A027960 -id:A027960 - cp's OEIS frontend

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

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

A027011 Triangular array T read by rows: T(n,k) = t(n, 2k+1) for 0 <= k <= floor((2n-1)/2), t given by A027960, n >= 0.

Original entry on oeis.org

3, 3, 4, 3, 7, 5, 3, 7, 15, 6, 3, 7, 18, 28, 7, 3, 7, 18, 44, 47, 8, 3, 7, 18, 47, 98, 73, 9, 3, 7, 18, 47, 120, 199, 107, 10, 3, 7, 18, 47, 123, 291, 373, 150, 11, 3, 7, 18, 47, 123, 319, 661, 654, 203, 12, 3, 7, 18, 47, 123, 322, 806, 1404, 1085, 267, 13, 3, 7, 18, 47

Offset: 1

Clark Kimberling

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

			                                          3
                                     3,   4
                                3,   7,   5
                           3,   7,  15,   6
                      3,   7,  18,  28,   7
                 3,   7,  18,  44,  47,   8
            3,   7,  18,  47,  98,  73,   9
       3,   7,  18,  47, 120, 199, 107,  10
  3,   7,  18,  47, 123, 291, 373, 150,  11

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

This is a bisection of the "Lucas array " A027960, see A026998 for the other bisection.
Right-edge columns include A027965, A027967, A027969, A027971.
An earlier version of this entry had (unjustifiably) each row starting with 1.

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:
T:=proc(n,k)return t(n,2*k+1):end:
for n from 0 to 8 do for k from 0 to floor((2*n-1)/2) do print(T(n,k));od:od: # Nathaniel Johnston, Apr 18 2011

t[n_, k_] := t[n, k] = If[k == 0 || k == 2*n, 1, If[k == 1, 3, t[n-1, k-2] + t[n-1, k-1]]]; T[n_, k_] := t[n, 2*k+1]; Table[T[n, k], {n, 1, 12}, {k, 0, (2*n-1)/2}] // Flatten (* Jean-François Alcover, Nov 18 2013, after Nathaniel Johnston *)

T(n, k) = Lucas(2n) = A005248(n) for 2k+1 <= n, otherwise the (2n-2k+1)-th coefficient of the power series for (1+2x)/((1-x-x^2)(1-x)^(2k-n+1)).

Edited by Ralf Stephan, May 05 2005

A027973 a(n) = T(n,n) + T(n,n+1) + ... + T(n,2n), T given by A027960.

Original entry on oeis.org

1, 4, 9, 21, 46, 99, 209, 436, 901, 1849, 3774, 7671, 15541, 31404, 63329, 127501, 256366, 514939, 1033449, 2072676, 4154701, 8324529, 16673534, 33386671, 66837421, 133778524, 267724809, 535721061, 1071881326, 2144473299, 4290096449, 8582053396, 17167117141

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000032, A000045, A027960.

List([0..40], n-> 2^(n+2) - Lucas(1,-1,n+2)[2]); # G. C. Greubel, Sep 26 2019

[2^n-Lucas(n): n in [2..40]]; // Vincenzo Librandi, May 05 2017

with(combinat): a[0]:=1: for n from 1 to 30 do a[n]:=2*a[n-1]+fibonacci(n+1)-fibonacci(n-3) od: seq(a[n],n=0..30); # Emeric Deutsch, Nov 29 2006

Table[2^n - LucasL[n], {n, 2, 50}] (* Vincenzo Librandi, May 05 2017 *)

vector(40, n, f=fibonacci; 2^(n+1) - f(n+2) - f(n) ) \\ G. C. Greubel, Sep 26 2019

[2^(n+2) - lucas_number2(n+2,1,-1) for n in (0..40)] # G. C. Greubel, Sep 26 2019

With a different offset: recurrence: a(-1)=a(0)=1 a(n+2) = a(n+1) + a(n) + 2^n; formula: a(n-2) = floor(2^n - phi^n) - (1-(-1)^n)/2. - Benoit Cloitre, Sep 02 2002
a(n) = A101220(4, 2, n+1) - A101220(4, 2, n). - Ross La Haye, Aug 05 2005
a(n) = 2*a(n-1) + Fibonacci(n+1) - Fibonacci(n-3) for n>=1; a(0)=1. - Emeric Deutsch, Nov 29 2006
O.g.f.: 4/(1-2*x) - (x+3)/(1-x-x^2). - R. J. Mathar, Nov 23 2007
a(n) = 2^(n+2) + F(n) - F(n+4) with F(n)=A000045(n). - Johannes W. Meijer, Aug 15 2010
Eigensequence of an infinite lower triangular matrix with the Lucas series (1, 3, 4, 7, ...) as the left border and the rest ones. - Gary W. Adamson, Jan 30 2012
a(n) = 2^(n+2) - Lucas(n+2). - Vincenzo Librandi, May 05 2017, corrected by Greg Dresden, Sep 13 2021

A027974 a(n) = Sum_{k=1..n+1} A027960(n+1, n+1+k).

Original entry on oeis.org

1, 5, 14, 35, 81, 180, 389, 825, 1726, 3575, 7349, 15020, 30561, 61965, 125294, 252795, 509161, 1024100, 2057549, 4130225, 8284926, 16609455, 33282989, 66669660, 133507081, 267285605, 535010414, 1070731475, 2142612801, 4287086100, 8577182549, 17159235945

Offset: 0

Clark Kimberling

The former name, a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A027960, was in error with the data given. [This double summation gives A023537(n+1), or A027960(n+2, n+4) for n >= 0]. - G. C. Greubel, Jun 08 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Philipp Emanuel Weidmann, The Sequencer OEIS Survey
Susanne Wienand, Suggestion for a proof of Philipp Emanuel Weidman's conjecture concerning A027983
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000032, A000045, A027960, A101220.

List([0..30], n-> 2^(n+3) - Lucas(1,-1,n+4)[2]); # G. C. Greubel, Sep 26 2019

[2^(n+3) - Lucas(n+4): n in [0..30]]; // G. C. Greubel, Sep 26 2019

with(combinat); f:=fibonacci; seq(2^(n+3) - f(n+5) - f(n+3), n=0..30); # G. C. Greubel, Sep 26 2019

Table[2^(n+3) - LucasL[n+4], {n,0,30}] (* G. C. Greubel, Sep 26 2019 *)

vector(31, n, f=fibonacci; 2^(n+2) - f(n+4) - f(n+2)) \\ G. C. Greubel, Sep 26 2019

def A027974(n): return 2**(n+3) - lucas_number2(n+4,1,-1)
[A027974(n) for n in range(31)] # G. C. Greubel, Sep 26 2019; Jun 08 2025

a(n) = 2^(n+3) - Fibonacci(n+5) - Fibonacci(n+3).
a(n) = A101220(4, 2, n+1).
G.f.: (1+2*x)/((1-2*x)*(1-x-x^2)). - R. J. Mathar, Sep 22 2008
a(n) = 2*a(n-1) + A000032(n+1). - David A. Corneth, Apr 16 2015
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3). - Colin Barker, Feb 17 2016
From G. C. Greubel, Jun 08 2025: (Start)
a(n) = 2^(n+3) - A000032(n+4).
E.g.f.: 8*exp(2*x) - exp(x/2)*( 7*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2) ). (End)

A027964 T(n,n+4), T given by A027960.

Original entry on oeis.org

1, 7, 26, 73, 174, 373, 743, 1404, 2552, 4506, 7784, 13226, 22193, 36889, 60882, 99947, 163430, 266455, 433495, 704150, 1142496, 1852212, 3001056, 4860468, 7869649, 12739243, 20619098, 33369709, 54001422, 87385081

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 (5,-9,6,1,-3,1).

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^4*(1+2*x)/((1-x)^4*(1-x-x^2)) )); // G. C. Greubel, Jun 29 2019

Drop[CoefficientList[Series[x^4*(1+2*x)/((1-x)^4*(1-x-x^2)), {x,0,40}], x], 4] (* G. C. Greubel, Jun 29 2019 *)

my(x='x+O('x^40)); Vec(x^4*(1+2*x)/((1-x)^4*(1-x-x^2))) \\ G. C. Greubel, Jun 29 2019

a=(x^4*(1+2*x)/((1-x)^4*(1-x-x^2))).series(x, 40).coefficients(x, sparse=False); a[4:] # G. C. Greubel, Jun 29 2019

G.f.: x^4*(1+2*x)/((1-x)^4*(1-x-x^2)). - Ralf Stephan, Feb 07 2004

A027963 T(n,n+3), T given by A027960.

Original entry on oeis.org

1, 6, 19, 47, 101, 199, 370, 661, 1148, 1954, 3278, 5442, 8967, 14696, 23993, 39065, 63483, 103025, 167040, 270655, 438346, 709716, 1148844, 1859412, 3009181, 4869594, 7879855, 12750611, 20631713, 33383659, 54016798, 87401977, 141420392, 228824086, 370246298, 599072310, 969320643

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000032.

List([3..40], n-> Lucas(1,-1,n+4)[2] - (3*n^2+5*n+14)/2 ) # G. C. Greubel, Jun 01 2019

[Lucas(n+4) -(3*n^2+5*n+14)/2: n in [3..40]]; // G. C. Greubel, Jun 01 2019

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, n+3], {n, 3, 33}]  (* Jean-François Alcover, Dec 27 2013 *)
Table[LucasL[n+4] -(3*n^2+5*n+14)/2, {n,3,40}] (* G. C. Greubel, Jun 01 2019 *)

{a(n) = fibonacci(n+5) + fibonacci(n+3) - (3*n^2+5*n+14)/2}; \\ G. C. Greubel, Jun 01 2019

[lucas_number2(n+4,1,-1) - (3*n^2+5*n+14)/2 for n in (3..40)] # G. C. Greubel, Jun 01 2019

G.f.: x^3*(1+2*x)/((1-x)^3*(1-x-x^2)). Differences of A027964. - Ralf Stephan, Feb 07 2004

a(n) = Lucas(n+4) - (3*n^2 + 5*n + 14)/2.

Terms a(34) onward added by G. C. Greubel, Jun 01 2019

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

A027966 T(n, 2*n-4), T given by A027960.

Original entry on oeis.org

1, 4, 11, 26, 54, 101, 174, 281, 431, 634, 901, 1244, 1676, 2211, 2864, 3651, 4589, 5696, 6991, 8494, 10226, 12209, 14466, 17021, 19899, 23126, 26729, 30736, 35176, 40079, 45476, 51399, 57881, 64956, 72659, 81026, 90094, 99901, 110486, 121889, 134151, 147314, 161421, 176516, 192644

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 (5,-10,10,-5,1).

A column of triangle A026998.

List([2..50], n-> (n^4+2*n^3-25*n^2+94*n-96)/24); # G. C. Greubel, Jun 30 2019

[(n^4+2*n^3-25*n^2+94*n-96)/24: n in [2..50]]; // G. C. Greubel, Jun 30 2019

LinearRecurrence[{5,-10,10,-5,1}, {1,4,11,26,54}, 50] (* G. C. Greubel, Jun 30 2019 *)

vector(50, n, n++; (n^4+2*n^3-25*n^2+94*n-96)/24) \\ G. C. Greubel, Jun 30 2019

[(n^4+2*n^3-25*n^2+94*n-96)/24 for n in (2..50)] # G. C. Greubel, Jun 30 2019

From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^2*(1 - x + x^2 + x^3 - x^4)/(1-x)^5.
Differences of A027967. (End)
From G. C. Greubel, Jun 30 2019: (Start)
a(n) = (n^4 + 2*n^3 - 25*n^2 + 94*n - 96)/24.
E.g.f.:  (96 +24*x - (96 - 72*x + 12*x^2 - 8*x^3 - x^4)*exp(x))/24. (End)

Terms a(31) 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

A027968 a(n) = T(n, 2*n-6), T given by A027960.

Original entry on oeis.org

1, 4, 11, 29, 73, 171, 370, 743, 1397, 2482, 4201, 6821, 10685, 16225, 23976, 34591, 48857, 67712, 92263, 123805, 163841, 214103, 276574, 353511, 447469, 561326, 698309, 862021, 1056469, 1286093, 1555796, 1870975, 2237553

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

A column of triangle A026998.

List([3..40], n-> (-7920 +7548*n -3176*n^2 +735*n^3 -65*n^4 -3*n^5 +n^6)/720) # G. C. Greubel, Jul 01 2019

[(-7920 +7548*n -3176*n^2 +735*n^3 -65*n^4 -3*n^5 +n^6)/720: n in [3..40]]; // G. C. Greubel, Jul 01 2019

Table[(-7920 +7548*n -3176*n^2 +735*n^3 -65*n^4 -3*n^5 +n^6)/720, {n,3,40}] (* G. C. Greubel, Jul 01 2019 *)

for(n=3,40, print1((-7920 +7548*n -3176*n^2 +735*n^3 -65*n^4 -3*n^5 +n^6)/720, ", ")) \\ G. C. Greubel, Jul 01 2019

[(-7920 +7548*n -3176*n^2 +735*n^3 -65*n^4 -3*n^5 +n^6)/720 for n in (3..40)] # G. C. Greubel, Jul 01 2019

From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^3*(1 -3*x +4*x^2 +x^3 -4*x^4 +3*x^5 -x^6)/(1-x)^7.
a(n) = A027969(n+1) - A027969(n). (End)
From G. C. Greubel, Jul 01 2019: (Start)
a(n) = (-7920 +7548*n -3176*n^2 +735*n^3 -65*n^4 -3*n^5 +n^6)/720.
E.g.f.: (7920 +2880*x +360*x^2 -(7920 -5040*x +1440*x^2 -360*x^3 +30*x^4 -12*x^5 -x^6)*exp(x))/6!. (End)

cp's OEIS Frontend

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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A027011 Triangular array T read by rows: T(n,k) = t(n, 2k+1) for 0 <= k <= floor((2n-1)/2), t given by A027960, n >= 0.

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A027973 a(n) = T(n,n) + T(n,n+1) + ... + T(n,2n), T given by A027960.

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A027974 a(n) = Sum_{k=1..n+1} A027960(n+1, n+1+k).

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A027964 T(n,n+4), T given by A027960.

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A027963 T(n,n+3), T given by A027960.

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