A027011 -id:A027011 - cp's OEIS frontend

A027960 'Lucas array': triangular array T read by rows.

1, 1, 3, 1, 1, 3, 4, 4, 1, 1, 3, 4, 7, 8, 5, 1, 1, 3, 4, 7, 11, 15, 13, 6, 1, 1, 3, 4, 7, 11, 18, 26, 28, 19, 7, 1, 1, 3, 4, 7, 11, 18, 29, 44, 54, 47, 26, 8, 1, 1, 3, 4, 7, 11, 18, 29, 47, 73, 98, 101, 73, 34, 9, 1, 1, 3, 4, 7, 11, 18, 29, 47, 76, 120, 171, 199, 174, 107, 43, 10, 1

Offset: 0

Clark Kimberling

The k-th row contains 2k+1 numbers.
Columns in the right half consist of convolutions of the Lucas numbers with the natural numbers.
T(n,k) = number of strings s(0),...,s(n) such that s(n)=n-k. s(0) in {0,1,2}, s(1)=1 if s(0) in {1,2}, s(1) in {0,1,2} if s(0)=0 and for 1 <= i <= n, s(i) = s(i-1)+d, with d in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0 <= s(i) <= 2i-2.

			                           1
                       1,  3,  1
                   1,  3,  4,  4,  1
               1,  3,  4,  7,  8,  5,   1
           1,  3,  4,  7, 11, 15, 13,   6,  1
        1, 3,  4,  7, 11, 18, 26, 28,  19,  7,  1
     1, 3, 4,  7, 11, 18, 29, 44, 54,  47, 26,  8, 1
  1, 3, 4, 7, 11, 18, 29, 47, 73, 98, 101, 73, 34, 9, 1

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Central column is the Lucas numbers without initial 2: A000204.
Columns in the right half include A027961, A027962, A027963, A027964, A053298.
Bisection triangles are in A026998 and A027011.
Row sums: A036563, A153881 (alternating sign).
Sums include: A023537, A027973, A027974, A027975, A027976, A027978, A027979, A027980, A027981, A027982, A027984, A027985, A027986, A027987, A053298.
Diagonals of the form T(n, 2*n-p): A000012 (p=0), A000027 (p=1), A034856 (p=2), A027965 (p=3), A027966 (p=4), A027967 (p=5), A027968 (p=6), A027969 (p=7), A027970 (p=8), A027971 (p=9), A027972 (p=10).
Diagonals of the form T(n, n+p): A000032 (p=0), A027961 (p=1), A023537 (p=2), A027963 (p=3), A027964 (p=4), A053298 (p=5), A027002 U A027018 (p=6), A027007 U A027014 (p=7), A027003 U A027019 (p=8).
Cf. A000032, A004396, A026998, A027011, A027977, A075193, A168616.

function T(n,k) // T = A027960
      if k le n then return Lucas(k+1);
      elif k gt 2*n then return 0;
      else return T(n-1, k-2) + T(n-1, k-1);
      end if;
end function;
[T(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Jun 08 2025

T:=proc(n,k)option remember:if(k=0 or k=2*n)then return 1:elif(k=1)then return 3:else return T(n-1,k-2) + T(n-1,k-1):fi:end:
for n from 0 to 6 do for k from 0 to 2*n do print(T(n,k));od:od: # Nathaniel Johnston, Apr 18 2011

(* First program *)
t[, 0] = 1; t[, 1] = 3; t[n_, k_] /; (k == 2*n) = 1; t[n_, k_] := t[n, k] = t[n-1, k-2] + t[n-1, k-1]; Table[t[n, k], {n, 0, 8}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Dec 27 2013 *)
(* Second program *)
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];
Table[A027960[n,k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Jun 08 2025 *)

T(r,n)=if(r<0||n>2*r,return(0)); if(n==0||n==2*r,return(1)); if(n==1,3,T(r-1,n-1)+T(r-1,n-2)) /* Ralf Stephan, May 04 2005 */

@CachedFunction
def T(n, k): # T = A027960
    if (k>2*n): return 0
    elif (kG. C. Greubel, Jun 01 2019; Jun 08 2025

T(n, k) = Lucas(k+1) for k <= n, otherwise the (2n-k)th coefficient of the power series for (1+2*x)/{(1-x-x^2)*(1-x)^(k-n)}.
Recurrence: T(n, 0)=T(n, 2n)=1 for n >= 0; T(n, 1)=3 for n >= 1; and for n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) for 2 <= k <= 2*n-1.
From G. C. Greubel, Jun 08 2025: (Start)
T(n, k) = A000032(k+1) - f(n, k)*[k > n], where f(n, k) = Sum_{j=0..k-n-1} binomial(2*n -k+j, j)*A000032(2*(k-n-j)).
Sum_{k=0..A004396(n+1)} T(n-k, k) = A027975(n).
Sum_{k=0..n} T(n, k) = A027961(n).
Sum_{k=0..2*n} T(n, k) = A168616(n+2) + 2.
Sum_{k=n+1..2*n} (-1)^k*T(n, k) = A075193(n-1), n >= 1. (End)

Edited by Ralf Stephan, May 04 2005

A026998 Triangular array T read by rows: T(n, k) = t(n, 2k), t given by A027960, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 8, 1, 1, 4, 11, 13, 1, 1, 4, 11, 26, 19, 1, 1, 4, 11, 29, 54, 26, 1, 1, 4, 11, 29, 73, 101, 34, 1, 1, 4, 11, 29, 76, 171, 174, 43, 1, 1, 4, 11, 29, 76, 196, 370, 281, 53, 1, 1, 4, 11, 29, 76, 199, 487, 743, 431, 64, 1

Offset: 0

Clark Kimberling

Right-edge columns are polynomials approximating Lucas(2n+1).

			  .................................... 1;
  ................................. 1, 1;
  ............................. 1,  4, 1;
  ........................ 1,   4,  8, 1;
  ................... 1,   4,  11, 13, 1;
  .............. 1,   4,  11,  26, 19, 1;
  .......... 1,  4,  11,  29,  54, 26, 1;
  ...... 1,  4, 11,  29,  73, 101, 34, 1;
  .. 1,  4, 11, 29,  76, 171, 174, 43, 1;
  1, 4, 11, 29, 76, 196, 370, 281, 53, 1;

G. C. Greubel, Table of n, a(n) for n = 0..1325

This is a bisection of the "Lucas array" A027960, see A027011 for the other bisection.
Row sums give A095121.
Signed row sums give A090132.
Diagonal sums give A027010.
Right-edge columns include A034856, A027966, A027968, A027970, A027972.
Cf. A000032.

function t(n, k) // t = A027960
      if k le n then return Lucas(k+1);
      elif k gt 2*n then return 0;
      else return t(n-1, k-2) + t(n-1, k-1);
      end if;
end function;
A026998:= func< n,k | t(n, 2*k) >;
[A026998(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2025

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];
A026998[n_, k_]:= A027960[n,2*k];
Table[A026998[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 09 2025 *)

@CachedFunction
def t(n, k): # t = A027960
    if (k>2*n): return 0
    elif (kA026998(n,k): return t(n, 2*k)
print(flatten([[A026998(n, k) for k in (0..n)] for n in (0..12)])) # G. C. Greubel, Jul 09 2025

T(n, k) = Lucas(2*n+1) = A002878(n) for 2*k <= n, otherwise the (2*n-2*k)-th coefficient of the power series for (1+2*x)/( (1-x-x^2)*(1-x)^(2*k-n) ).

Edited by Ralf Stephan, May 05 2005

A027965 T(n, 2*n-3), T given by A027960.

Original entry on oeis.org

3, 7, 15, 28, 47, 73, 107, 150, 203, 267, 343, 432, 535, 653, 787, 938, 1107, 1295, 1503, 1732, 1983, 2257, 2555, 2878, 3227, 3603, 4007, 4440, 4903, 5397, 5923, 6482, 7075, 7703, 8367, 9068, 9807, 10585, 11403, 12262, 13163, 14107, 15095, 16128, 17207, 18333, 19507, 20730, 22003

Offset: 2

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A column of triangle A027011.

List([2..50], n-> (18-10*n+3*n^2+n^3)/6) # G. C. Greubel, Jun 30 2019

[(18-10*n+3*n^2+n^3)/6: n in [2..50]]; // G. C. Greubel, Jun 30 2019

LinearRecurrence[{4,-6,4,-1}, {3,7,15,28}, 50] (* G. C. Greubel, Jun 30 2019 *)

vector(50, n, n++; (18-10*n+3*n^2+n^3)/6) \\ G. C. Greubel, Jun 30 2019

[(18-10*n+3*n^2+n^3)/6 for n in (2..50)] # G. C. Greubel, Jun 30 2019

a(n+2) = A074742(n-1) = A008778(n) + 2 = A000297(n-1) + 3.
From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^2*(3 - 2*x)*(1 - x + x^2)/(1-x)^4.
Differences of A027966. (End)
From G. C. Greubel, Jun 30 2019: (Start)
a(n) = (18 - 10*n + 3*n^2 + n^3)/6.
E.g.f.: (-18 - 12*x + (18 - 6*x + 6*x^2 + x^3)*exp(x))/6. (End)

Terms a(32) onward added by G. C. Greubel, Jun 30 2019

A027967 T(n, 2*n-5), T given by A027960.

Original entry on oeis.org

3, 7, 18, 44, 98, 199, 373, 654, 1085, 1719, 2620, 3864, 5540, 7751, 10615, 14266, 18855, 24551, 31542, 40036, 50262, 62471, 76937, 93958, 113857, 136983, 163712, 194448, 229624, 269703, 315179, 366578, 424459, 489415, 562074, 643100, 733194, 833095, 943581, 1065470, 1199621

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 (6,-15,20,-15,6,-1).

A column of triangle A027011.

List([3..50], n-> (840-736*n+300*n^2-45*n^3+n^5)/120) G. C. Greubel, Jun 30 2019

[(840-736*n+300*n^2-45*n^3+n^5)/120: n in [3..50]]; // G. C. Greubel, Jun 30 2019

LinearRecurrence[{6,-15,20,-15,6,-1}, {3,7,18,44,98,199}, 50] (* G. C. Greubel, Jun 30 2019 *)

for(n=3,50, print1((840-736*n+300*n^2-45*n^3+n^5)/120, ", ")) \\ G. C. Greubel, Jun 30 2019

[(840-736*n+300*n^2-45*n^3+n^5)/120 for n in (3..50)] # G. C. Greubel, Jun 30 2019

From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^3*(3-2*x)*(1-3*x+5*x^2-3*x^3+x^4)/(1-x)^6.
Differences of A027968. (End)
From G. C. Greubel, Jun 30 2019: (Start)
a(n) = (840 - 736*n + 300*n^2 - 45*n^3 + n^5)/120.
E.g.f.: (-120*(7 + 3*x + x^2) + (840 - 480*x + 180*x^2 - 20*x^3 + 10*x^4 + x^5)*exp(x))/120. (End)

Terms a(37) onward added by G. C. Greubel, Jun 30 2019

A027969 a(n) = T(n, 2*n-7), T given by A027960.

Original entry on oeis.org

3, 7, 18, 47, 120, 291, 661, 1404, 2801, 5283, 9484, 16305, 26990, 43215, 67191, 101782, 150639, 218351, 310614, 434419, 598260, 812363, 1088937, 1442448, 1889917, 2451243, 3149552, 4011573, 5068042, 6354135, 7909931, 9780906, 12018459, 14680471, 17831898, 21545399, 25902000, 30991795

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 (8,-28,56,-70,56,-28,8,-1).

A column of triangle A027011.

List([4..50], n-> (90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040) # G. C. Greubel, Jul 01 2019

[(90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040: n in [4..50]]; // G. C. Greubel, Jul 01 2019

Table[(90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040, {n,4,50}] (* G. C. Greubel, Jul 01 2019 *)

for(n=4,50, print1((90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040, ", ")) \\ G. C. Greubel, Jul 01 2019

[(90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040 for n in (4..50)] # G. C. Greubel, Jul 01 2019

From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^4*(3-2x)*(1-x+x^2)*(1-4x+7x^2-4x^3+x^4)/(1-x)^8.
First differences of A027970. (End)
From G. C. Greubel, Jul 01 2019: (Start)
a(n) = (90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040.
E.g.f.: (-90720 - 35280*x - 7560*x^2 - 1680*x^3 + (90720 - 55440*x + 17640*x^2 - 3360*x^3 + 630*x^4 - 42*x^5 + 14*x^6 + x^7)*exp(x))/5040. (End)

Terms a(35) onward added by G. C. Greubel, Jul 01 2019

A027971 T(n, 2n-9), T given by A027960.

Original entry on oeis.org

3, 7, 18, 47, 123, 319, 806, 1954, 4506, 9859, 20495, 40615, 77040, 140455, 247085, 420906, 696509, 1122751, 1767344, 2722551, 4112177, 6100063, 8900312, 12789498, 18121132, 25342683, 35015477, 47837823, 64671742

Offset: 5

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

A column of triangle A027011.

a:=[3,7,18,47, 123,319,806,1954,4506,9859];; for n in [11..40] do a[n]:=10*a[n-1]-45*a[n-2]+120*a[n-3]-210*a[n-4]+252*a[n-5]-210*a[n-6] +120*a[n-7] -45*a[n-8]+10*a[n-9]-a[n-10]; od; a; # G. C. Greubel, Sep 26 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^5*(3 -2*x)*(1-7*x+23*x^2-44*x^3+55*x^4-44*x^5+23*x^6-7*x^7+x^8)/(1-x)^10 )); // G. C. Greubel, Sep 26 2019

seq(coeff(series(x^5*(3-2*x)*(1 -7*x +23*x^2 -44*x^3 +55*x^4 -44*x^5 +23*x^6 -7*x^7 +x^8)/(1-x)^10, x, n+1), x, n), n = 5..40); # G. C. Greubel, Sep 26 2019

LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1}, {3,7,18,47, 123,319,806,1954,4506,9859},40] (* Harvey P. Dale, Aug 04 2017 *)

Vec(-x^5*(2*x-3)*(x^8-7*x^7+23*x^6-44*x^5+55*x^4-44*x^3+23*x^2 -7*x+1)/(x-1)^10 + O(x^40)) \\ Colin Barker, Nov 25 2014

def A027971_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^5*(3-2*x)*(1-7*x+23*x^2-44*x^3+55*x^4-44*x^5+23*x^6-7*x^7 +x^8)/(1-x)^10 ).list()
a=A027971_list(40); a[5:] # G. C. Greubel, Sep 26 2019

a(n) = (17055360 -16329024*n +7697736*n^2 -2299060*n^3 +462798*n^4 -60207*n^5 +4284*n^6 -30*n^7 -18*n^8 +n^9)/362880. - Colin Barker, Nov 25 2014

G.f.: x^5*(3-2*x)*(1 -7*x +23*x^2 -44*x^3 +55*x^4 -44*x^5 +23*x^6 -7*x^7 +x^8)/(1-x)^10. - Colin Barker, Nov 25 2014

cp's OEIS Frontend

A027021 a(n) = T(n,n) + T(n,m+1) + ... + T(n,n), where m=[ (n+2)/2 ], T given by A027011.

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A027960 'Lucas array': triangular array T read by rows.

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A026998 Triangular array T read by rows: T(n, k) = t(n, 2k), t given by A027960, 0 <= k <= n, n >= 0.

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A027965 T(n, 2*n-3), T given by A027960.

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A027967 T(n, 2*n-5), T given by A027960.

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GAP

Magma

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A027969 a(n) = T(n, 2*n-7), T given by A027960.

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Magma

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A027971 T(n, 2n-9), T given by A027960.

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