cf. A001539, A006752, A016742, A016802, A016813.

Cf. A016826, A016838, A017126, A068467, A087197.

Cf. A272399, A014105.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A016862

a(n) = (5*n + 1)^2.

+10
14

1, 36, 121, 256, 441, 676, 961, 1296, 1681, 2116, 2601, 3136, 3721, 4356, 5041, 5776, 6561, 7396, 8281, 9216, 10201, 11236, 12321, 13456, 14641, 15876, 17161, 18496, 19881, 21316, 22801, 24336, 25921

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Polygamma Function.

Wikipedia, Polygamma Function.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Harvey P. Dale, Jul 11 2012

Sum_{n>=0} 1/a(n) = polygamma(1, 1/5)/25 = 1.050695088216... - Amiram Eldar, Oct 02 2020

G.f.: (1 +33*x +16*x^2)/(1-x)^3. - Wesley Ivan Hurt, Oct 02 2020

From G. C. Greubel, Dec 28 2022: (Start)

a(2*n) = A017282(n).

a(2*n+1) = 4*A016886(n).

E.g.f.: (1 + 35*x + 25*x^2)*exp(x). (End)

MATHEMATICA

(5*Range[0, 40]+1)^2 (* or *) LinearRecurrence[{3, -3, 1}, {1, 36, 121}, 40] (* Harvey P. Dale, Jul 11 2012 *)

PROG

(PARI) a(n)=(5*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

(Magma) [(5*n+1)^2: n in [0..40]]; // G. C. Greubel, Dec 28 2022

(SageMath) [(5*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

CROSSREFS

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), A016814 (m=4), this sequence (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A017282

a(n) = (10*n + 1)^2.

+10
12

1, 121, 441, 961, 1681, 2601, 3721, 5041, 6561, 8281, 10201, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40401, 44521, 48841, 53361, 58081, 63001, 68121, 73441, 78961, 84681, 90601, 96721, 103041, 109561, 116281, 123201

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: (1+118*x+81*x^2)/(1-x)^3. - Bruno Berselli, Jul 30 2011

a(n) = a(n-1) + 40*(5*n-2), n > 0; a(0)=1. - Miquel Cerda, Oct 30 2016

a(n) = A017281(n)^2. - Michel Marcus, Oct 30 2016

E.g.f.: (1 +120*x +100*x^2)*exp(x). - G. C. Greubel, Dec 24 2022

MATHEMATICA

(* Programs from Michael De Vlieger, Mar 30 2017 *)

Table[(10 n+1)^2, {n, 0, 35}]

FoldList[#1 + 200 #2 - 80 &, 1, Range@ 35]

CoefficientList[Series[(1+118x+81x^2)/(1-x)^3, {x, 0, 35}], x] (* End *)

LinearRecurrence[{3, -3, 1}, {1, 121, 441}, 40] (* Harvey P. Dale, Sep 21 2017 *)

PROG

(Magma) [(10*n+1)^2: n in [0..35]]; // Vincenzo Librandi, Jul 30 2011

(PARI) for(n=0, 35, print1((10*n+1)^2", ")); \\ Bruno Berselli, Jul 30 2011

(SageMath) [(10*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

CROSSREFS

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), this sequence (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

Cf. A017281.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Bruno Berselli, Jul 30 2011

STATUS

approved

A016780

a(n) = (3*n+1)^4.

+10
11

1, 256, 2401, 10000, 28561, 65536, 130321, 234256, 390625, 614656, 923521, 1336336, 1874161, 2560000, 3418801, 4477456, 5764801, 7311616, 9150625, 11316496, 13845841, 16777216, 20151121, 24010000, 28398241, 33362176, 38950081, 45212176, 52200625, 59969536, 68574961

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

From Harvey P. Dale, Oct 21 2015: (Start)

a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5).

G.f.: -((1+251*x+1131*x^2+545*x^3+16*x^4)/(-1+x)^5). (End)

a(n) = A000583(A016777(n)). - Michel Marcus, Nov 06 2015

E.g.f.: exp(x)*(1+255*x+945*x^2+594*x^3+81*x^4). - Wolfdieter Lang, Apr 02 2017

Sum_{n>=0} 1/a(n) = PolyGamma(3, 1/3)/486. - Amiram Eldar, Mar 29 2022

MATHEMATICA

(3*Range[0, 30]+1)^4 (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 256, 2401, 10000, 28561}, 30] (* Harvey P. Dale, Oct 21 2015 *)

PROG

(Magma) [(3*n+1)^4: n in [0..30]]; // Vincenzo Librandi, Sep 21 2011

CROSSREFS

Cf. A000583 (n^4), A016777 (3n+1), A016778, A016779, A016781.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

A016781

a(n) = (3*n+1)^5.

+10
11

1, 1024, 16807, 100000, 371293, 1048576, 2476099, 5153632, 9765625, 17210368, 28629151, 45435424, 69343957, 102400000, 147008443, 205962976, 282475249, 380204032, 503284375, 656356768, 844596301, 1073741824, 1350125107, 1680700000, 2073071593, 2535525376

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

In general the e.g.f. of {(1 + 3*m)^n}_{m>=0} is E(n,x) = exp(x)*Sum_{m=0..n} A282629(n, m)*x^m, and the o.g.f. is G(n, x) = (Sum_{m=0..n} A225117(n, n-m)*x^m)/(1-x)^(n+1). - Wolfdieter Lang, Apr 02 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

FORMULA

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, May 13 2012

From Wolfdieter Lang, Apr 02 2017: (Start)

O.g.f.: (1+1018*x+10678*x^2+14498*x^3+2933*x^4+32*x^5)/(1-x)^6.

E.g.f: exp(x)*(1+1023*x+7380*x^2+8775*x^3+2835*x^4+243*x^5). (End)

a(n) = A000584(A016777(n)). - Michel Marcus, Apr 06 2017

Sum_{n>=0} 1/a(n) = 2*Pi^5/(3^6*sqrt(3)) + 121*zeta(5)/3^5. - Amiram Eldar, Mar 29 2022

MATHEMATICA

(3Range[0, 20]+1)^5 (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 1024, 16807, 100000, 371293, 1048576}, 30] (* Harvey P. Dale, May 13 2012 *)

PROG

(Magma) [(3*n+1)^5: n in [0..30]]; // Vincenzo Librandi, Sep 21 2011

(Maxima) A016781(n):=(3*n+1)^5$

makelist(A016781(n), n, 0, 20); /* Martin Ettl, Nov 12 2012 */

CROSSREFS

Cf. A016777, A016778, A016779, A016780, A225117, A282629.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

A016994

a(n) = (7*n + 1)^2.

+10
11

1, 64, 225, 484, 841, 1296, 1849, 2500, 3249, 4096, 5041, 6084, 7225, 8464, 9801, 11236, 12769, 14400, 16129, 17956, 19881, 21904, 24025, 26244, 28561, 30976, 33489, 36100, 38809, 41616, 44521, 47524

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: (1 + 61*x + 36*x^2)/(1-x)^3. - Vincenzo Librandi, Jan 27 2013

From G. C. Greubel, Dec 28 2022: (Start)

a(2*n) = A134934(n).

a(2*n+1) = 4*A017030(n).

E.g.f.: (1 + 63*x + 49*x^2)*exp(x). (End)

Sum_{n>=0} 1/a(n) = Psi'(1/7)/49 = 1.027703498712483534.. - R. J. Mathar, May 07 2024

MATHEMATICA

(7Range[0, 50]+1)^2 (* Harvey P. Dale, Mar 05 2011 *)

CoefficientList[Series[(1+61*x+36*x^2)/(1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 27 2013 *)

PROG

(Magma) [(7*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Jul 13 2011

(PARI) a(n)=(7*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

(SageMath) [(7*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

CROSSREFS

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), A016814 (m=4), A016862 (m=5), A016922 (m=6), this sequence (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A017078

a(n) = (8*n + 1)^2.

+10
11

1, 81, 289, 625, 1089, 1681, 2401, 3249, 4225, 5329, 6561, 7921, 9409, 11025, 12769, 14641, 16641, 18769, 21025, 23409, 25921, 28561, 31329, 34225, 37249, 40401, 43681, 47089, 50625, 54289, 58081

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: (1 + 78*x + 49*x^2)/(1-x)^3. - R. J. Mathar, Mar 21 2016

From G. C. Greubel, Dec 28 2022: (Start)

a(2*n) = A016754(8*n).

E.g.f.: (1 + 80*x + 64*x^2)*exp(x). (End)

Sum_{n>=0} 1/a(n) = psi'(1/8)/64 = 1.02168958507793.. - R. J. Mathar, May 07 2024

MATHEMATICA

(8*Range[0, 40] +1)^2 (* G. C. Greubel, Dec 28 2022 *)

PROG

(Magma) [(8*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Jul 11 2011

(PARI) a(n)=(8*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

(SageMath) [(8*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

CROSSREFS

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), this sequence (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A017174

a(n) = (9*n + 1)^2.

+10
11

1, 100, 361, 784, 1369, 2116, 3025, 4096, 5329, 6724, 8281, 10000, 11881, 13924, 16129, 18496, 21025, 23716, 26569, 29584, 32761, 36100, 39601, 43264, 47089, 51076, 55225, 59536, 64009, 68644

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: x*(1 + 97*x + 64*x^2)/(1-x)^3. - Bruno Berselli, Aug 25 2011

From G. C. Greubel, Dec 28 2022: (Start)

a(2*n) = A016754(9*n).

a(2*n+1) = 4*A017222(n).

E.g.f.: (1 + 99*x + 81*x^2)*exp(x). (End)

MATHEMATICA

(9*Range[0, 40] +1)^2 (* G. C. Greubel, Dec 28 2022 *)

LinearRecurrence[{3, -3, 1}, {1, 100, 361}, 50] (* Harvey P. Dale, Feb 25 2024 *)

PROG

(Magma) [(9*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Aug 25 2011

(PARI) a(n)=(9*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

(SageMath) [(9*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

CROSSREFS

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), this sequence (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

Cf. A017222.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A017402

a(n) = (11*n+1)^2.

+10
11

1, 144, 529, 1156, 2025, 3136, 4489, 6084, 7921, 10000, 12321, 14884, 17689, 20736, 24025, 27556, 31329, 35344, 39601, 44100, 48841, 53824, 59049, 64516, 70225, 76176, 82369, 88804, 95481, 102400

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 05 2014

From G. C. Greubel, Dec 24 2022: (Start)

G.f.: (1 + 141*x + 100*x^2)/(1-x)^3.

E.g.f.: (1 + 143*x + 121*x^2)*exp(x). (End)

MATHEMATICA

(11*Range[0, 30]+1)^2 (* or *) LinearRecurrence[{3, -3, 1}, {1, 144, 529}, 30] (* Harvey P. Dale, May 05 2014 *)

PROG

(PARI) a(n)=(11*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

(Magma) [(11*n+1)^2: n in [0..50]]; // G. C. Greubel, Dec 24 2022

(SageMath) [(11*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

CROSSREFS

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), this sequence (m=11), A017534 (m=12), A134934 (m=14).

Cf. A017534, A082043.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved