A027960 -id:A027960 - cp's OEIS frontend

A053298 Partial sums of A027964.

1, 8, 34, 107, 281, 654, 1397, 2801, 5353, 9859, 17643, 30869, 53062, 89951, 150833, 250780, 414210, 680665, 1114160, 1818310, 2960806, 4813018, 7814074, 12674542, 20544191, 33283434, 53902532, 87272241, 141273663, 228658744

Offset: 0

Barry E. Williams, Mar 04 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1)

Cf. A027964 and A000204.
A column in triangular array A027960.
Cf. A137176 (row k=5).

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x)/((1-x-x^2)*(1-x)^5))); // G. C. Greubel, May 24 2018

LinearRecurrence[{6,-14,15,-5,-4,4,-1},{1,8,34,107,281,654,1397},30] (* Harvey P. Dale, May 09 2018 *)
CoefficientList[Series[(1+2x)/((1-x-x^2)(1-x)^5), {x,0,50}], x] (* G. C. Greubel, May 24 2018 *)

x='x+O('x^30); Vec((1+2*x)/((1-x-x^2)*(1-x)^5)) \\ G. C. Greubel, May 24 2018

a(n) = 3*F(n+10) + F(n+9) - (3*n^4 + 58*n^3 + 489*n^2 + 2234*n + 4752)/24, where F(.) are the Fibonacci numbers (A000045).
a(n) = a(n-1) + a(n-2) + (3*n+4)*C(n+3, 3)/4.
G.f.: (1 + 2*x)/((1 - x - x^2)*(1 - x)^5). - R. J. Mathar, Nov 28 2008

A360278 Determinant of the matrix [L(j+k) + d(j,k)]_{1<=j, k<=n}, where L(n) denotes the Lucas number A000032(n), and d(j,k) is 1 or 0 according as j = k or not.

Original entry on oeis.org

4, 16, 44, 121, 319, 841, 2204, 5776, 15124, 39601, 103679, 271441, 710644, 1860496, 4870844, 12752041, 33385279, 87403801, 228826124, 599074576, 1568397604, 4106118241, 10749957119, 28143753121, 73681302244, 192900153616, 505019158604, 1322157322201, 3461452807999, 9062201101801, 23725150497404, 62113250390416, 162614600673844, 425730551631121, 1114577054219519

Offset: 1

Zhi-Wei Sun, Feb 01 2023

nonn

Conjecture 1: Let v(0) = 2, v(1) = A, and v(n+1) = A*v(n) + v(n-1) for n > 0. Then A^2*det[v(j+k) + d(j,k)]_{1<=j, k<=n} = v(n+1)^2 - (A^2 + 4)*(n mod 2) for any positive integer n. In particular, a(n) = L(n+1)^2 - 5*(n mod 2) for all n > 0.
Conjecture 2: Let v(0) = 2, v(1) = A, and v(n+1) = A*v(n) - v(n-1) for n > 0. Then det[v(j+k) + d(j,k)]_{1<=j, k<=n} = u(n+1)^2 - n^2 for any positive integer n, where u(0) = 0, u(1) = 1, and u(n+1) = A*u(n) - u(n-1) for all n > 0.
Conjecture 3: Let F(n) denote the Fibonacci number A000045(n). Then, for any positive integer n, we have det[F(j+k) + d(j,k)]_{1<=j, k<=n} = F(n+1)^2 + (n mod 2).

			a(2) = 16 since the determinant of the 2 X 2 matrix [L(1+1)+1, L(1+2); L(2+1), L(2+2)+1] = [4, 4; 4, 8] is 16.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Han Wang and Zhi-Wei Sun, Evaluations of some Toeplitz-type determinants, arXiv:2206.12317 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A000032, A000045, A027960.

A360278:= func< n | Lucas(n+1)^2 - 5*(n mod 2) >;
[A360278(n): n in [1..40]]; // G. C. Greubel, Jun 10 2025

(* First program *)
a[n_]:=a[n]=Det[Table[LucasL[j+k]+Boole[j==k],{j,1,n},{k,1,n}]];
Table[a[n],{n,1,25}]
(* Second program *)
LinearRecurrence[{3, 0, -3, 1}, {4, 16, 44, 121}, 41] (* G. C. Greubel, Jun 10 2025 *)

def A360278(n): return lucas_number2(n+1,1,-1)^2 - 5*(n%2)
print([A360278(n) for n in range(1,41)]) # G. C. Greubel, Jun 10 2025

From G. C. Greubel, Jun 10 2025: (Start)
a(n) = A000032(2*n+2) - 2 - (n mod 2) = A000032(n+1)^2 - 5*(n mod 2).
a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
a(n) = 1 + Sum_{k=0..n-1} A027960(n, k)*A027960(n, k+1).
G.f.: (1 + x + 4*x^2 - x^3)/((1-x^2)*(1-3*x+x^2)) - 1.
E.g.f.: exp(3*x/2)*( 3*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2) ) - 2*cosh(x) - 3*sinh(x) - 1. (End)

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

nonn

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

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

A053298 Partial sums of A027964.

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A360278 Determinant of the matrix [L(j+k) + d(j,k)]_{1<=j, k<=n}, where L(n) denotes the Lucas number A000032(n), and d(j,k) is 1 or 0 according as j = k or not.

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A026999 Uniquification of A026998.

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

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