A001911 -id:A001911 - cp's OEIS frontend

A108617 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n, T(n,0) = T(n,n) = n-th Fibonacci number.

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 5, 6, 5, 3, 5, 8, 11, 11, 8, 5, 8, 13, 19, 22, 19, 13, 8, 13, 21, 32, 41, 41, 32, 21, 13, 21, 34, 53, 73, 82, 73, 53, 34, 21, 34, 55, 87, 126, 155, 155, 126, 87, 55, 34, 55, 89, 142, 213, 281, 310, 281, 213, 142, 89, 55, 89, 144, 231, 355, 494, 591, 591, 494, 355, 231, 144, 89

Offset: 0

Reinhard Zumkeller, Jun 12 2005

Sum of n-th row = 2*A027934(n). - Reinhard Zumkeller, Oct 07 2012

			Triangle begins:
   0;
   1,   1;
   1,   2,   1;
   2,   3,   3,   2;
   3,   5,   6,   5,   3;
   5,   8,  11,  11,   8,   5;
   8,  13,  19,  22,  19,  13,   8;
  13,  21,  32,  41,  41,  32,  21,  13;
  21,  34,  53,  73,  82,  73,  53,  34,  21;
  34,  55,  87, 126, 155, 155, 126,  87,  55,  34;
  55,  89, 142, 213, 281, 310, 281, 213, 142,  89,  55;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Hacéne Belbachir and László Szalay, On the Arithmetic Triangles, Šiauliai Mathematical Seminar, Vol. 9 (17), 2014. See Fig. 1 p. 18.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Pascal's Triangle.
Wikipedia, Fibonacci number.
Wikipedia, Pascal's triangle.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A074829, A108037.

T(2n,n) gives 2*A176085(n).

a108617 n k = a108617_tabl !! n !! k
a108617_row n = a108617_tabl !! n
a108617_tabl = [0] : iterate f [1,1] where
   f row@(u:v:_) = zipWith (+) ([v - u] ++ row) (row ++ [v - u])
-- Reinhard Zumkeller, Oct 07 2012

function T(n,k) // T = A108617
  if k eq 0 or k eq n then return Fibonacci(n);
  else return T(n-1,k-1) + T(n-1,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 20 2023

A108617 := proc(n,k) option remember;
    if k = 0 or k=n then
        combinat[fibonacci](n) ;
    elif k <0 or k > n then
        0 ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Oct 05 2012

a[1]:={0}; a[n_]:= a[n]= Join[{Fibonacci[#]}, Map[Total, Partition[a[#],2,1]], {Fibonacci[#]}]&[n-1]; Flatten[Map[a, Range[15]]] (* Peter J. C. Moses, Apr 11 2013 *)

def T(n,k): # T = A108617
    if (k==0 or k==n): return fibonacci(n)
    else: return T(n-1,k-1) + T(n-1,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 20 2023

T(n,0) = T(n,n) = A000045(n);
T(n,1) = T(n,n-1) = A000045(n+1) for n>0;
T(n,2) = T(n,n-2) = A000045(n+2) - 2 = A001911(n-1) for n>1;
Sum_{k=0..n} T(n,k) = 2*A027934(n-1) for n>0.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*((n+1 mod 2)*Fibonacci(n-2) + [n=0]). - G. C. Greubel, Oct 20 2023

A385886 Irregular triangle read by rows listing the lengths of maximal anti-runs (sequences of distinct consecutive elements increasing by more than 1) of binary indices, duplicate rows removed.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 14 2025

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

This is the triangle A384877, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A384877.

			The binary indices of 27 are {1,2,4,5}, with maximal anti-runs ((1),(2,4),(5)), with lengths (1,2,1). After removing duplicates, this is our row 10.
The binary indices of 53 are {1,3,5,6}, with maximal anti-runs ((1,3,5),(6)), with lengths (3,1). After removing duplicates, this is our row 16.
Triangle begins:
   0: .
   1: 1
   2: 1 1
   3: 2
   4: 1 1 1
   5: 1 2
   6: 2 1
   7: 1 1 1 1
   8: 3
   9: 1 1 2
  10: 1 2 1
  11: 2 1 1
  12: 1 1 1 1 1
  13: 1 3
  14: 2 2
  15: 1 1 1 2
  16: 3 1
  17: 1 1 2 1
  18: 1 2 1 1
  19: 2 1 1 1
  20: 1 1 1 1 1 1

In the following references, "before" is short for "before removing duplicate rows".
Positions of singleton rows appear to be A001906 = A055588 - 1.
Positions of rows of the form (1,1,...) appear to be A001911-2, before A023758.
Row sums appear to be A200648, before A000120.
Row lengths appear to be A200649, before A384890.
Standard composition numbers of each row appear to be A348366.
Before we had A384877, ranks A385816, firsts A052499.
For runs instead of anti-runs we have A385817, see A245563, A245562, A246029.
Cf. A044813, A048793, A069010, A164707, A242882, A328592, A384175, A384177, A384878, A384879, A384893.

DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2!=#1+1&],{n,0,100}]]

A050143 A(n,k) = Sum_{h=0..n-1, m=0..k} A(h,m) for n >= 1 and k >= 1, with A(n,0) = 1 for n >= 0 and A(0,k) = 0 for k >= 1; square array A, read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 4, 7, 1, 0, 1, 5, 12, 15, 1, 0, 1, 6, 18, 32, 31, 1, 0, 1, 7, 25, 56, 80, 63, 1, 0, 1, 8, 33, 88, 160, 192, 127, 1, 0, 1, 9, 42, 129, 280, 432, 448, 255, 1, 0, 1, 10, 52, 180, 450, 832, 1120, 1024, 511, 1

Offset: 1

Clark Kimberling

The triangular version of this square array is defined by T(n,k) = A(k,n-k) for 0 <= k <= n. Conversely, A(n,k) = T(n+k,n) for n,k >= 0. We have [o.g.f of T](x,y) = [o.g.f. of A](x*y, x) and [o.g.f. of A](x,y) = [o.g.f. of T](y,x/y). - Petros Hadjicostas, Feb 11 2021
Formatted as a triangular array with offset (0,8), it is [0, 1, 0, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 1, 0, 0, 0, 0, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2006
The sum of the first two columns [of the rectangular array] gives the powers of 2; that is, Sum_{j=0..1} A(i,j) = 2^i, i >= 0. On the other hand, for i >= 1 and j >= 2, A(i,j) is the number of lattice paths of i-1 upsteps (1,1) and j-1 downsteps (1,-1) in which each downstep-free vertex is colored red or blue. A downstep-free vertex is one not incident with a downstep. For example, dots indicate the downstep-free vertices in the path .U.U.UDU.UDDU., and with i = j = 2, A(2,2) = 4 counts UD, *UD, DU, DU*, where asterisks indicate the red vertices. - David Callan, Aug 27 2011

			Square array A(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1,   0,   0,    0,    0,    0,    0,     0,     0,     0, ...
  1,   1,   1,    1,    1,    1,    1,     1,     1,     1, ...
  1,   3,   4,    5,    6,    7,    8,     9,    10,    11, ...
  1,   7,  12,   18,   25,   33,   42,    52,    63,    75, ...
  1,  15,  32,   56,   88,  129,  180,   242,   316,   403, ...
  1,  31,  80,  160,  280,  450,  681,   985,  1375,  1865, ...
  1,  63, 192,  432,  832, 1452, 2364,  3653,  5418,  7773, ...
  1, 127, 448, 1120, 2352, 4424, 7700, 12642, 19825, 29953, ...
  ...
If we read the above square array by descending antidiagonals, we get the following triangular array T(n,k) (with rows n >= 0 and columns 0 <= k <= n):
   1;
   0, 1;
   0, 1, 1;
   0, 1, 3,  1;
   0, 1, 4,  7,   1;
   0, 1, 5, 12,  15,   1;
   0, 1, 6, 18,  32,  31,   1;
   0, 1, 7, 25,  56,  80,  63,   1;
   0, 1, 8, 33,  88, 160, 192, 127,   1;
   0, 1, 9, 42, 129, 280, 432, 448, 255, 1;
   ...

Yumin Cho, Jaehyun Kim, Jang Soo Kim, and Nakyung Lee, Enumeration of multiplex juggling card sequences using generalized q-derivatives, arXiv:2402.09903 [math.CO], 2024. See p. 9.
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40(4) (2002), 328-338; see Example 3A and Eq. (8) on p. 333.

Antidiagonal sums are odd-indexed Fibonacci numbers (A001519).
Signed alternating antidiagonal sums are Fibonacci(n)-2, as in A001911.
Cf. A000225, A001792, A050147, A050148, A055807 (mirror array of triangle), A084938.

T[n_, k_] := If[n == k, 1, JacobiP[k - 1, 1, n - 2*k - 1, 3]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 25 2021 *)

Formulas for the square array (A(n,k): n,k >= 0):
A(n,1) = -1 + 2^n = A000225(n) for n >= 1.
A(n+2,2) = 4*A001792(n) for n >= 0.
From Petros Hadjicostas, Feb 11 2021: (Start)
Recurrence: A(n,k) = 2*A(n-1,k) + A(n,k-1) - A(n-1,k-1) for n >= 1 and k >= 2; with A(n,0) = 1 for n >= 0, A(0,k) = 0 for k >= 1, and A(n,1) = -1 + 2^n for n >= 1.
Bivariate o.g.f.: Sum_{n,k>=0} A(n,k)*x^n*y^k = (1 - 2*x)*(1 - y)/((1 - x)*(1 - 2*x - y + x*y)).
A(n,k) = Sum_{s=1..n} binomial(n,s)*binomial(s+k-2,k-1) for n >= 0 and k >= 1. (It can be proved by using a partial fraction decomposition on the bivariate o.g.f. above.)
A(n,k) = n*hypergeom([-n + 1, k], [2], -1) for n >= 0 and k >= 1. (End)
Formulas for the triangular array (T(n,k): 0 <= k <= n):
Sum_{k=0..n} T(n,k) = Fibonacci(2*n-1) = A001519(n) with Fibonacci(-1) = 1.
Sum_{k=0..n} (-1)^(n+k-1)*T(n,k) = Fibonacci(n+1) - 2 = A001911(n-2) with A001911(-2) = A001911(-1) = -1.
T(n,k) = A055807(n,n-k) for 0 <= k <= n.
From Petros Hadjicostas, Feb 12 2021: (Start)
Recurrence: T(n,k) = 2*T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) for n >= 3 and 1 <= k <= n-2; with T(n,n) = 1 for n >= 0, T(n,0) = 0 for n >= 1, and T(n+1, n) = 2^n - 1 for n >= 1.
Bivariate o.g.f: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 - x)*(1 - 2*x*y)/((1 - x*y)*(1 - x - 2*x*y + x^2*y)).
T(n,k) = Sum_{s=1..k} binomial(k,s)*binomial(s+n-k-2, s-1) = k*hypergeom([-k+1, n-k], [2], -1) for n >= 1 and 0 <= k <= n - 1. (End)
T(n, k) = JacobiP(k - 1, 1, n - 2*k - 1, 3) n >= 0 and 0 <= k < n. - Peter Luschny, Nov 25 2021

Various sections edited by Petros Hadjicostas, Feb 12 2021

A200763 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with nondecreasing average value.

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 5, 10, 11, 5, 6, 15, 23, 19, 6, 7, 21, 42, 51, 32, 7, 8, 28, 69, 113, 110, 53, 8, 9, 36, 106, 219, 297, 233, 87, 9, 10, 45, 154, 388, 679, 767, 488, 142, 10, 11, 55, 215, 638, 1387, 2070, 1957, 1013, 231, 11, 12, 66, 290, 995, 2583, 4874, 6235, 4947, 2088

Offset: 1

R. H. Hardin Nov 22 2011

Table starts
..2...3....4.....5......6......7.......8.......9.......10.......11........12
..3...6...10....15.....21.....28......36......45.......55.......66........78
..4..11...23....42.....69....106.....154.....215......290......381.......489
..5..19...51...113....219....388.....638.....995.....1483.....2133......2975
..6..32..110...297....679...1387....2583....4500.....7410....11669.....17687
..7..53..233...767...2070...4874...10283...20012....36412....62780....103412
..8..87..488..1957...6235..16919...40437...87914...176767...333702....597390
..9.142.1013..4947..18608..58198..157577..382720...850389..1757813...3420112
.10.231.2088.12419..55148.198807..609826.1654657..4062796..9195619..19445435
.11.375.4278.31006.162532.675372.2347039.7114665.19304047.47842607.109955586

			Some solutions for n=8 k=8
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..2....1....1....4....3....3....0....3....2....3....2....5....3....1....0....2
..2....7....6....2....1....2....3....2....1....4....7....3....4....5....3....6
..2....3....4....7....7....7....5....6....1....6....3....3....5....2....8....4
..4....7....6....6....4....4....4....5....1....6....4....3....3....7....3....4
..7....4....8....4....5....6....2....4....6....7....6....8....8....7....7....6
..5....5....5....6....6....5....4....7....6....4....4....4....7....4....6....8

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 2 is A001911(n+1)
Column 7 is A200707
Row 2 is A000217(n+1)
Row 3 is A019298(n+1)

A259222 T(n,k) is the number of (n+1) X (k+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

7, 13, 13, 24, 23, 24, 45, 40, 40, 45, 85, 71, 66, 71, 85, 162, 127, 112, 112, 127, 162, 311, 230, 192, 183, 192, 230, 311, 601, 421, 334, 303, 303, 334, 421, 601, 1168, 779, 588, 510, 487, 510, 588, 779, 1168, 2281, 1456, 1048, 869, 798, 798, 869, 1048, 1456, 2281

Offset: 1

R. H. Hardin, Jun 21 2015

Table starts
 13   24   45   85  162   311   601  1168  2281  4473   8802  17371
 23   40   71  127  230   421   779  1456  2747  5227  10022  19345
 40   66  112  192  334   588  1048  1890  3448  6360  11854  22308
 71  112  183  303  510   869  1499  2616  4619  8251  14910  27249
127  192  303  487  798  1325  2227  3784  6499 11283  19806  35161
230  334  510  798 1278  2078  3422  5694  9566 16222  27774  48030
421  588  869 1325 2078  3319  5377  8804 14545 24225  40670  68843
779 1048 1499 2227 3422  5377  8591 13888 22655 37231  61598 102589
1456 1890 2616 3784 5694  8804 13888 22210 35872 58368  95550 157276
2747 3448 4619 6499 9566 14545 22655 35872 57455 92767 150686 245965
Each row (and each column, by symmetry) has a rational generating function (and therefore a linear recurrence with constant coefficients) because the growth from an array to the next larger one is described by the transfer matrix method. - R. J. Mathar, Oct 09 2020

			Some solutions for n=4, k=4:
  0 0 1 0 1      1 1 1 0 1      0 0 0 0 1      0 0 0 1 0
  0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0
  1 1 0 1 0      0 0 0 1 0      1 1 1 1 0      0 0 0 1 0
  0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      1 1 1 0 1
  1 1 0 1 0      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0

R. H. Hardin, Table of n, a(n) for n = 1..480

Cf. A259215, A259216, A259217, A259218, A259219, A259220, A259221.

Empirical for diagonal and column k (k=3..7 recurrences work also for k=1,2):
diagonal: a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 16*a(n-4) - 6*a(n-5) - 5*a(n-6) + 2*a(n-7).
k=1: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)
k=2: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)
k=3: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=4: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=5: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=6: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=7: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
Empirical: T(n,k) = 2^(k+1) + 2^(n+1) + F(n+3)*F(k+3) - 2*F(n+3) - 2*F(k+3) + 2 = 2^(n+1) + A001911(k)*F(n+3) + A234933(k+1) = A234933(n+1) + A234933(k+1) + A143211(n+3,k+3) - 2, F=A000045. - Ehren Metcalfe, Dec 27 2018

A200534 T(n,k)=Number of nXk 0..2 arrays with every row and column running average nondecreasing rightwards and downwards.

Original entry on oeis.org

3, 6, 6, 11, 20, 11, 19, 57, 57, 19, 32, 146, 243, 146, 32, 53, 354, 905, 905, 354, 53, 87, 825, 3135, 4848, 3135, 825, 87, 142, 1873, 10292, 23925, 23925, 10292, 1873, 142, 231, 4169, 32556, 111085, 167171, 111085, 32556, 4169, 231, 375, 9144, 100065, 494555

Offset: 1

R. H. Hardin Nov 18 2011

Table starts
...3.....6.....11.......19.........32..........53............87............142
...6....20.....57......146........354.........825..........1873...........4169
..11....57....243......905.......3135.......10292.........32556.........100065
..19...146....905.....4848......23925......111085........494555........2132979
..32...354...3135....23925.....167171.....1092645.......6820330.......41135050
..53...825..10292...111085....1092645....10000162......87098434......731106792
..87..1873..32556...494555....6820330....87098434....1056792824....12348422785
.142..4169.100065..2132979...41135050...731106792...12348422785...201092476918
.231..9144.300847..8981773..241717185..5964324991..140068079857..3178929109296
.375.19825.888587.37125420.1391776867.47562454949.1551453118671.49064050909052

			Some solutions for n=6 k=4
..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..1....0..0..0..1
..0..0..1..2....0..0..0..0....0..0..0..1....0..0..1..2....0..0..1..1
..0..1..1..2....0..2..2..2....0..0..0..1....0..2..1..2....0..1..2..1
..0..2..2..2....1..1..1..2....1..1..1..2....0..1..2..2....0..1..1..1
..0..2..1..2....1..1..2..2....2..2..2..2....2..2..2..2....0..2..2..2
..0..2..2..2....2..2..2..2....1..1..2..2....1..2..2..2....1..1..2..2

R. H. Hardin, Table of n, a(n) for n = 1..198

Column 1 is A001911(n+1)

A201155 T(n,k)=Number of nXk 0..2 arrays with every diagonal, row and column running average nondecreasing rightwards and downwards and diagonally.

Original entry on oeis.org

3, 6, 6, 11, 20, 11, 19, 56, 56, 19, 32, 140, 235, 140, 32, 53, 330, 860, 860, 330, 53, 87, 745, 2910, 4572, 2910, 745, 87, 142, 1634, 9287, 22183, 22183, 9287, 1634, 142, 231, 3504, 28452, 100362, 153796, 100362, 28452, 3504, 231, 375, 7388, 84473, 432048

Offset: 1

R. H. Hardin Nov 27 2011

Table starts
...3.....6.....11.......19.........32..........53............87............142
...6....20.....56......140........330.........745..........1634...........3504
..11....56....235......860.......2910........9287.........28452..........84473
..19...140....860.....4572......22183......100362........432048........1791740
..32...330...2910....22183.....153796......988216.......6002677.......34907951
..53...745...9287...100362.....988216.....9005581......77316192......632331257
..87..1634..28452...432048....6002677....77316192.....938925282....10844322561
.142..3504..84473..1791740...34907951...632331257...10844322561...177169920997
.231..7388.244933..7225068..196209473..4971371594..120031503310..2771232663847
.375.15366.697065.28510387.1073245835.37852928161.1282802588833.41769721151444

			Some solutions for n=3 k=7
..0..0..0..0..0..0..1....0..0..0..0..0..0..0....0..0..0..0..1..1..1
..0..1..1..1..2..1..1....0..0..1..2..2..2..2....0..0..1..1..1..2..2
..1..1..1..1..2..2..2....0..0..1..1..2..1..1....1..1..2..2..2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..161

Column 1 is A001911(n+1)

A370174 Triangle read by rows: Riordan array (1/(1 - x), x*(1 + x)/(1 - x - x^2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 11, 15, 7, 1, 1, 19, 37, 28, 9, 1, 1, 32, 82, 87, 45, 11, 1, 1, 53, 170, 234, 169, 66, 13, 1, 1, 87, 337, 573, 535, 291, 91, 15, 1, 1, 142, 647, 1314, 1511, 1061, 461, 120, 17, 1, 1, 231, 1213, 2871, 3933, 3398, 1904, 687, 153, 19, 1

Offset: 0

Philippe Deléham, Feb 29 2024

			Triangle T(n,k) begins:
      k=0   1   2   3   4   5    6
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  3,  1;
  n=3:  1,  6,  5,  1;
  n=4:  1, 11, 15,  7,  1;
  n=5:  1, 19, 37, 28,  9,  1;
  n=6:  1, 32, 82, 87, 45, 11,  1;
  ...
87 = 28 + 37 + 7 + 15.

Cf. A000012 (column k=0), A000384, A001911, A005408.

Cf. A057960 (row sums), A196472, A218988.

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(n,0) = 1, T(n,k) = 0 if k > n.

Sum_{k = 0..n} T(n,k)* x^k = A000012(n), A057960(n), A196472(n+1), A218988(n-1) for x = 0, 1, 2, 3 respectively.

A180671 a(n) = Fibonacci(n+6) - Fibonacci(6).

Original entry on oeis.org

0, 5, 13, 26, 47, 81, 136, 225, 369, 602, 979, 1589, 2576, 4173, 6757, 10938, 17703, 28649, 46360, 75017, 121385, 196410, 317803, 514221, 832032, 1346261, 2178301, 3524570, 5702879, 9227457, 14930344, 24157809, 39088161, 63245978, 102334147, 165580133

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn15 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+6)-8); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+6)-Fibonacci(6): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+6)-fibonacci(6) od: seq(a(n),n=0..nmax);

f[n_]:= Fibonacci[n+6] - Fibonacci[6]; Array[f, 40, 0] (* or *)
LinearRecurrence[{2,0,-1}, {0,5,13}, 41] (* or *)
CoefficientList[Series[x(3x+5)/(x^3-2x+1), {x,0,40}], x] (* Robert G. Wilson v, Apr 11 2017 *)

for(n=1,40,print(fibonacci(n+6)-fibonacci(6))); \\ Anton Mosunov, Mar 02 2017

concat(0, Vec(x*(5+3*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Apr 20 2017

[fibonacci(n+6)-8 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+6) - F(6) with F = A000045.
a(n) = a(n-1) + a(n-2) + 8 for n>1, a(0)=0, a(1)=5, and where 8 = F(6).
From Colin Barker, Apr 13 2012: (Start)
G.f.: x*(5 + 3*x)/((1 - x)*(1 - x - x^2)).
a(n) = 2*a(n-1) - a(n-3). (End)
a(n) = (-8 + (2^(-n)*((1-sqrt(5))^n*(-9+4*sqrt(5)) + (1+sqrt(5))^n*(9+4*sqrt(5)))) / sqrt(5)). - Colin Barker, Apr 20 2017

A180672 a(n) = Fibonacci(n+7) - Fibonacci(7).

Original entry on oeis.org

0, 8, 21, 42, 76, 131, 220, 364, 597, 974, 1584, 2571, 4168, 6752, 10933, 17698, 28644, 46355, 75012, 121380, 196405, 317798, 514216, 832027, 1346256, 2178296, 3524565, 5702874, 9227452, 14930339, 24157804, 39088156, 63245973

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn16 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+7)-13 ); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+7) - Fibonacci(7): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+7)-fibonacci(7) od: seq(a(n),n=0..nmax);

Fibonacci[7 +Range[0, 40]] -13 (* G. C. Greubel, Jul 13 2019 *)

concat(0, Vec(x*(8+5*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n)=fibonacci(n+7)-fibonacci(7) \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+7)-13 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+7) - F(7) with F = A000045.
a(n) = a(n-1) + a(n-2) + 13 for n>1, a(0)=0, a(1)=8, and where 13 = F(7).
G.f.: x*(8 + 5*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-13 + (2^(-1-n)*((1-sqrt(5))^n*(-29+13*sqrt(5)) + (1+sqrt(5))^n*(29+13*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
a(n) = 8*A000071(n+2) + 5*A000071(n+1). - Bruno Berselli, Feb 24 2017

cp's OEIS Frontend

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A370174 Triangle read by rows: Riordan array (1/(1 - x), x*(1 + x)/(1 - x - x^2)).

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A180671 a(n) = Fibonacci(n+6) - Fibonacci(6).

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