A026998 -id:A026998 - cp's OEIS frontend

A027002 a(n) = T(2*n, n+3), T given by A026998.

1, 43, 431, 2482, 10636, 38138, 122069, 362853, 1027843, 2822668, 7601784, 20228876, 53447609, 140633575, 369179479, 967898846, 2535852052, 6641420806, 17390705661, 45533644161, 119213967867, 312112955384, 817130734512, 2139286435768, 5600737350897

Offset: 3

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 3..2370
Index entries for linear recurrences with constant coefficients, signature (9,-34,71,-90,71,-34,9,-1).

Cf. A000032, A026998.

A027002 := func< n | Lucas(2*n+7) -(435+433*n+155*n^2+125*n^3-20*n^4+12*n^5)/15 >;
[A027002(n): n in [3..50]]; // G. C. Greubel, Jun 16 2025

gf:= x^3*(1+34*x+78*x^2-6*x^3-11*x^4) / ((1-x)^6*(1-3*x+x^2)):
S:= series(gf,x,100):
seq(coeff(S,x,n),n=3..100); # Robert Israel, Feb 18 2016

LinearRecurrence[{9, -34, 71, -90, 71, -34, 9, -1}, {1, 43, 431, 2482, 10636, 38138, 122069, 362853}, 30] (* Vincenzo Librandi, Feb 19 2016 *)

Vec(x^3*(1+34*x+78*x^2-6*x^3-11*x^4)/((1-x)^6*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

def A027002(n): return lucas_number2(2*n+7,1,-1) -(435+433*n+155*n^2+125*n^3 -20*n^4+12*n^5)/15
print([A027002(n) for n in range(3,51)]) # G. C. Greubel, Jun 16 2025

G.f.: x^3*(1+34*x+78*x^2-6*x^3-11*x^4) / ((1-x)^6*(1-3*x+x^2)). - Colin Barker, Feb 18 2016
From Robert Israel, Feb 18 2016: (Start)
By definition, a(n) is the coefficient of x^(2*n-6) in the Maclaurin series of (1+2*x)/((1-x-x^2)*(1-x)^6).  This can be written explicitly:
a(n) = ((29-13*sqrt(5))/2)*((3-sqrt(5))/2)^n + ((29+13*sqrt(5))/2)*((3+sqrt(5))/2)^n - (4/5)*n^5 + (4/3)*n^4 - (25/3)*n^3 - (31/3)*n^2 - (433/15)*n - 29.
This confirms Colin Barker's g.f. (End)
From G. C. Greubel, Jun 16 2025: (Start)
a(n) = A000032(2*n+7) - (1/15)*(435 + 433*n + 155*n^2 + 125*n^3 - 20*n^4 + 12*n^5).
E.g.f.: exp(3*x/2)*(29*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)) - (1/15)*(435 + 705*x + 570*x^2 + 305*x^3 + 100*x^4 + 12*x^5)*exp(x). (End)

A027003 a(n) = A026998(2*n, n+4).

Original entry on oeis.org

1, 64, 901, 6821, 36425, 155793, 573382, 1899933, 5844446, 17056486, 47974934, 131553646, 354615679, 945220982, 2501450971, 6590435731, 17316698039, 45428211431, 119066290172, 311909267867, 816853717452, 2138914514428, 5600243896572, 14662288678348, 38387242941837

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (11,-53,148,-266,322,-266,148,-53,11,-1).

Cf. A000032, A026998.

A027003:= func< n | Lucas(2*n+9) -(48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n +47880)/630 >;
[A027003(n): n in [4..45]]; // G. C. Greubel, Jul 20 2025

f[n_]:= (48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n+47880)/630;
A027003[n_]:= LucasL[2*n+9] -f[n];
Table[A027003[n], {n,4,50}] (* G. C. Greubel, Jul 20 2025 *)

def A027003(n): return lucas_number2(2*n+9,1,-1) -(48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n +47880)//630
print([A027003(n) for n in range(4,46)]) # G. C. Greubel, Jul 20 2025

From G. C. Greubel, Jul 20 2025: (Start)
a(n) = Lucas(2*n+9) - f(n), where f(n) = (48*n^7 - 280*n^6 + 1596*n^5 - 910*n^4 + 10122*n^3 + 20405*n^2 + 46509*n + 47880)/630.
G.f.: x^4*(1 + 53*x + 250*x^2 + 154*x^3 - 59*x^4 - 15*x^5)/((1-x)^8*(1-3*x+x^2)).
E.g.f.: 4*exp(3*x/2)*(19*cosh(p*x) + 17*p*sinh(p*x)) - (1/630)*(47880 + 77490*x + 62685*x^2 + 33810*x^3 + 13650*x^4 + 4116*x^5 + 728*x^6 + 48*x^7)*exp(x), where 2*p = sqrt(5). (End)

More terms from Sean A. Irvine, Oct 21 2019

A027007 a(n) = A026998(2n+1, n+4).

Original entry on oeis.org

1, 53, 634, 4201, 20120, 78753, 269829, 844702, 2486178, 7017354, 19260116, 51903794, 138254821, 365619439, 962704734, 2528441803, 6631057180, 17376467099, 45514392201, 119188310928, 312079208726, 817086876180, 2139230058328, 5600665608772, 14662845807193

Offset: 3

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (10,-43,105,-161,161,-105,43,-10,1).

Cf. A000032, A026998.

A027007:= func< n | Lucas(2*n+9) - (1/30)*(8*n^6 - 4*n^5 + 110*n^4 + 325*n^3 + 1052*n^2 + 2199*n + 2280) >;
[A027007(n): n in [3..45]]; // G. C. Greubel, Jul 23 2025

Table[LucasL[2*n+9] -(1/30)*(8*n^6 -4*n^5 +110*n^4 +325*n^3 +1052*n^2 +2199*n +2280), {n,3,45}] (* G. C. Greubel, Jul 23 2025 *)

def A027007(n): return lucas_number2(2*n+9,1,-1) - (1/30)*(8*n^6 - 4*n^5 + 110*n^4 + 325*n^3 + 1052*n^2 + 2199*n + 2280)
print([A027007(n) for n in range(3,46)]) # G. C. Greubel, Jul 23 2025

From G. C. Greubel, Jul 23 2025: (Start)
a(n) = Lucas(2*n+9) - (1/30)*(8*n^6 - 4*n^5 + 110*n^4 + 325*n^3 + 1052*n^2 + 2199*n + 2280).
G.f.: x^3*(1 + 43*x + 147*x^2 + 35*x^3 - 32*x^4 - 2*x^5)/((1-x)^7*(1-3*x+x^2)).
E.g.f.: 4*exp(3*x/2)*( 19*cosh(p*x) + 17*p*sinh(p*x) ) - (1/30)*(2280 + 3690*x + 2985*x^2 + 1605*x^3 + 590*x^4 + 116*x^5 + 8*x^6)*exp(x), where 2*p = sqrt(5). (End)

More terms from Sean A. Irvine, Oct 21 2019

A027010 a(n) = Sum_{k=floor((n+1)/2)..n} T(k,n-k); i.e., a(n) is the n-th diagonal sum of left-justified array T given by A026998.

Original entry on oeis.org

1, 1, 2, 5, 6, 13, 17, 29, 43, 64, 100, 144, 223, 326, 492, 733, 1089, 1634, 2421, 3626, 5389, 8041, 11985, 17847, 26624, 39640, 59112, 88059, 131242, 195592, 291433, 434369, 647218, 964581, 1437374, 2142013, 3192113, 4756821

Offset: 1

Clark Kimberling

nonn

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

Cf. A026998, A122514.

R:= PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(1-x^2+2*x^3)/((1-x)*(1-2*x^2-x^3+x^4)) )); // G. C. Greubel, Jul 11 2025

CoefficientList[Series[(1-x^2+2 x^3)/((1-x)(1-2 x^2 -x^3 +x^4)), {x,0,40}], x] (* Vincenzo Librandi, Aug 03 2017 *)

def A027010_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x^2+2*x^3)/((1-x)*(1-2*x^2-x^3+x^4)) ).list()
a=A027010_list(40); a[1:] # G. C. Greubel, Jul 11 2025

G.f.: x*(1 - x^2 + 2*x^3)/((1-x)*(1-2*x^2-x^3+x^4)).

a(n) = 2*b(n+2) + 3*b(n+1) - b(n) - 4*b(n-1) - 2, where b(n) = A122514(n). - G. C. Greubel, Jul 11 2025

A026999 Uniquification of A026998.

Original entry on oeis.org

1, 4, 8, 11, 13, 19, 26, 29, 34, 43, 53, 54, 64, 73, 76, 89, 101, 103, 118, 134, 151, 169, 171, 174, 188, 196, 199, 208, 229, 251, 274, 281, 298, 323, 349, 370, 376, 404, 431, 433, 463, 487, 494, 518, 521, 526, 559, 593, 628, 634, 664, 701, 739, 743, 778, 818, 859, 901, 944, 988, 1033, 1079, 1126, 1148

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A026998, A027960.

f[n_, k_]:= f[n, k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] -f[n,k]*Boole[k>n];
A026999= Table[A027960[n,2*k], {n,0,225}, {k,0,n}]//Flatten//Union;
Table[A026999[[n]], {n,120}] (* G. C. Greubel, Aug 21 2025 *)

@CachedFunction
def t(n, k): # t = A027960
    if (k>2*n): return 0
    elif (kA026998(n, k): return t(n, 2*k)
A026999 = sorted(set( flatten([[A026998(n,k) for k in range(n+1)] for n in range(103)]) ))
print([A026999[n] for n in range(100)]) # G. C. Greubel, Aug 21 2025

More terms added by G. C. Greubel, Aug 21 2025

A027001 a(n) = T(2*n, n+2), T given by A026998.

Original entry on oeis.org

1, 26, 174, 743, 2552, 7784, 22193, 60882, 163430, 433495, 1142496, 3001056, 7869649, 20619098, 54001422, 141401879, 370224248, 969294632, 2537687585, 6643800690, 17393752166, 45537499111, 119218794624, 312118940928, 817138091617, 2139295405274

Offset: 2

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (7,-19,26,-19,7,-1).

Bisection of A027964.

Cf. A000032, A000045, A026998.

[3*Fibonacci(2*n+10)-2*Fibonacci(2*n+9)-Fibonacci(2*n+8)-4*n^3-26*n^2-68*n-75: n in [0..30]]; // Vincenzo Librandi, Feb 19 2016

LinearRecurrence[{7, -19, 26, -19, 7, -1}, {1, 26, 174, 743, 2552, 7784}, 30] (* Vincenzo Librandi, Feb 19 2016 *)

Vec(x^2*(1+x)*(1+18*x-7*x^2)/((1-x)^4*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 18 2016

def A027001(n): return lucas_number2(2*n+5,1,-1) -(4*(n+1)**3 -10*n**2 +7)
print([A027001(n) for n in range(2,41)]) # G. C. Greubel, Jul 20 2025

a(n+2) = 3*F(2*n+10) - 2*F(2*n+9) - F(2*n+8) -(4*n^3 +26*n^2 +68*n +75), n >= 0, F(n) = A000045(n). - Ralf Stephan, Feb 07 2004
From Colin Barker, Feb 18 2016: (Start)
a(n) = 2^(-1-n)*( (11-5*sqrt(5))*(3-sqrt(5))^n + (11+5*sqrt(5))*(3+sqrt(5))^n ) - 11 - 12*n - 2*n^2 - 4*n^3.
G.f.: x^2*(1+x)*(1+18*x-7*x^2) / ((1-x)^4*(1-3*x+x^2)). (End)
From G. C. Greubel, Jul 20 2025: (Start)
a(n) = Lucas(2*n+5) - (4*(n+1)^3 - 10*n^2 + 7), n >= 2.
E.g.f.: exp(3*x/2)*(11*cosh(p*x) + 10*p*sinh(p*x)) - (4*x^3 + 14*x^2 + 18*x + 11)*exp(x), where 2*p = sqrt(5). (End)

A027004 a(n) = T(2*n+1,n+1), T given by A026998.

Original entry on oeis.org

1, 8, 26, 73, 196, 518, 1361, 3568, 9346, 24473, 64076, 167758, 439201, 1149848, 3010346, 7881193, 20633236, 54018518, 141422321, 370248448, 969323026, 2537720633, 6643838876, 17393795998, 45537549121, 119218851368, 312119004986, 817138163593, 2139295485796

Offset: 0

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A000032, A002878, A026998.

[Fibonacci(2*n+3) + 2*Fibonacci(2*n+2) - 3: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

LucasL[2*Range[0,40] +3] -3 (* G. C. Greubel, Jul 21 2025 *)

Vec((1+4*x-2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 18 2016

def A027004(n): return lucas_number2(2*n+3,1,-1) -3 # G. C. Greubel, Jul 21 2025

a(n) = Fibonacci(2*n+3) + 2*Fibonacci(2*n+2) - 3.
a(n) = A002878(n+1) - 3.
From Colin Barker, Feb 18 2016: (Start)
a(n) = 2^(-n)*((2-sqrt(5))*(3-sqrt(5))^n + (2+sqrt(5))*(3+sqrt(5))^n) - 3.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n > 2.
G.f.: (1+4*x-2*x^2) / ((1-x)*(1-3*x+x^2)). (End)
From G. C. Greubel, Jul 21 2025: (Start)
a(n) = Lucas(2*n+3) - 3.
E.g.f.: 2*exp(3*x/2)*(2*cosh(p*x) + p*sinh(p*x)) - 3*exp(x), where 2*p = sqrt(5). (End)

A027005 a(n) = T(2*n+1,n+2), T given by A026998.

Original entry on oeis.org

1, 19, 101, 370, 1148, 3278, 8967, 23993, 63483, 167040, 438346, 1148844, 3009181, 7879855, 20631713, 54016798, 141420392, 370246298, 969320643, 2537718005, 6643835991, 17393792844, 45537545686, 119218847640, 312119000953, 817138159243, 2139295481117

Offset: 1

Clark Kimberling

Bisection of A027963.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,13,-6,1).

Cf. A000032, A026998, A027963.

A027005:= func< n | Lucas(2*n+5) -(6*n^2+11*n+11) >;
[A027005(n): n in [1..40]]; // G. C. Greubel, Jul 21 2025

LinearRecurrence[{6,-13,13,-6,1},{1,19,101,370,1148},30] (* Harvey P. Dale, Aug 19 2020 *)

Vec(x*(1+13*x-2*x^3)/((1-x)^3*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

def A027005(n): return lucas_number2(2*n+5,1,-1) -(6*n**2 +11*n +11)
print([A027005(n) for n in range(1,41)]) # G. C. Greubel, Jul 21 2025

From Colin Barker, Feb 19 2016: (Start)
a(n) = 2^(-1-n)*((25+11*sqrt(5))*(3+sqrt(5))^n - (25-11*sqrt(5))*(3-sqrt(5))^n )/sqrt(5) + 7*(1+n) - 6*(n+1)*(n+2) + 7*(n+1) - 6.
a(n) = 6*a(n-1) - 13*a(n-2) + 13*a(n-3) - 6*a(n-4) + a(n-5) for n>5.
G.f.: x*(1+13*x-2*x^3) / ((1-x)^3*(1-3*x+x^2)). (End)
From G. C. Greubel, Jul 21 2025: (Start)
a(n) = Lucas(2*n+5) - (6*n^2 + 11*n + 11).
E.g.f.: exp(3*x/2)*(11*cosh(p*x) + 10*p*sinh(p*x)) - (11 + 17*x + 6*x^2)*exp(x), where 2*p = sqrt(5). (End)

A027006 a(n) = T(2*n+1, n+3), T given by A026998.

Original entry on oeis.org

1, 34, 281, 1397, 5353, 17643, 53062, 150833, 414210, 1114160, 2960806, 7814074, 20544191, 53902532, 141273663, 370060623, 969088727, 2537431693, 6643486220, 17393369595, 45537037936, 119218243314, 312118286876, 817137321092, 2139294503373, 5600747154678

Offset: 2

Clark Kimberling

Bisection of A053298.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (8,-26,45,-45,26,-8,1).

Cf. A000032, A026998, A053298.

A027006:= func< n | Lucas(2*n+7) -(12*n^4 +20*n^3 +81*n^2 +169*n +174)/6 >;
[A027006(n): n in [2..40]]; // G. C. Greubel, Jul 22 2025

A027006[n_]:= LucasL[2*n+7] -(12*n^4 +20*n^3 +81*n^2 +169*n +174)/6;
Table[A027006[n], {n,2,42}] (* G. C. Greubel, Jul 22 2025 *)

Vec(x^2*(1+26*x+35*x^2-12*x^3-2*x^4) / ((1-x)^5*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

def A027006(n): return lucas_number2(2*n+7,1,-1) -(12*n^4 +20*n^3 +81*n^2 +169*n +174)//6
print([A027006(n) for n in range(2,41)]) # G. C. Greubel, Jul 22 2025

G.f.: x^2*(1+26*x+35*x^2-12*x^3-2*x^4) / ((1-x)^5*(1-3*x+x^2)). - Colin Barker, Feb 19 2016
From G. C. Greubel, Jul 22 2025: (Start)
a(n) = Lucas(2*n+7) - (12*n^4 + 20*n^3 + 81*n^2 + 169*n + 174)/6.
E.g.f.: exp(3*x/2)*(29*cosh(p*x) + 26*p*sinh(p*x)) - (1/6)*(174 + 282*x + 225*x^2 + 92*x^3 + 12*x^4)*exp(x), where 2*p = sqrt(5). (End)

A027008 a(n) = greatest number in row n of array T given by A026998.

Original entry on oeis.org

1, 1, 4, 8, 13, 26, 54, 101, 174, 370, 743, 1397, 2552, 5353, 10636, 20120, 38138, 78753, 155793, 296248, 573382, 1173183, 2316317, 4423690, 8673078, 17641499, 34801731, 66705394, 131894869, 267203186, 526966454, 1013155981

Offset: 0

Clark Kimberling

nonn

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

Cf. A026998.

f[n_, k_]:= f[n, k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] -f[n,k]*Boole[k>n];
b[n_]:= b[n]= Table[A027960[n,2*k], {k,0,n}];
A027008[n_]:= Max[b[n]];
Table[A027008[n], {n,0,50}] (* G. C. Greubel, Jul 24 2025 *)

@CachedFunction
def t(n, k): # t = A027960
    if (k>2*n): return 0
    elif (kA026998(n, k): return t(n, 2*k)
def b(n): return flatten([A026998(n,k) for k in range(n+1)])
def A027008(n): return max(b(n))
print([A027008(n) for n in range(51)]) # G. C. Greubel, Jul 24 2025

Offset changed by G. C. Greubel, Jul 24 2025

cp's OEIS Frontend

A027002 a(n) = T(2*n, n+3), T given by A026998.

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A027003 a(n) = A026998(2*n, n+4).

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A027007 a(n) = A026998(2n+1, n+4).

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A027010 a(n) = Sum_{k=floor((n+1)/2)..n} T(k,n-k); i.e., a(n) is the n-th diagonal sum of left-justified array T given by A026998.

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A026999 Uniquification of A026998.

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A027001 a(n) = T(2*n, n+2), T given by A026998.

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A027004 a(n) = T(2*n+1,n+1), T given by A026998.

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