A001911 -id:A001911 - cp's OEIS frontend

A011794 Triangle defined by T(n+1, k) = T(n, k-1) + T(n-1, k), T(n,1) = 1, T(1,k) = 1, T(2,k) = min(2,k).

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 8, 1, 4, 7, 11, 12, 13, 1, 4, 10, 14, 19, 20, 21, 1, 5, 11, 21, 26, 32, 33, 34, 1, 5, 15, 25, 40, 46, 53, 54, 55, 1, 6, 16, 36, 51, 72, 79, 87, 88, 89, 1, 6, 21, 41, 76, 97, 125, 133, 142, 143, 144, 1, 7, 22, 57, 92, 148, 176, 212, 221, 231, 232, 233

Offset: 1

N. J. A. Sloane and David Broadhurst

			matrix(10,10,n,k,a(n-1,k-1))
  [ 0 0 0 0 0 0 0 0 0 0 ]
  [ 0 1 1 1 1 1 1 1 1 1 ]
  [ 0 1 2 2 2 2 2 2 2 2 ]
  [ 0 1 2 3 3 3 3 3 3 3 ]
  [ 0 1 3 4 5 5 5 5 5 5 ]
  [ 0 1 3 6 7 8 8 8 8 8 ]
Triangle begins as:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  6,  7,  8;
  1, 4,  7, 11, 12, 13;
  1, 4, 10, 14, 19, 20, 21;
  1, 5, 11, 21, 26, 32, 33, 34;
  1, 5, 15, 25, 40, 46, 53, 54, 55;
  1, 6, 16, 36, 51, 72, 79, 87, 88, 89;

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.

Columns include A008619 and (essentially) A055802, A055803, A055804, A055805, A055806.
Right-hand columns 1-14 are A000045, A000071, A001911, A001924, A001891, A014162, A053808, A014166, A053809, A053739, A054469, A053295, A054470, A053296.
Essentially a reflected version of A055801.
Sums include: A039834 (signed row), A131913 (row).
Cf. A008466, A074878, A103609.

function T(n,k) // T = A011794(n,k)
  if k eq 1 or n eq 1 then return 1;
  elif n eq 2 then return Min(2, k);
  else return T(n-1,k-1) + T(n-2,k);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 21 2024

T[n_, k_]:= T[n, k]= T[n-1, k-1] + T[n-2, k]; T[n_, 1] = 1; T[1, k_] = 1; T[2, k_] := Min[2, k]; Table[T[n, k], {n,15}, {k,n}]//Flatten (* Jean-François Alcover, Feb 26 2013 *)

T(n,k)=if(n<=0 || k<=0,0, if(n<=2 || k==1, min(n,k), T(n-1,k-1)+T(n-2,k)))

def T(n, k): # T = A011794
    if (k==1 or n==1): return 1
    elif (n==2): return min(2,k)
    else: return T(n-1, k-1) + T(n-2, k)
flatten([[T(n, k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Oct 21 2024

T(n,n) = Fibonacci(n+1). - Jean-François Alcover, Feb 26 2013
From G. C. Greubel, Oct 21 2024: (Start)
Sum_{k=1..n} T(n, k) = A131913(n-1).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A039834(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1,k) = (1/2)*((1-(-1)^n)*A074878((n+3)/2) + (1+(-1)^n)*A008466((n+6)/2)) (diagonal row sums).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1,k) = (-1)^floor((n-1)/2)*A103609(n) + [n=1] (signed diagonal row sums). (End)

Entry improved by comments from Michael Somos

More terms added by G. C. Greubel, Oct 21 2024

A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 8, 1, 1, 2, 3, 5, 8, 13, 1, 1, 2, 3, 5, 8, 13, 21, 1, 1, 2, 3, 5, 8, 13, 21, 34, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

Offset: 1

Gary W. Adamson, Mar 23 2005

Triangle of A104762, Fibonacci sequence in each row starts from the right.
The triangle or chess sums, see A180662 for their definitions, link the Fibonacci(n) triangle to sixteen different sequences, see the crossrefs. The knight sums Kn14 - Kn18 have been added. As could be expected all sums are related to the Fibonacci numbers. - Johannes W. Meijer, Sep 22 2010
Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A104763 is reluctant sequence of Fibonacci numbers (A000045), except 0. - Boris Putievskiy, Dec 13 2012

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3;
  1, 1, 2, 3, 5;
  1, 1, 2, 3, 5, 8;
  1, 1, 2, 3, 5, 8, 13; ...

Reinhard Zumkeller, Rows n = 1..100 of table, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A104762, A000045.
Cf. A000071 (row sums). - R. J. Mathar, Jul 22 2009
Triangle sums (see the comments): A000071 (Row1; Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A008346 (Row2); A131524 (Kn11); A001911 (Kn12); A006327 (Kn13); A167616 (Kn14); A180671 (Kn15); A180672 (Kn16); A180673 (Kn17); A180674 (Kn18); A052952 (Kn21 & Kn22 & Kn23 & Fi2 & Ze2); A001906 (Kn3 &Fi1 & Ze3); A004695 (Ca2 & Ze4); A001076 (Ca3 & Ze1); A080239 (Gi2); A081016 (Gi3). - Johannes W. Meijer, Sep 22 2010

Flat(List([1..15], n -> List([1..n], k -> Fibonacci(k)))); # G. C. Greubel, Jul 13 2019

a104763 n k = a104763_tabl !! (n-1) !! (k-1)
a104763_row n = a104763_tabl !! (n-1)
a104763_tabl = map (flip take $ tail a000045_list) [1..]
-- Reinhard Zumkeller, Aug 15 2013

[Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019

Table[Fibonacci[k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 13 2019 *)

for(n=1,15, for(k=1,n, print1(fibonacci(k), ", "))) \\ G. C. Greubel, Jul 13 2019

[[fibonacci(k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019

F(1) through F(n) starting from the left in n-th row.
T(n,k) = A000045(k), 1<=k<=n. - R. J. Mathar, May 02 2008
a(n) = A000045(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
G.f.: (x*y)/((x-1)*(x^2*y^2+x*y-1)). - Vladimir Kruchinin, Jun 21 2025

Edited by R. J. Mathar, May 02 2008

Extended by R. J. Mathar, Aug 27 2008

A052499 If n is in the sequence then so are 2n and 4n-1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 15, 16, 22, 23, 24, 27, 28, 30, 31, 32, 43, 44, 46, 47, 48, 54, 55, 56, 59, 60, 62, 63, 64, 86, 87, 88, 91, 92, 94, 95, 96, 107, 108, 110, 111, 112, 118, 119, 120, 123, 124, 126, 127, 128, 171, 172, 174, 175, 176, 182, 183, 184, 187

Offset: 0

Henry Bottomley, Mar 15 2000

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = 1 + A003754.

			a(9)=14 is in the sequence because 14=2*(4*(2*1)-1).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
T. Karki, A. Lacroix, M. Rigo, On the recognizability of self-generating sets, JIS 13 (2010) #10.2.2.
C. Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.

Cf. A001911, A003754.

import Data.Set (singleton, deleteFindMin, insert)
a052499 n = a052499_list !! n
a052499_list = f $ singleton 1 where
   f s = m : f (insert (2*m) $ insert (4*m-1) s') where
      (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jul 06 2011

1 + Select[ Range[0, 200], FreeQ[ IntegerDigits[#, 2], {_, 0, 0, _} ] & ] (* Jean-François Alcover, Jan 20 2012, after J.-P. Allouche *)
a[1] = 1; a[n_] := a[n] = a[n - 1] + Ceiling[2^IntegerExponent[a[n - 1], 2]/3]; Array[a, 200] (* Birkas Gyorgy, May 30 2012 *)

from itertools import count, islice
def A052499_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not '00' in bin(n-1),count(max(startvalue,1)))
A052499_list = list(islice(A052499_gen(),20)) # Chai Wah Wu, Feb 12 2025

a(A001911(n)) = 2^n.

A120562 Sum of binomial coefficients binomial(i+j, i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 3, 2, 4, 3, 5, 1, 4, 3, 4, 2, 5, 3, 5, 2, 5, 4, 6, 3, 7, 5, 8, 1, 6, 4, 5, 3, 7, 4, 7, 2, 6, 5, 7, 3, 8, 5, 8, 2, 7, 5, 7, 4, 9, 6, 10, 3, 9, 7, 10, 5, 12, 8, 13, 1

Offset: 0

Sam Northshield (samuel.northshield(AT)plattsburgh.edu), Aug 07 2006

a(n) is the number of 'vectors' (..., e_k, e_{k-1}, ..., e_0) with e_k in {0,1,3} such that Sum_{k} e_k 2^k = n. a(2^n-1) = F(n+1)*a(2^{k+1}+j) + a(j) = a(2^k+j) + a(2^{k-1}+j) if 2^k > 4j. This sequence corresponds to the pair (3,1) as Stern's diatomic sequence [A002487] corresponds to (2,1) and Gould's sequence [A001316] corresponds to (1,1). There are many interesting similarities to A000119, the number of representations of n as a sum of distinct Fibonacci numbers.
A120562 can be generated from triangle A177444. Partial sums of A120562 = A177445. - Gary W. Adamson, May 08 2010
The Ca1 and Ca2 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence. Some A120562(2^n-p) sequences, 0 <= p <= 32, lead to known sequences, see the crossrefs. - Johannes W. Meijer, Jun 05 2011

			a(2^n)=1 since a(2n)=a(n).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 6.
Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.
S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.

Cf. A001316 (1,1), A002487 (2,1), A120562 (3,1), A112970 (4,1), A191373 (5,1).
Cf. A177444, A177445. - Gary W. Adamson, May 08 2010
Cf. A000012 (p=0), A000045 (p=1, p=2, p=4, p=8, p=16, p=32), A000071 (p=3, p=6, p=12, p=13, p=24, p=26), A001610 (p=5, p=10, p=20), A001595 (p=7, p=14, p=28), A014739 (p=11, p=22, p=29), A111314 (p=15, p=30), A027961 (p=19), A154691 (p=21), A001911 (p=23). - Johannes W. Meijer, Jun 05 2011
Same recurrence for odd n as A000930.

p := product((1+x^(2^i)+x^(3*2^i)), i=0..25): s := series(p, x, 1000): for k from 0 to 250 do printf(`%d, `, coeff(s, x, k)) od:
A120562:=proc(n) option remember; if n <0 then A120562(n):=0 fi: if (n=0 or n=1) then 1 elif n mod 2 = 0 then A120562(n/2) else A120562((n-1)/2) + A120562((n-3)/2); fi; end: seq(A120562(n),n=0..64); # Johannes W. Meijer, Jun 05 2011

a[0] = a[1] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_?OddQ] := a[n] = a[(n-1)/2] + a[(n-1)/2 - 1]; Table[a[n], {n, 0, 64}] (* Jean-François Alcover, Sep 29 2011 *)
Nest[Append[#1, If[EvenQ@ #2, #1[[#2/2 + 1]], Total@ #1[[#2 ;; #2 + 1]] & @@ {#1, (#2 - 1)/2}]] & @@ {#, Length@ #} &, {1, 1}, 10^4 - 1] (* Michael De Vlieger, Feb 19 2019 *)

Recurrence; a(0)=a(1)=1, a(2*n)=a(n) and a(2*n+1)=a(n)+a(n-1).
G.f.: A(x) = Product_{i>=0} (1 + x^(2^i) + x^(3*2^i)) = (1 + x + x^3)*A(x^2).
a(n-1) << n^x with x = log_2(phi) = 0.69424... - Charles R Greathouse IV, Dec 27 2011

Reference edited and link added by Jason G. Wurtzel, Aug 22 2010

A157727 a(n) = Fibonacci(n) + 4.

Original entry on oeis.org

4, 5, 5, 6, 7, 9, 12, 17, 25, 38, 59, 93, 148, 237, 381, 614, 991, 1601, 2588, 4185, 6769, 10950, 17715, 28661, 46372, 75029, 121397, 196422, 317815, 514233, 832044, 1346273, 2178313, 3524582, 5702891, 9227469, 14930356, 24157821, 39088173, 63245990, 102334159

Offset: 0

N. J. A. Sloane, Jun 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..285
Ivana Jovović and Branko Malešević, Some enumerations of non-trivial composition of the differential operations and the directional derivative, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 1, 28-38.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616.

a157727 = (+ 4) . a000045
a157727_list = 4 : 5 : map (subtract 4)
                       (zipWith (+) a157727_list $ tail a157727_list)
-- Reinhard Zumkeller, Jul 30 2013

[ Fibonacci(n) + 4: n in [0..40] ]; // Vincenzo Librandi, Apr 24 2011

Fibonacci[Range[0,50]]+4 (* Harvey P. Dale, Jun 17 2011 *)

a(n)=fibonacci(n)+4 \\ Charles R Greathouse IV, Jul 02 2013

a(0) = 4, a(1) = 5, a(n) = a(n - 2) + a(n - 1) - 4. - Reinhard Zumkeller, Jul 30 2013

G.f.: (4 - 3*x - 5*x^2)/((1 - x)*(1 - x - x^2)). - Stefano Spezia, Jul 21 2024

A261019 Irregular triangle read by rows: T(n,k) (0 <= k <= A261017(n)) = number of binary strings of length n such that the smallest number whose binary representation is not visible in the string is k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 6, 4, 1, 1, 1, 4, 11, 10, 5, 1, 1, 5, 19, 21, 15, 0, 2, 1, 1, 6, 32, 40, 35, 2, 9, 2, 1, 1, 7, 53, 72, 73, 6, 31, 10, 2, 1, 1, 8, 87, 125, 144, 15, 79, 40, 12, 1, 1, 9, 142, 212, 274, 32, 185, 116, 52, 1, 1, 10, 231, 354, 509, 64, 408, 296, 168, 2, 4

Offset: 1

Alois P. Heinz and N. J. A. Sloane, Aug 17 2015

This is a more compact version of the triangle in A261015, ending each row at the last nonzero entry.

			The first 16 rows are:
1, 1,
1, 1,  1,    1,
1, 1,  2,    3,    1,
1, 1,  3,    6,    4,    1,
1, 1,  4,   11,   10,    5,
1, 1,  5,   19,   21,   15,    0,     2,
1, 1,  6,   32,   40,   35,    2,     9,     2,
1, 1,  7,   53,   72,   73,    6,    31,    10,     2,
1, 1,  8,   87,  125,  144,   15,    79,    40,    12,
1, 1,  9,  142,  212,  274,   32,   185,   116,    52,
1, 1, 10,  231,  354,  509,   64,   408,   296,   168,    2,    4,
1, 1, 11,  375,  585,  931,  120,   864,   699,   461,   24,   24,
1, 1, 12,  608,  960, 1685,  218,  1771,  1557,  1133,  130,  110,   2,   4,
1, 1, 13,  985, 1568, 3027,  385,  3555,  3325,  2612,  471,  387,  14,  24,  0,  16,
1, 1, 14, 1595, 2553, 5409,  668,  7021,  6893,  5759, 1401, 1135,  92, 120,  0,  90, 16,
1, 1, 15, 2582, 4148, 9628, 1142, 13696, 13964, 12309, 3734, 2972, 373, 439, 28, 390, 98, 16,
...

Alois P. Heinz, N. J. A. Sloane, R. Zumkeller and Hiroaki Yamanouchi, Rows n = 1..58, flattened (rows 17..25 from R. Zumkeller, rows 26..36 from Alois P. Heinz)
R. Zumkeller, The first 25 rows of the triangle, displayed as a triangle (similar to the way the rows are shown in the Example section, but showing 25 rows).

Cf. A261015, A261016.
The row lengths are given by A261017.
Columns k=3-10 give: A001911, A001891, A261441, A261442, A261443, A261473, A261474, A261475.
Cf. A076478, A030308, A000079 (row sums), A261392 (max per row).

import Data.List (isInfixOf, sort, group)
a261019 n k = a261019_tabf !! (n-1) !! k
a261019_row n = a261019_tabf !! (n-1)
a261019_tabf = map (i 0 . group . sort . map f) a076478_tabf
   where f bs = g a030308_tabf where
           g (cs:css) | isInfixOf cs bs = g css
                      | otherwise = foldr (\d v -> 2 * v + d) 0 cs
         i _ [] = []
         i r gss'@(gs:gss) | head gs == r = (length gs) : i (r + 1) gss
                           | otherwise    = 0 : i (r + 1) gss'
-- Reinhard Zumkeller, Aug 18 2015

A110813 A triangle of pyramidal numbers.

Original entry on oeis.org

1, 3, 1, 5, 4, 1, 7, 9, 5, 1, 9, 16, 14, 6, 1, 11, 25, 30, 20, 7, 1, 13, 36, 55, 50, 27, 8, 1, 15, 49, 91, 105, 77, 35, 9, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1, 23, 121, 385, 825

Offset: 0

Paul Barry, Aug 05 2005

Triangle A029653 less first column. In general, the product (1/(1-x),x/(1-x))*(1+m*x,x) yields the Riordan array ((1+(m-1)x)/(1-x)^2,x/(1-x)) with general term T(n,k)=(m*n-(m-1)*k+1)*C(n+1,k+1)/(n+1). This is the reversal of the (1,m)-Pascal triangle, less its first column. - Paul Barry, Mar 01 2006
The column sequences give, for k=0..10: A005408 (odd numbers), A000290 (squares), A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.
Linked to Chebyshev polynomials by the fact that this triangle with interpolated zeros in the rows and columns is a scaled version of A053120.
Row sums are A033484. Diagonal sums are A001911(n+1) or F(n+4)-2. Factors as (1/(1-x),x/(1-x))*(1+2x,x). Inverse is A110814 or (-1)^(n-k)*A104709.
This triangle is a subtriangle of the [2,1] Pascal triangle A029653 (omit there the first column).
Subtriangle of triangles in A029653, A131084, A208510. - Philippe Deléham, Mar 02 2012
This is the iterated partial sums triangle of A005408 (odd numbers). Such iterated partial sums of arithmetic progression sequences have been considered by Narayana Pandit (see the Mar 20 2015 comment on A000580 where the MacTutor History of Mathematics archive link and the Gottwald et al. reference, p. 338, are given). - Wolfdieter Lang, Mar 23 2015

			The number triangle T(n, k) begins
n\k  0   1   2   3    4    5    6   7   8  9 10 11
0:   1
1:   3   1
2:   5   4   1
3:   7   9   5   1
4:   9  16  14   6    1
5:  11  25  30  20    7    1
6:  13  36  55  50   27    8    1
7:  15  49  91 105   77   35    9   1
8:  17  64 140 196  182  112   44  10   1
9:  19  81 204 336  378  294  156  54  11  1
10: 21 100 285 540  714  672  450 210  65 12  1
11: 23 121 385 825 1254 1386 1122 660 275 77 13  1
... reformatted by _Wolfdieter Lang_, Mar 23 2015
As a number square S(n, k) = T(n+k, k), rows begin
  1,   1,   1,   1,   1,   1, ...
  3,   4,   5,   6,   7,   8, ...
  5,   9,  14,  20,  27,  35, ...
  7,  16,  30,  50,  77, 112, ...
  9,  25,  55, 105, 182, 294, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000290, A000330, A002415, A005408, A005585, A029655, A040977, A050486, A053347, A054333, A054334, A057788.

Table[2*Binomial[n + 1, k + 1] - Binomial[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 19 2017 *)

for(n=0,10, for(k=0,n, print1(2*binomial(n+1, k+1) - binomial(n,k), ", "))) \\ G. C. Greubel, Oct 19 2017

Number triangle T(n, k) = C(n, k)*(2n-k+1)/(k+1) = 2*C(n+1, k+1) - C(n, k); Riordan array ((1+x)/(1-x)^2, x/(1-x)); As a number square read by antidiagonals, T(n, k)=C(n+k, k)(2n+k+1)/(k+1).
Equals A007318 * an infinite bidiagonal matrix with 1's in the main diagonal and 2's in the subdiagonal. - Gary W. Adamson, Dec 01 2007
Binomial transform of an infinite lower triangular matrix with all 1's in the main diagonal, all 2's in the subdiagonal and the rest zeros. - Gary W. Adamson, Dec 12 2007
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0)=T(1,1)=1, T(1,0)=3, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 30 2013
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(7 + 9*x + 5*x^2/2! + x^3/3!) = 7 + 16*x + 30*x^2/2! + 50*x^3/3! + 77*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
T(n, k) = ps(1, 2; k, n-k) with ps(a, d; k, n) = sum(ps(a, d; k-1, j), j=0..n) and input ps(a, d; 0, j) = a + d*j. See the iterated partial sums comment from Mar 23 2015 above. - Wolfdieter Lang, Mar 23 2015
From Franck Maminirina Ramaharo, May 21 2018: (Start)
T(n,k) = coefficients in the expansion of ((x + 2)*(x + 1)^n - 2)/x.
T(n,k) = A135278(n,k) + A135278(n-1,k).
T(n,k) = A097207(n,n-k).
G.f.: (y + 1)/((y - 1)*(x*y + y - 1)).
E.g.f.: ((x + 2)*exp(x*y + y) - 2*exp(y))/x.
(End)

A157726 a(n) = Fibonacci(n) + 3.

Original entry on oeis.org

3, 4, 4, 5, 6, 8, 11, 16, 24, 37, 58, 92, 147, 236, 380, 613, 990, 1600, 2587, 4184, 6768, 10949, 17714, 28660, 46371, 75028, 121396, 196421, 317814, 514232, 832043, 1346272, 2178312, 3524581, 5702890, 9227468, 14930355, 24157820, 39088172, 63245989, 102334158

Offset: 0

N. J. A. Sloane, Jun 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..285
Ivana Jovović and Branko Malešević, Some enumerations of non-trivial composition of the differential operations and the directional derivative, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 1, 28-38.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616.

a157726 = (+ 3) . a000045
a157726_list = 3 : 4 : map (subtract 3)
                       (zipWith (+) a157726_list $ tail a157726_list)
-- Reinhard Zumkeller, Jul 30 2013

[ Fibonacci(n) + 3: n in [0..40] ]; // Vincenzo Librandi, Apr 24 2011

Fibonacci[Range[0,45]]+3 (* Harvey P. Dale, Oct 26 2011 *)

a(n)=fibonacci(n)+3 \\ Charles R Greathouse IV, Jul 02 2013

G.f.: ( 3-2*x-4*x^2 ) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Aug 08 2012

a(0) = 3, a(1) = 4, a(n) = a(n - 2) + a(n - 1) - 3. - Reinhard Zumkeller, Jul 30 2013

A157728 a(n) = Fibonacci(n) - 4.

Original entry on oeis.org

1, 4, 9, 17, 30, 51, 85, 140, 229, 373, 606, 983, 1593, 2580, 4177, 6761, 10942, 17707, 28653, 46364, 75021, 121389, 196414, 317807, 514225, 832036, 1346265, 2178305, 3524574, 5702883, 9227461, 14930348, 24157813, 39088165, 63245982, 102334151

Offset: 5

N. J. A. Sloane, Jun 26 2010

Partial sums of A071679. - R. J. Mathar, Oct 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 5..287
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 5, 10.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616.

a157728 = subtract 4 . a000045  -- Reinhard Zumkeller, Oct 31 2013

[ Fibonacci(n) - 4: n in [5..50] ]; // Vincenzo Librandi, Apr 24 2011

Fibonacci[Range[5,40]]-4 (* or *) LinearRecurrence[{2,0,-1},{1,4,9},40] (* Harvey P. Dale, Jan 17 2022 *)

a(n)=fibonacci(n) - 4 \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Oct 12 2010: (Start)
a(n) = 2*a(n-1) - a(n-3).
G.f.: x^5*(1+x)^2/((x-1)*(x^2+x-1)). (End)

A157729 a(n) = Fibonacci(n) + 5.

Original entry on oeis.org

5, 6, 6, 7, 8, 10, 13, 18, 26, 39, 60, 94, 149, 238, 382, 615, 992, 1602, 2589, 4186, 6770, 10951, 17716, 28662, 46373, 75030, 121398, 196423, 317816, 514234, 832045, 1346274, 2178314, 3524583, 5702892, 9227470, 14930357, 24157822, 39088174, 63245991, 102334160

Offset: 0

N. J. A. Sloane, Jun 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..285
Ivana Jovović and Branko Malešević, Some enumerations of non-trivial composition of the differential operations and the directional derivative, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 1, 28-38.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616.

a157729 = (+ 5) . a000045
a157729_list = 5 : 6 : map (subtract 5)
                       (zipWith (+) a157729_list $ tail a157729_list)
-- Reinhard Zumkeller, Jul 30 2013

[ Fibonacci(n) + 5: n in [0..40] ]; // Vincenzo Librandi, Apr 24 2011

Fibonacci[Range[0,40]]+5 (* or *) LinearRecurrence[{2,0,-1},{5,6,6},50] (* Harvey P. Dale, Aug 17 2012 *)

a(n)=fibonacci(n)+5 \\ Charles R Greathouse IV, Jul 02 2013

G.f.: ( 5-4*x-6*x^2 ) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Aug 09 2012
a(0)=5, a(1)=6, a(2)=6, a(n)=2*a(n-1)+0*a(n-2)-a(n-3). - Harvey P. Dale, Aug 17 2012
a(0) = 5, a(1) = 6, a(n) = a(n - 2) + a(n - 1) - 5. - Reinhard Zumkeller, Jul 30 2013

cp's OEIS Frontend

A011794 Triangle defined by T(n+1, k) = T(n, k-1) + T(n-1, k), T(n,1) = 1, T(1,k) = 1, T(2,k) = min(2,k).

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A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.

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A052499 If n is in the sequence then so are 2n and 4n-1.

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A120562 Sum of binomial coefficients binomial(i+j, i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n.

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A157727 a(n) = Fibonacci(n) + 4.

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A261019 Irregular triangle read by rows: T(n,k) (0 <= k <= A261017(n)) = number of binary strings of length n such that the smallest number whose binary representation is not visible in the string is k.

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