A001023 -id:A001023 - cp's OEIS frontend

A180696 14^a(n) is smallest power of 14 beginning with n.

0, 3, 4, 11, 5, 26, 6, 20, 34, 7, 14, 28, 35, 1, 15, 22, 29, 36, 2, 9, 16, 23, 71, 30, 37, 44, 3, 10, 58, 17, 65, 24, 31, 120, 38, 86, 45, 4, 52, 11, 59, 18, 107, 25, 114, 73, 32, 121, 80, 39, 87, 46, 5, 94, 53, 12, 101, 60, 19, 108, 67, 26, 115, 204, 74, 33, 122, 81, 170, 40

Offset: 1

Daniel Mondot, Sep 18 2010

Daniel Mondot, Table of n, a(n) for n = 1..32699

Cf. A001023, A180694, A180695, A180698.

A067414 Seventh column of triangle A067410.

Original entry on oeis.org

1, 8, 112, 1568, 21952, 307328, 4302592, 60236288, 843308032, 11806312448, 165288374272, 2314037239808, 32396521357312, 453551299002368, 6349718186033152, 88896054604464128, 1244544764462497792

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (14).

Cf. A067413 (sixth column), A067415 (eighth column), A001023 (powers of 14).

a(n) = A067410(n+6, 6).
a(n) = 8*14^(n-1), n>=1, a(0)=1.
G.f.: (1-6*x)/(1-14*x).

A013719 a(n) = 14^(2*n + 1).

Original entry on oeis.org

14, 2744, 537824, 105413504, 20661046784, 4049565169664, 793714773254144, 155568095557812224, 30491346729331195904, 5976303958948914397184, 1171355575953987221848064, 229585692886981495482220544

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (196).

Bisection of A001023 (14^n).

Cf. A013708-A013718, A013720-A013729.

[14^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(14^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

a(n)=14^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 196*a(n-1), a(0)=14.
G.f.: 14/(1-196*x). (End)

A013754 a(n) = 14^(3*n + 1).

Original entry on oeis.org

14, 38416, 105413504, 289254654976, 793714773254144, 2177953337809371136, 5976303958948914397184, 16398978063355821105872896, 44998795805848373114515226624, 123476695691247935826229781856256, 338820052976784335907174521413566464, 929722225368296217729286886758826377216

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2744).

Subsequence of A001023.

[14^(3*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

14^(3*Range[0,20]+1) (* or *) NestList[2744#&,14,20] (* Harvey P. Dale, May 16 2020 *)

a(n)=14^(3*n+1) \\ Charles R Greathouse IV, Jun 27 2011

From Philippe Deléham, Nov 30 2008: (Start)
a(n) = 2744*a(n-1); a(0)=14.
G.f.: 14/(1-2744*x).
a(n) = A013755(n)/14. (End)

A013755 a(n) = 14^(3*n + 2).

Original entry on oeis.org

196, 537824, 1475789056, 4049565169664, 11112006825558016, 30491346729331195904, 83668255425284801560576, 229585692886981495482220544, 629983141281877223603213172736, 1728673739677471101567216945987584, 4743480741674980702700443299789930496

Offset: 0

N. J. A. Sloane

			From _Philippe Deléham_, Dec 02 2008: (Start)
a(n) = 2744*a(n-1); a(0)=196.
G.f.: 196/(1-2744*x).
a(n) = 14*A013754(n). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2744).

Subsequence of A001023.

[14^(3*n+2): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

14^(3*Range[0,10]+2) (* or *) NestList[2744#&,196,10] (* Harvey P. Dale, Jan 12 2016 *)

a(n)=14^(3*n+2) \\ Charles R Greathouse IV, Jun 27 2011

A160193 Numerator of Hermite(n, 5/28).

Original entry on oeis.org

1, 5, -367, -5755, 402817, 11037925, -734331695, -29632858075, 1866841880705, 102262852326725, -6074903893493615, -431244900588230075, 24038761085803317505, 2148769817796050860325, -111757677404273451703855, -12351237147086094379982875, 595378957401697424118753025

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerator of 1, 5/14, -367/196, -5755/2744, 402817/38416, 11037925/537824,..

G. C. Greubel, Table of n, a(n) for n = 0..417
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Cf. A001023 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(5/14)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 09 2018

A160193 := proc(n)
        orthopoly[H](n,5/28) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator/@HermiteH[Range[0,20],5/28] (* Harvey P. Dale, Jul 11 2011 *)
Table[14^n*HermiteH[n, 5/28], {n,0,30}] (* G. C. Greubel, Jul 09 2018 *)

a(n)=numerator(polhermite(n,5/28)) \\ Charles R Greathouse IV, Jan 29 2016

D-finite with recurrence a(n) -5*a(n-1) +392*(n-1)*a(n-2)=0. [DLMF] - R. J. Mathar, Feb 16 2014
From G. C. Greubel, Jul 09 2018: (Start)
a(n) = 14^n * Hermite(n, 5/28).
E.g.f.: exp(5*x - 196*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(5/14)^(n-2*k)/(k!*(n-2*k)!)). (End)

A227871 Sum of digits of 14^n.

Original entry on oeis.org

1, 5, 16, 17, 22, 29, 37, 23, 52, 44, 67, 65, 73, 68, 52, 80, 85, 83, 100, 122, 106, 116, 130, 137, 118, 140, 124, 152, 166, 173, 136, 158, 178, 179, 202, 128, 199, 176, 187, 206, 220, 227, 244, 230, 232, 224, 256, 272, 253, 275, 268, 278, 301, 272, 298, 257

Offset: 0

Irene Sermon, Oct 25 2013

			For n=9, 14^9=20661046784 and the sum of the digits is 44.

Irene Sermon, Table of n, a(n) for n = 0..10000

Cf. A001023, A007953, A175527.

Total[IntegerDigits[#]]&/@(14^Range[0,60]) (* Harvey P. Dale, Jul 18 2019 *)

a(n) = A007953(A001023(n)).

A123187 Triangle of coefficients in expansion of (1+13x)^n.

Original entry on oeis.org

1, 1, 13, 1, 26, 169, 1, 39, 507, 2197, 1, 52, 1014, 8788, 28561, 1, 65, 1690, 21970, 142805, 371293, 1, 78, 2535, 43940, 428415, 2227758, 4826809, 1, 91, 3549, 76895, 999635, 7797153, 33787663, 62748517, 1, 104, 4732, 123032, 1999270, 20792408

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 03 2006

T(n,k) equals the number of n-length words on {0,1,...,13} having n-k zeros. - Milan Janjic, Jul 24 2015

			1
1, 13
1, 26, 169
1, 39, 507, 2197
1, 52, 1014, 8788, 28561
1, 65, 1690, 21970, 142805, 371293

Cf. A013609, A013610, A013611, A013621, A038220, A038222.

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+13*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 24 2015

p[0, x] = 1; p[1, x] = 13*x + 1; p[k_, x_] := p[k, x] = (13*x + 1)*p[k - 1, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

p(k, x) = (13*x + 1)*p(k - 1, x).

T(n,k) = 13^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*12^(n-i). Row sums are 14^n = A001023. G.f.: 1 / [1 - x(1+13y)]. - Mircea Merca, Apr 28 2012

A160184 Numerator of Hermite(n, 1/28).

Original entry on oeis.org

1, 1, -391, -1175, 458641, 2301041, -896635319, -6308683751, 2454058631585, 22238090874721, -8635680761357159, -95808996990263479, 37141246445981806129, 487826768288181211345, -188783965120435102822039, -2865977269485973590683399, 1107183737638672431002905921

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 1/14, -391/196, -1175/2744, 458641/38416, ...

G. C. Greubel, Table of n, a(n) for n = 0..417

Cf. A001023 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(1/14)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 24 2018

Table[14^n*HermiteH[n, 1/28], {n, 0, 30}] (* G. C. Greubel, Sep 24 2018 *)

a(n)=numerator(polhermite(n, 1/28)) \\ Charles R Greathouse IV, Jan 29 2016

x='x+O('x^30); Vec(serlaplace(exp(x - 196*x^2))) \\ G. C. Greubel, Sep 24 2018

From G. C. Greubel, Sep 24 2018: (Start)
a(n) = 14^n * Hermite(n, 1/28).
E.g.f.: exp(x - 196*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/14)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160192 Numerator of Hermite(n, 3/28).

Original entry on oeis.org

1, 3, -383, -3501, 439905, 6809283, -841785951, -18540791469, 2254238275137, 64906636872195, -7758232724066751, -277708714711204653, 32620373362042216353, 1404202914087633336771, -162020813910704234524575, -8192328034245044455772973, 928105401692205765637182081

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 3/14, -383/196, -3501/2744, 439905/38416, ...

G. C. Greubel, Table of n, a(n) for n = 0..417

Cf. A001023 (denominators)

[Numerator((&+[(-1)^k*Factorial(n)*(3/14)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 24 2018

Table[14^n*HermiteH[n, 3/28], {n, 0, 30}] (* G. C. Greubel, Sep 24 2018 *)

a(n)=numerator(polhermite(n, 3/28)) \\ Charles R Greathouse IV, Jan 29 2016

x='x+O('x^30); Vec(serlaplace(exp(3*x - 196*x^2))) \\ G. C. Greubel, Sep 24 2018

From G. C. Greubel, Sep 24 2018: (Start)
a(n) = 14^n * Hermite(n, 3/28).
E.g.f.: exp(3*x - 196*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(3/14)^(n-2*k)/(k!*(n-2*k)!)). (End)

cp's OEIS Frontend

A180696 14^a(n) is smallest power of 14 beginning with n.

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A067414 Seventh column of triangle A067410.

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A013719 a(n) = 14^(2*n + 1).

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A013754 a(n) = 14^(3*n + 1).

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A013755 a(n) = 14^(3*n + 2).

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Magma

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A160193 Numerator of Hermite(n, 5/28).

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Magma

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A227871 Sum of digits of 14^n.

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Mathematica

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A123187 Triangle of coefficients in expansion of (1+13x)^n.

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Maple

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