A098158 -id:A098158 - cp's OEIS frontend

A098172 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 4, 1, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 1, 20, 1, 0, 0, 0, 0, 0, 7, 35, 1, 0, 0, 0, 0, 0, 0, 28, 56, 1, 0, 0, 0, 0, 0, 0, 1, 84, 84, 1, 0, 0, 0, 0, 0, 0, 0, 10, 210, 120, 1, 0, 0, 0, 0, 0, 0, 0, 0, 55, 462, 165, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 220, 924, 220, 1

Offset: 0

Paul Barry, Aug 30 2004

Row sums are A024493.
From R. J. Mathar, Mar 22 2013: (Start)
The matrix inverse starts
  1;
  0, 1;
  0, 0,      1;
  0, 0,     -1,     1;
  0, 0,      4,    -4,     1;
  0, 0,    -40,    40,   -10,   1;
  0, 0,    796,  -796,   199, -20,   1;
  0, 0, -27580, 27580, -6895, 693, -35, 1;
  ... (End)

			Rows begin
  {1},
  {0,1},
  {0,0,1},
  {0,0,1,1},
  {0,0,0,4,1},
  {0,0,0,0,10,1},
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A098158.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 3*(n-k)) ))); # G. C. Greubel, Mar 15 2019

[[Binomial(n, 3*(n-k)): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Mar 15 2019

Table[Binomial[n, 3(n-k)], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 15 2019 *)

{T(n, k) = binomial(n, 3*(n-k))}; \\ G. C. Greubel, Mar 15 2019

[[binomial(n, 3*(n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 15 2019

Triangle T(n, k) = binomial(n, 3(n-k)).

A152262 a(n) = 14a(n-1) - 43a(n-2), n > 1; a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 55, 469, 4201, 38647, 360415, 3383989, 31878001, 300780487, 2840172775, 26828857909, 253476581401, 2395031249527, 22630944493135, 213846879174229, 2020725695234401, 19094743928789767, 180435210107977495

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

Binomial transform of A145301. Inverse binomial transform of A152263. - Philippe Deléham, Dec 03 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Index entries for linear recurrences with constant coefficients, signature (14,-43).

Cf. A027907, A098158, A145301, A152263.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((7+r6)^n+(7-r6)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

[n le 2 select 7^(n-1) else 14*Self(n-1) -43*Self(n-2): n in [1..30]]; // G. C. Greubel, May 23 2023

LinearRecurrence[{14,-43},{1,7},30] (* Harvey P. Dale, Apr 26 2015 *)

@CachedFunction
def a(n): # a = A152262
    if (n<2): return 7^n
    else: return 14*a(n-1) - 43*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, May 23 2023

a(n) = ((7 + sqrt(6))^n + (7 - sqrt(6))^n)/2.
From Philippe Deléham, Dec 03 2008: (Start)
G.f.: (1-7*x)/(1-14*x+43*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*7^(2k-n)*6^(n-k). (End)
a(n) = Sum_{k=0..n} A027907(n,2k)*6^k. - J. Conrad, Aug 24 2016
E.g.f.: cosh(sqrt(6)*x)*exp(7*x). - Ilya Gutkovskiy, Aug 24 2016
a(n) = m^n*(ChebyshevU(n, 7/m) - (7/m)*ChebyshevU(n-1, 7/m)), with m = sqrt(43). - G. C. Greubel, May 23 2023

Name from Philippe Deléham, Dec 03 2008

A098173 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 4k).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 15, 1, 0, 0, 0, 0, 0, 0, 35, 1, 0, 0, 0, 0, 0, 0, 1, 70, 1, 0, 0, 0, 0, 0, 0, 0, 9, 126, 1, 0, 0, 0, 0, 0, 0, 0, 0, 45, 210, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 165, 330, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 495, 495, 1

Offset: 0

Paul Barry, Aug 30 2004

Row sums are A038503.

			Rows begin
  {1},
  {0,1},
  {0,0,1},
  {0,0,0,1},
  {0,0,0,1,1},
  {0,0,0,0,5,1},
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A098158, A098172.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 4*(n-k)) ))); # G. C. Greubel, Mar 15 2019

[[Binomial(n, 4*(n-k)): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Mar 15 2019

Table[Binomial[n, 4(n-k)], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 15 2019 *)

{T(n, k) = binomial(n, 4*(n-k))}; \\ G. C. Greubel, Mar 15 2019

[[binomial(n, 4*(n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 15 2019

Triangle T(n, k) = binomial(n, 4(n-k)).

A143647 a(n) = ((5 + sqrt(3))^n + (5 - sqrt(3))^n)/2.

Original entry on oeis.org

1, 5, 28, 170, 1084, 7100, 47152, 315320, 2115856, 14221520, 95666368, 643790240, 4333242304, 29169037760, 196359046912, 1321871638400, 8898817351936, 59906997474560, 403295993003008, 2715005985589760, 18277548009831424

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Oct 27 2008

nonn

Binomial transform of A083882. - R. J. Mathar, Nov 01 2008

Inverse binomial transform of A147961.

Index entries for linear recurrences with constant coefficients, signature (10, -22).

Cf. A083882, A147961, A098158.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r3)^n+(5-r3)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 01 2008

Simplify[With[{c=Sqrt[3]},Table[((5+c)^n+(5-c)^n)/2,{n,0,25}]]] (* or *) LinearRecurrence[{10,-22},{1,5},25] (* Harvey P. Dale, Jun 04 2011 *)

From Philippe Deléham, Klaus Brockhaus and R. J. Mathar, Nov 01 2008: (Start)
a(n) = 10*a(n-1) - 22*a(n-2), a(0)=1, a(1)=5.
G.f.: (1-5x)/(1-10x+22*x^2). (End)
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2*k)*3^(n-k))/5^n. - Philippe Deléham, Nov 06 2008

More terms from Klaus Brockhaus and R. J. Mathar, Nov 01 2008

Edited by Klaus Brockhaus, Jul 15 2009

A146962 a(n) = 10a(n-1) - 19a(n-2) with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 31, 215, 1561, 11525, 85591, 636935, 4743121, 35329445, 263175151, 1960492055, 14604592681, 108796577765, 810478516711, 6037650189575, 44977410078241, 335058747180485, 2496016680318271, 18594050606753495

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

Binomial transform of A143648.

Inverse binomial transform of A145301.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-19).

Cf. A098158, A143648, A145301.

a:=[1,5];; for n in [3..30] do a[n]:=10*a[n-1]-19*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((5+r6)^n+(5-r6)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008

seq(coeff(series((1-5*x)/(1-10*x+19*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 08 2020

LinearRecurrence[{10,-19},{1,5},30] (* Harvey P. Dale, Apr 27 2014 *)
CoefficientList[Series[(1-5x)/(1-10x+19x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 28 2014 *)

my(x='x+O('x^30)); Vec((1-5*x)/(1-10*x+19*x^2)) \\ G. C. Greubel, Jan 08 2020

def A146962_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x)/(1-10*x+19*x^2) ).list()
A146962_list(30) # G. C. Greubel, Jan 08 2020

a(n) = ((5 + sqrt(6))^n + (5 - sqrt(6))^n)/2.
G.f.: (1-5*x)/(1-10*x+19*x^2). - Philippe Deléham and Klaus Brockhaus, Nov 05 2008
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2*k)*6^(n-k))/5^n. - Philippe Deléham, Nov 06 2008
E.g.f.: exp(5*x)*cosh(sqrt(6)*x). - G. C. Greubel, Jan 08 2020

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008
Edited by Klaus Brockhaus, Jul 15 2009
Name from Philippe Deléham and Klaus Brockhaus, Nov 05 2008

A146964 a(n) = ((4 + sqrt(7))^n + (4 - sqrt(7))^n)/2.

Original entry on oeis.org

1, 4, 23, 148, 977, 6484, 43079, 286276, 1902497, 12643492, 84025463, 558412276, 3711069041, 24662841844, 163903113383, 1089259330468, 7238946623297, 48108239012164, 319715392487639, 2124748988791636, 14120553377944337

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

nonn

Binomial transform of A146963.

Inverse binomial transform of A146965.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-9).

Cf. A098158, A146963, A146965.

a:=[1,4];; for n in [3..25] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((4+r7)^n+(4-r7)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008

seq(coeff(series((1-4*x)/(1-8*x+9*x^2), x, n+1), x, n), n = 0..25); # G. C. Greubel, Jan 08 2020

LinearRecurrence[{8,-9}, {1,4}, 25] (* G. C. Greubel, Jan 08 2020 *)

my(x='x+O('x^25)); Vec((1-4*x)/(1-8*x+9*x^2)) \\ G. C. Greubel, Jan 08 2020

def A146964_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-4*x)/(1-8*x+9*x^2) ).list()
A146964_list(25) # G. C. Greubel, Jan 08 2020

From Philippe Deléham and Klaus Brockhaus, Nov 05 2008: (Start)
a(n) = 8*a(n-1) - 9*a(n-2) with a(0)=1, a(1)=4.
G.f.: (1-4*x)/(1-8*x+9*x^2). (End)
a(n) = (Sum_{k=0..n} A098158(n,k)*4^(2*k)*7^(n-k))/4^n. - Philippe Deléham, Nov 06 2008
E.g.f.: exp(4*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008

Edited by Klaus Brockhaus, Jul 16 2009

A146965 a(n) = 10a(n-1) - 18a(n-2) with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 32, 230, 1724, 13100, 99968, 763880, 5839376, 44643920, 341330432, 2609713760, 19953189824, 152557050560, 1166413088768, 8918103977600, 68185604178176, 521330170184960, 3985960826642432, 30475665203095040

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

nonn

The Mathematica program implements the formula provided by Deleham and Brockhaus. - Harvey P. Dale, Feb 17 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..145
Index entries for linear recurrences with constant coefficients, signature (10,-18).

a:=[1,5];; for n in [3..30] do a[n]:=10*a[n-1]-18*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((5+r7)^n+(5-r7)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008

seq(coeff(series((1-5*x)/(1-10*x+18*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 08 2020

Transpose[NestList[{#[[2]],10#[[2]]-18#[[1]]}&,{1,5},20]][[1]]  (* Harvey P. Dale, Feb 17 2011 *)
LinearRecurrence[{10,-18},{1,5},30] (* Harvey P. Dale, Aug 27 2013 *)

my(x='x+O('x^30)); Vec((1-5*x)/(1-10*x+18*x^2)) \\ G. C. Greubel, Jan 08 2020

def A146965_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x)/(1-10*x+18*x^2) ).list()
A146965_list(30) # G. C. Greubel, Jan 08 2020

a(n) = ((5 + sqrt(7))^n + (5 - sqrt(7))^n)/2.
G.f.: (1-5*x)/(1-10*x+18*x^2). - Philippe Deléham and Klaus Brockhaus, Nov 05 2008
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2*k)*7^(n-k))/5^n. - Philippe Deléham, Nov 06 2008
E.g.f.: exp(5*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008

Name from Philippe Deléham and Klaus Brockhaus, Nov 05 2008

A146966 a(n) = ((6 + sqrt(7))^n + (6 - sqrt(7))^n) / 2.

Original entry on oeis.org

1, 6, 43, 342, 2857, 24366, 209539, 1807854, 15617617, 134983638, 1166892763, 10088187654, 87218361721, 754062898686, 6519422294323, 56365243469982, 487319675104417, 4213244040623526, 36426657909454219, 314935817735368374

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

nonn

			a(3) = ((6 + sqrt(7))^3 + (6 - sqrt(7))^3) / 2 = 342.

Robert Israel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-29).

a:=[1,6];; for n in [3..20] do a[n]:12*a[n-1]-29*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((6+r7)^n+(6-r7)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008

I:=[1,6]; [n le 2 select I[n] else 12*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 31 2016

f:= gfun:-rectoproc({a(n) = 12*a(n-1)-29*a(n-2), a(0)=1, a(1)=6},a(n),remember):
map(f, [$0..100]); # Robert Israel, Feb 01 2016

RecurrenceTable[{a[1]==1, a[2]==6, a[n]== 12 a[n-1] - 29 a[n-2]}, a, {n, 20}] (* Vincenzo Librandi, Jan 31 2016 *)
LinearRecurrence[{12,-29},{1,6},20] (* Harvey P. Dale, Apr 17 2018 *)

Vec((1-6*x)/(1-12*x+29*x^2) + O(x^30)); \\ Michel Marcus, Jan 31 2016

def A146966_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-6*x)/(1-12*x+29*x^2) ).list()
A146966_list(20) # G. C. Greubel, Jan 08 2020

a(n) = 12*a(n-1)-29*a(n-2), a(0)=1, a(1)=6. - Philippe Deléham, Nov 05 2008
G.f.: (1-6*x)/(1-12*x+29*x^2). - Klaus Brockhaus, Nov 05 2008
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2*k)*7^(n-k))/6^n. - Philippe Deléham, Nov 06 2008
E.g.f.: exp(6*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008

Typo in name corrected by Sean Reeves, Dec 19 2015

A147689 a(n) = ((7 + sqrt(8))^n + (7 - sqrt(8))^n)/2.

Original entry on oeis.org

1, 7, 57, 511, 4817, 46487, 453321, 4440527, 43581217, 428075431, 4206226137, 41336073247, 406249753841, 3992717550647, 39241805801577, 385683861645551, 3790660025173057, 37256202024955207, 366169767317277561

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 10 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14, -41).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((7+r8)^n+(7-r8)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 13 2008

LinearRecurrence[{14,-41},{1,7},20] (* Harvey P. Dale, Sep 11 2020 *)

From Philippe Deléham, Nov 13 2008: (Start)
a(n) = 14*a(n-1) - 41*a(n-2), a(0)=1, a(1)=7.
G.f.: (1-7x)/(1-14x+41x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2k)*8^(n-k))/7^n. (End)

Extended beyond a(6) by Klaus Brockhaus, Nov 13 2008

A147961 a(n) = ((6+sqrt(3))^n + (6-sqrt(3))^n)/2.

Original entry on oeis.org

1, 6, 39, 270, 1953, 14526, 109863, 838998, 6442497, 49623030, 382873959, 2956927518, 22848289569, 176600866734, 1365216845031, 10554773538150, 81605126571777, 630953992102374, 4878478728359847, 37720263000939822, 291653357975402913, 2255071616673820830, 17436298586897553831

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

			a(3)=270

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-33).

Cf. A098158.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((6+r3)^n+(6-r3)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

CoefficientList[Series[(1-6x)/(1-12x+33x^2),{x,0,30}],x] (* or *) LinearRecurrence[{12,-33},{1,6},30] (* Harvey P. Dale, Jul 30 2021 *)

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 12*a(n-1) - 33*a(n-2) for n > 1, with a(0)=1, a(1)=6.
G.f.: (1-6x)/(1-12x+33x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*3^(n-k))/6^n. (End)
E.g.f.: exp(6*x)*cosh(sqrt(3)*x). - Stefano Spezia, Apr 23 2025

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

cp's OEIS Frontend

A098172 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A152262 a(n) = 14*a(n-1) - 43*a(n-2), n > 1; a(0)=1, a(1)=7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

SageMath

Formula

Extensions

A098173 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 4k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A143647 a(n) = ((5 + sqrt(3))^n + (5 - sqrt(3))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A146962 a(n) = 10*a(n-1) - 19*a(n-2) with a(0)=1, a(1)=5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A146964 a(n) = ((4 + sqrt(7))^n + (4 - sqrt(7))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

A152262 a(n) = 14a(n-1) - 43a(n-2), n > 1; a(0)=1, a(1)=7.

A146962 a(n) = 10a(n-1) - 19a(n-2) with a(0)=1, a(1)=5.

A146965 a(n) = 10a(n-1) - 18a(n-2) with a(0)=1, a(1)=5.