A000071 -id:A000071 - cp's OEIS frontend

A001924 Apply partial sum operator twice to Fibonacci numbers.

0, 1, 3, 7, 14, 26, 46, 79, 133, 221, 364, 596, 972, 1581, 2567, 4163, 6746, 10926, 17690, 28635, 46345, 75001, 121368, 196392, 317784, 514201, 832011, 1346239, 2178278, 3524546, 5702854, 9227431, 14930317, 24157781, 39088132, 63245948, 102334116, 165580101

Offset: 0

N. J. A. Sloane

Leading coefficients in certain rook polynomials (for n>=2; see p. 18 of the Riordan paper). - Emeric Deutsch, Mar 08 2004
(1, 3, 7, 14, ...) = row sums of triangle A141289. - Gary W. Adamson, Jun 22 2008
a(n) is the number of nonempty subsets of {1,2,...,n} such that the difference of successive elements is at most 2. See example below. Generally, the o.g.f. for the number of nonempty subsets of {1,2,...,n} such that the difference of successive elements is <= k is: x/((1-x)*(1-2*x+x^(k+1))). Cf. A000217 the case for k=1, A001477 the case for k=0 (counts singleton subsets). - Geoffrey Critzer, Feb 17 2012
-Fibonacci(n-2) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Dec 31 2012
a(n) is the number of bit strings of length n+1 with the pattern 00 and without the pattern 011, see example. - John M. Campbell, Feb 10 2013
From Jianing Song, Apr 28 2025: (Start)
For n >= 2, a(n-2) is the number of subsets of {1,2,...,n} with 2 or more elements that contain no consecutive elements (i.e., such that the difference of successive elements is at least 2). Note that the number of such subsets with k elements is binomial(n+1-k,k), and Sum_{k=2..floor((n+1)/2)} binomial(n+1-k,k) = F(n+2) - binomial(n+1,0) - binomial(n,1) = F(n+2) - (n+1).
If subsets of {1,2,...,n} are required to contain no consecutive elements module n, then the result is A023548(n-3). (End)

			a(5) = 26 because there are 31 nonempty subsets of {1,2,3,4,5} but 5 of these have successive elements that differ by 3 or more: {1,4}, {1,5}, {2,5}, {1,2,5}, {1,4,5}. - _Geoffrey Critzer_, Feb 17 2012
From _John M. Campbell_, Feb 10 2013: (Start)
There are a(5) = 26 bit strings with the pattern 00 and without the pattern 011 of length 5+1:
   000000, 000001, 000010, 000100, 000101, 001000,
   001001, 001010, 010000, 010001, 010010, 010100,
   100000, 100001, 100010, 100100, 100101, 101000, 101001,
   110000, 110001, 110010, 110100, 111000, 111001, 111100.
(End)

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Bader AlBdaiwi, On the Number of Cycles in a Graph, arXiv preprint arXiv:1603.01807 [cs.DM], 2016.
Jean-Luc Baril and Jean-Marcel Pallo, Motzkin subposet and Motzkin geodesics in Tamari lattices, 2013.
Ning-Ning Cao and Feng-Zhen Zhao, Some Properties of Hyperfibonacci and Hyperlucas Numbers, J. Int. Seq. 13 (2010) # 10.8.8.
Hung Viet Chu, Various Sequences from Counting Subsets, arXiv:2005.10081 [math.CO], 2020.
Hung Viet Chu, Partial Sums of the Fibonacci Sequence, arXiv:2106.03659 [math.CO], 2021.
Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, arXiv:1606.06228 [math.CO], 2016.
Emrah Kiliç and Pantelimon Stănică, Generating matrices for weighted sums of second order linear recurrences, JIS 12 (2009) 09.2.7.
Wolfdieter Lang, Problem B-858, Fibonacci Quarterly, 36 (1998), 373-374, Solution, ibid. 37 (1999) 183-184.
Candice A. Marshall, Construction of Pseudo-Involutions in the Riordan Group, Dissertation, Morgan State University, 2017.
Igor Pak, Boris Shapiro, Ilya Smirnov, and Ken-ichi Yoshida, Hilbert-Kunz multiplicity of quadrics via the Ehrhart theory, Stockholm Univ. (Sweden, 2025). See p. 6.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Stacey Wagner, Enumerating Alternating Permutations with One Alternating Descent, DePaul Discoveries: Vol. 2: Iss. 1, Article 2.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A023548, A001891, A133640, A141289.
Right-hand column 4 of triangle A011794.
Cf. A014162, A039834, A077880, A122595, A129696.
Cf. A065220.

List([0..40], n-> Fibonacci(n+4) -n-3); # G. C. Greubel, Jul 08 2019

a001924 n = a001924_list !! n
a001924_list = drop 3 $ zipWith (-) (tail a000045_list) [0..]
-- Reinhard Zumkeller, Nov 17 2013

[Fibonacci(n+4)-(n+3): n in [0..40]]; // Vincenzo Librandi, Jun 23 2016

A001924:=-1/(z**2+z-1)/(z-1)**2; # Conjectured by Simon Plouffe in his 1992 dissertation.
##
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|-1|-2|3>>^n.
         <<0, 1, 3, 7>>)[1, 1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 05 2012

a[n_]:= Fibonacci[n+4] -3-n; Array[a, 40, 0]  (* Robert G. Wilson v *)
LinearRecurrence[{3,-2,-1,1},{0,1,3,7},40] (* Harvey P. Dale, Jan 24 2015 *)
Nest[Accumulate,Fibonacci[Range[0,40]],2] (* Harvey P. Dale, Jun 15 2016 *)

a(n)=fibonacci(n+4)-n-3 \\ Charles R Greathouse IV, Feb 24 2011

[fibonacci(n+4) -n-3 for n in (0..40)] # G. C. Greubel, Jul 08 2019

From Wolfdieter Lang: (Start)
G.f.: x/((1-x-x^2)*(1-x)^2).
Convolution of natural numbers n >= 1 with Fibonacci numbers F(k).
a(n) = Fibonacci(n+4) - (3+n). (End)
From Henry Bottomley, Jan 03 2003: (Start)
a(n) = a(n-1) + a(n-2) + n = a(n-1) + A000071(n+2).
a(n) = A001891(n) - a(n-1) = n + A001891(n-1).
a(n) = A065220(n+4) + 1 = A000126(n+1) - 1. (End)
a(n) = Sum_{k=0..n} Sum_{i=0..k} Fibonacci(i). - Benoit Cloitre, Jan 26 2003
a(n) = (sqrt(5)/2 + 1/2)^n*(7*sqrt(5)/10 + 3/2) + (3/2 - 7*sqrt(5)/10)*(sqrt(5)/2 - 1/2)^n*(-1)^n - n - 3. - Paul Barry, Mar 26 2003
a(n) = Sum_{k=0..n} Fibonacci(k)*(n-k). - Benoit Cloitre, Jun 07 2004
A107909(a(n)) = A000225(n) = 2^n - 1. - Reinhard Zumkeller, May 28 2005
a(n) - a(n-1) = A101220(1,1,n). - Ross La Haye, May 31 2006
F(n) + a(n-3) = A133640(n). - Gary W. Adamson, Sep 19 2007
a(n) = A077880(-3-n) = 2*a(n-1) - a(n-3) + 1. - Michael Somos, Dec 31 2012
INVERT transform is A122595. PSUM transform is A014162. PSUMSIGN transform is A129696. BINOMIAL transform of A039834 with 0,1 prepended is this sequence. - Michael Somos, Dec 31 2012
a(n) = A228074(n+1,3) for n > 1. - Reinhard Zumkeller, Aug 15 2013
a(n) = Sum_{k=0..n} Sum_{i=0..n} i * C(n-k,k-i). - Wesley Ivan Hurt, Sep 21 2017
E.g.f.: exp(x/2)*(15*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x)*(3 + x). - Stefano Spezia, Jun 25 2022

Description improved by N. J. A. Sloane, Jan 01 1997

A003754 Numbers with no adjacent 0's in binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 170, 171, 173, 174, 175, 181

Offset: 1

N. J. A. Sloane

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.
Ahnentafel numbers of ancestors contributing the X-chromosome to a female. A280873 gives the male inheritance. - Floris Strijbos, Jan 09 2017 [Equivalence with this sequence pointed out by John Blythe Dobson, May 09 2018]
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order has no parts greater than two. See the corresponding example below. - Gus Wiseman, Apr 04 2020
The binary representation of a(n+1) has the same string of digits as the lazy Fibonacci (also known as dual Zeckendorf) representation of n that uses 0s and 1s. (The "+1" is essentially an adjustment for the offset of this sequence.) - Peter Munn, Sep 06 2022

			21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - _Indranil Ghosh_, Feb 11 2017
From _Gus Wiseman_, Apr 04 2020: (Start)
The terms together with the corresponding compositions begin:
    0: ()            30: (1,1,1,2)         90: (2,1,2,2)
    1: (1)           31: (1,1,1,1,1)       91: (2,1,2,1,1)
    2: (2)           42: (2,2,2)           93: (2,1,1,2,1)
    3: (1,1)         43: (2,2,1,1)         94: (2,1,1,1,2)
    5: (2,1)         45: (2,1,2,1)         95: (2,1,1,1,1,1)
    6: (1,2)         46: (2,1,1,2)        106: (1,2,2,2)
    7: (1,1,1)       47: (2,1,1,1,1)      107: (1,2,2,1,1)
   10: (2,2)         53: (1,2,2,1)        109: (1,2,1,2,1)
   11: (2,1,1)       54: (1,2,1,2)        110: (1,2,1,1,2)
   13: (1,2,1)       55: (1,2,1,1,1)      111: (1,2,1,1,1,1)
   14: (1,1,2)       58: (1,1,2,2)        117: (1,1,2,2,1)
   15: (1,1,1,1)     59: (1,1,2,1,1)      118: (1,1,2,1,2)
   21: (2,2,1)       61: (1,1,1,2,1)      119: (1,1,2,1,1,1)
   22: (2,1,2)       62: (1,1,1,1,2)      122: (1,1,1,2,2)
   23: (2,1,1,1)     63: (1,1,1,1,1,1)    123: (1,1,1,2,1,1)
   26: (1,2,2)       85: (2,2,2,1)        125: (1,1,1,1,2,1)
   27: (1,2,1,1)     86: (2,2,1,2)        126: (1,1,1,1,1,2)
   29: (1,1,2,1)     87: (2,2,1,1,1)      127: (1,1,1,1,1,1,1)
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Tomi Kärki, Anne Lacroix, and Michel Rigo, On the recognizability of self-generating sets, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.
Wikipedia, Ahnentafel.
Witzel, Stefan On panel-regular ~A2 lattices Geom. Dedicata 191, 85-135 (2017).
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A104326(n) = A007088(a(n)); A023416(a(n)) = A087116(a(n)); A107782(a(n)) = 0; A107345(a(n)) = 1; A107359(n) = a(n+1) - a(n); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - Reinhard Zumkeller, May 25 2005
Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Complement of A004753.
Cf. A023705, A196168, A280873.
Positions of numbers <= 2 in A333766 (see this and A066099 for other sequences about compositions in standard order).
Cf. A318928.

a003754 n = a003754_list !! (n-1)
a003754_list = filter f [0..] where
   f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)
-- Reinhard Zumkeller, Dec 07 2012, Oct 19 2011

isA003754 := proc(n) local bdgs ; bdgs := convert(n,base,2) ; for i from 2 to nops(bdgs) do if op(i,bdgs)=0 and op(i-1,bdgs)= 0 then return false; end if; end do; return true; end proc:
A003754 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do if isA003754(a) then return a; end if; end do: end if; end proc:
# R. J. Mathar, Oct 23 2010

Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {_, 0, 0, _}]&] (* Jean-François Alcover, Oct 25 2011 *)
Select[Range[0,200],SequenceCount[IntegerDigits[#,2],{0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2015 *)

is(n)=n=bitor(n,n>>1)+1; n>>=valuation(n,2); n==1 \\ Charles R Greathouse IV, Feb 06 2017

i=0
while i<=500:
    if "00" not in bin(i)[2:]:
        print(str(i), end=',')
    i+=1 # Indranil Ghosh, Feb 11 2017

Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - Gus Wiseman, Apr 04 2020

A001610 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 2.

Original entry on oeis.org

0, 2, 3, 6, 10, 17, 28, 46, 75, 122, 198, 321, 520, 842, 1363, 2206, 3570, 5777, 9348, 15126, 24475, 39602, 64078, 103681, 167760, 271442, 439203, 710646, 1149850, 1860497, 3010348, 4870846, 7881195, 12752042, 20633238, 33385281, 54018520

Offset: 0

N. J. A. Sloane

For prime p, p divides a(p-1). - T. D. Noe, Apr 11 2009 [This result follows immediately from the fact that A032190(n) = (1/n)*Sum_{d|n} a(d-1)*phi(n/d). - Petros Hadjicostas, Sep 11 2017]
Generalization. If a(0,x)=0, a(1,x)=2 and, for n>=2, a(n,x)=a(n-1,x)+x*a(n-2,x)+1, then we obtain a sequence of polynomials Q_n(x)=a(n,x) of degree floor((n-1)/2), such that p is prime iff all coefficients of Q_(p-1)(x) are multiple of p (sf. A174625). Thus a(n) is the sum of coefficients of Q_(n-1)(x). - Vladimir Shevelev, Apr 23 2010
Odd composite numbers n such that n divides a(n-1) are in A005845. - Zak Seidov, May 04 2010; comment edited by N. J. A. Sloane, Aug 10 2010
a(n) is the number of ways to modify a circular arrangement of n objects by swapping one or more adjacent pairs. E.g., for 1234, new arrangements are 2134, 2143, 1324, 4321, 1243, 4231 (taking 4 and 1 to be adjacent) and a(4) = 6. - Toby Gottfried, Aug 21 2011
For n>2, a(n) equals the number of Markov equivalence classes with skeleton the cycle on n+1 nodes.  See Theorem 2.1 in the article by A. Radhakrishnan et al. below. - Liam Solus, Aug 23 2018
From Gus Wiseman, Feb 12 2019: (Start)
For n > 0, also the number of nonempty subsets of {1, ..., n + 1} containing no two cyclically successive elements (cyclically successive means 1 succeeds n + 1). For example, the a(5) = 17 stable subsets are:
  {1}, {2}, {3}, {4}, {5}, {6},
  {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},
  {1,3,5}, {2,4,6}.
(End)
Also the rank of the n-Lucas cube graph. - Eric W. Weisstein, Aug 01 2023

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv:1505.06339 [math.NT], 2015.
Ning-Ning Cao and Feng-Zhen Zhao, Some Properties of Hyperfibonacci and Hyperlucas Numbers, J. Int. Seq. 13 (2010) # 10.8.8.
Ligia L. Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, Journal of Integer Sequences, Vol. 19, 2016, Issue 7, #16.7.6.
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26.
Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discrete Math., 311 (2011), 2368-2383. See Table 2.1.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 96.
Kantaphon Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1.
Rui Liu and Feng-Zhen Zhao, On the Sums of Reciprocal Hyperfibonacci Numbers and Hyperlucas Numbers, Journal of Integer Sequences, Vol. 15 (2012), #12.4.5. - From _N. J. A. Sloane_, Oct 05 2012
Richard J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, sequence a_{1,s}, arXiv:0903.2514 [math.NT], 2009-2011.
El-Mehdi Mehiri, Saad Mneimneh, and Hacène Belbachir, The Towers of Fibonacci, Lucas, Pell, and Jacobsthal, arXiv:2502.11045 [math.CO], 2025. See p. 12.
Natascha Neumärker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, Journal of Integer Sequences 12 (2009) 09.4.5, Example 10.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Adityanarayanan Radhakrishnan, Liam Solus, and Caroline Uhler. Counting Markov equivalence classes for DAG models on trees, arXiv:1706.06091 [math.CO], 2017; Discrete Applied Mathematics 244 (2018): 170-185.
Vladimir Shevelev, On divisibility of C(n-i-1,i-1) by i, Int. J. of Number Theory, 3 (2007), no.1, 119-139. [_Vladimir Shevelev_, Apr 23 2010]
Eric Weisstein's World of Mathematics, Graph Rank
Eric Weisstein's World of Mathematics, Lucas Cube Graph
John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398.
Li-Na Zheng, Rui Liu, and Feng-Zhen Zhao, On the Log-Concavity of the Hyperfibonacci Numbers and the Hyperlucas Numbers, Journal of Integer Sequences, Vol. 17 (2014), #14.1.4.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000032 (first differences), A000071, A000126, A000204, A000296, A001644, A032190, A169985, A174625, A306357.

List([0..40], n-> Lucas(1,-1,n+1)[2] -1); # G. C. Greubel, Jul 12 2019

a001610 n = a001610_list !! n
a001610_list =
   0 : 2 : map (+ 1) (zipWith (+) a001610_list (tail a001610_list))
-- Reinhard Zumkeller, Aug 21 2011

I:=[0,2]; [n le 2 select I[n] else Self(n-1)+Self(n-2)+1: n in [1..40]]; // Vincenzo Librandi, Mar 20 2015

[Lucas(n+1) -1: n in [0..40]]; // G. C. Greubel, Jul 12 2019

t = {0, 2}; Do[AppendTo[t, t[[-1]] + t[[-2]] + 1], {n, 2, 40}]; t
RecurrenceTable[{a[n] == a[n - 1] +a[n - 2] +1, a[0] == 0, a[1] == 2}, a, {n, 0, 40}] (* Robert G. Wilson v, Apr 13 2013 *)
CoefficientList[Series[x (2 - x)/((1 - x - x^2) (1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
Table[Fibonacci[n] + Fibonacci[n + 2] - 1, {n, 0, 40}] (* Eric W. Weisstein, Feb 13 2018 *)
LinearRecurrence[{2, 0, -1}, {2, 3, 6}, 20] (* Eric W. Weisstein, Feb 13 2018 *)
Table[LucasL[n] - 1, {n, 20}] (* Eric W. Weisstein, Aug 01 2023 *)
LucasL[Range[20]] - 1 (* Eric W. Weisstein, Aug 01 2023 *)

a(n)=([0,1,0; 0,0,1; -1,0,2]^n*[0;2;3])[1,1] \\ Charles R Greathouse IV, Sep 08 2016

vector(40, n, f=fibonacci; f(n+1)+f(n-1)-1) \\ G. C. Greubel, Jul 12 2019

[lucas_number2(n+1,1,-1) -1 for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = A000204(n)-1 = A000032(n+1)-1 = A000071(n+1) + A000045(n).
G.f.: x*(2-x)/((1-x-x^2)*(1-x)) = (2*x-x^2)/(1-2*x+x^3). [Simon Plouffe in his 1992 dissertation]
a(n) = F(n) + F(n+2) - 1 where F(n) is the n-th Fibonacci number. - Zerinvary Lajos, Jan 31 2008
a(n) = A014217(n+1) - A000035(n+1). - Paul Curtz, Sep 21 2008
a(n) = Sum_{i=1..floor((n+1)/2)} ((n+1)/i)*C(n-i,i-1). In more general case of polynomials Q_n(x)=a(n,x) (see our comment) we have Q_n(x) = Sum_{i=1..floor((n+1)/2)}((n+1)/i)*C(n-i,i-1)*x^(i-1). - Vladimir Shevelev, Apr 23 2010
a(n) = Sum_{k=0..n-1} Lucas(k), where Lucas(n) = A000032(n). - Gary Detlefs, Dec 07 2010
a(0)=0, a(1)=2, a(2)=3; for n>=3, a(n) = 2*a(n-1) - a(n-3). - George F. Johnson, Jan 28 2013
For n > 1, a(n) = A048162(n+1) + 3. - Toby Gottfried, Apr 13 2013
For n > 0, a(n) = A169985(n + 1) - 1. - Gus Wiseman, Feb 12 2019

A008949 Triangle read by rows of partial sums of binomial coefficients: T(n,k) = Sum_{i=0..k} binomial(n,i) (0 <= k <= n); also dimensions of Reed-Muller codes.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 6, 16, 26, 31, 32, 1, 7, 22, 42, 57, 63, 64, 1, 8, 29, 64, 99, 120, 127, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256, 1, 10, 46, 130, 256, 382, 466, 502, 511, 512, 1, 11, 56, 176, 386, 638, 848, 968, 1013, 1023, 1024, 1, 12, 67, 232, 562, 1024, 1486, 1816, 1981, 2036, 2047, 2048

Offset: 0

N. J. A. Sloane

The second-left-from-middle column is A000346: T(2n+2, n) = A000346(n). - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
T(n,k) is the maximal number of regions into which n hyperplanes of co-dimension 1 divide R^k (the Cake-Without-Icing numbers). - Rob Johnson, Jul 27 2008
T(n,k) gives the number of vertices within distance k (measured along the edges) of an n-dimensional unit cube, (i.e., the number of vertices on the hypercube graph Q_n whose distance from a reference vertex is <= k). - Robert Munafo, Oct 26 2010
A triangle formed like Pascal's triangle, but with 2^n for n >= 0 on the right border instead of 1. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013
Consider each "1" as an apex of two sequences: the first is the set of terms in the same row as the "1", but the rightmost term in the row repeats infinitely. Example: the row (1, 4, 7, 8) becomes (1, 4, 7, 8, 8, 8, ...). The second sequence begins with the same "1" but is the diagonal going down and to the right, thus: (1, 5, 16, 42, 99, 219, 466, ...). It appears that for all such sequence pairs, the binomial transform of the first, (1, 4, 7, 8, 8, 8, ...) in this case; is equal to the second: (1, 5, 16, 42, 99, ...). - Gary W. Adamson, Aug 19 2015
Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let q(n) be the sum of polynomials in the n-th generation of T*. For n >= 0, row n of A008949 gives the coefficients of q(n+1); e.g., (row 3) = (1, 4, 7, 8) matches x^3 + 4*x^2 + 7*x + 9, which is the sum of the 8 polynomials in the 4th generation of T*. - Clark Kimberling, Jun 16 2016
T(n,k) is the number of subsets of [n]={1,...,n} of at most size k. Equivalently, T(n,k) is the number of subsets of [n] of at least size n-k. Counting the subsets of at least size (n-k) by conditioning on the largest element m of the smallest (n-k) elements of such a subset provides the formula T(n,k) = Sum_{m=n-k..n} C(m-1,n-k-1)*2^(n-m), and, by letting j=m-n+k, we obtain T(n,k) = Sum_{j=0..k} C(n+j-k-1,j)*2^(k-j). - Dennis P. Walsh, Sep 25 2017
If the interval of integers 1..n is shifted up or down by k, making the new interval 1+k..n+k or 1-k..n-k, then T(n-1,n-1-k) (= 2^(n-1)-T(n-1,k-1)) is the number of subsets of the new interval that contain their own cardinal number as an element. - David Pasino, Nov 01 2018

			Triangle begins:
  1;
  1,  2;
  1,  3,  4;
  1,  4,  7,   8;
  1,  5, 11,  15,  16;
  1,  6, 16,  26,  31,  32;
  1,  7, 22,  42,  57,  63,  64;
  1,  8, 29,  64,  99, 120, 127, 128;
  1,  9, 37,  93, 163, 219, 247, 255,  256;
  1, 10, 46, 130, 256, 382, 466, 502,  511,  512;
  1, 11, 56, 176, 386, 638, 848, 968, 1013, 1023, 1024;
  ...

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 376.

Harvey P. Dale, Table of n, a(n) for n = 0..10000
Milica Andelic, C. M. da Fonseca and A. Pereira, The mu-permanent, a new graph labeling, and a known integer sequence, arXiv:1609.04208 [math.CO], 2016.
Stefan Forcey, Planes and axioms, Univ. Akron (2024). See p. 2.
Stefan Forcey, Counting plane arrangements via oriented matroids, arXiv:2504.11461 [math.HO], 2025. See p. 18.
Rob Johnson, Dividing Space.
Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Dennis P. Walsh, A note on counting subsets of restricted size
Wikipedia, Bernoulli's triangle
Index entries for triangles and arrays related to Pascal's triangle

Diagonals are given by A000079, A000225, A000295, A002662, A002663, A002664, A035038-A035042.
Columns are given by A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863. - Ken Shirriff, Jun 28 2011
Row sums sequence is A001792.
T(n, m)= A055248(n, n-m).
Cf. A110555, A007318, A000346, A171886, A228196, A228576.
Cf. also A027306, A249111, A163866, A261363.
Cf. A000071, A001924

T:=Flat(List([0..11],n->List([0..n],k->Sum([0..k],j->Binomial(n+j-k-1,j)*2^(k-j))))); # Muniru A Asiru, Nov 25 2018

a008949 n k = a008949_tabl !! n !! k
a008949_row n = a008949_tabl !! n
a008949_tabl = map (scanl1 (+)) a007318_tabl
-- Reinhard Zumkeller, Nov 23 2012

[[(&+[Binomial(n,j): j in [0..k]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 25 2018

A008949 := proc(n,k) local i; add(binomial(n,i),i=0..k) end; # Typo corrected by R. J. Mathar, Oct 26 2010

Table[Length[Select[Subsets[n], (Length[ # ] <= k) &]], {n, 0, 12}, {k, 0, n}] // Grid (* Geoffrey Critzer, May 13 2009 *)
Flatten[Accumulate/@Table[Binomial[n,i],{n,0,20},{i,0,n}]] (* Harvey P. Dale, Aug 08 2015 *)
T[ n_, k_] := If[ n < 0 || k > n, 0, Binomial[n, k] Hypergeometric2F1[1, -k, n + 1 - k, -1]]; (* Michael Somos, Aug 05 2017 *)

A008949(n)=T8949(t=sqrtint(2*n-sqrtint(2*n)),n-t*(t+1)/2)
T8949(r,c)={ 2*c > r || return(sum(i=0,c,binomial(r,i))); 1<M. F. Hasler, May 30 2010

{T(n, k) = if(k>n, 0, sum(i=0, k, binomial(n, i)))}; /* Michael Somos, Aug 05 2017 */

row(n) = my(v=vector(n+1, k, binomial(n,k-1))); vector(#v, k, sum(i=1, k, v[i])); \\ Michel Marcus, Apr 13 2025

[[sum(binomial(n,j) for j in range(k+1)) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Nov 25 2018

From partial sums across rows of Pascal triangle A007318.
T(n, 0) = 1, T(n, n) = 2^n, T(n, k) = T(n-1, k-1) + T(n-1, k), 0 < k < n.
G.f.: (1 - x*y)/((1 - y - x*y)*(1 - 2*x*y)). - Antonio Gonzalez (gonfer00(AT)gmail.com), Sep 08 2009
T(2n,n) = A032443(n). - Philippe Deléham, Sep 16 2009
T(n,k) = 2 T(n-1,k-1) + binomial(n-1,k) = 2 T(n-1,k) - binomial(n-1,k). - M. F. Hasler, May 30 2010
T(n,k) = binomial(n,n-k)* 2F1(1, -k; n+1-k; -1). - Olivier Gérard, Aug 02 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
T(n,floor(n/2)) = A027306(n). - Reinhard Zumkeller, Nov 14 2014
T(n,n) = 2^n, otherwise for 0 <= k <= n-1, T(n,k) = 2^n - T(n,n-k-1). - Bob Selcoe, Mar 30 2017
For fixed j >= 0, lim_{n -> oo} T(n+1,n-j+1)/T(n,n-j) = 2. - Bob Selcoe, Apr 03 2017
T(n,k) = Sum_{j=0..k} C(n+j-k-1,j)*2^(k-j). - Dennis P. Walsh, Sep 25 2017

More terms from Larry Reeves (larryr(AT)acm.org), Mar 23 2000

A101220 a(n) = Sum_{k=0..n} Fibonacci(n-k)*n^k.

Original entry on oeis.org

0, 1, 3, 14, 91, 820, 9650, 140601, 2440317, 49109632, 1123595495, 28792920872, 816742025772, 25402428294801, 859492240650847, 31427791175659690, 1234928473553777403, 51893300561135516404, 2322083099525697299278

Offset: 0

Ross La Haye, Dec 14 2004

In what follows a(i,j,k) denotes a three-dimensional array, the terms a(n) are defined as a(n,n,n) in that array. - Joerg Arndt, Jan 03 2021
Previous name was: Three-dimensional array: a(i,j,k) = expansion of x*(1 + (i-j)*x)/((1-j*x)*(1-x-x^2)), read by a(n,n,n).
a(i,j,k) = the k-th value of the convolution of the Fibonacci numbers (A000045) with the powers of i = Sum_{m=0..k} a(i-1,j,m), both for i = j and i > 0; a(i,j,k) = a(i-1,j,k) + a(j,j,k-1), for i,k > 0; a(i,1,k) = Sum_{m=0..k} a(i-1,0,m), for i > 0. With F = Fibonacci and L = Lucas, then a(1,1,k) = F(k+2) - 1; a(2,1,k) = F(k+3) - 2; a(3,1,k) = L(k+2) - 3; a(4,1,k) = 4*F(k+1) + F(k) - 4; a(1,2,k) = 2^k - F(k+1); a(2,2,k) = 2^(k+1) - F(k+3); a(3,2,k) = 3(2^k - F(k+2)) + F(k); a(4,2,k) = 2^(k+2) - F(k+4) - F(k+2); a(1,3,k) = (3^k + L(k-1))/5, for k > 0; a(2,3,k) = (2 * 3^k - L(k)) /5, for k > 0; a(3,3,k) = (3^(k+1) - L(k+2))/5; a(4,3,k) = (4 * 3^k - L(k+2) - L(k+1))/5, etc..

			a(1,3,3) = 6 because a(1,3,0) = 0, a(1,3,1) = 1, a(1,3,2) = 2 and 4*2 - 2*1 - 3*0 = 6.

G. C. Greubel, Table of n, a(n) for n = 0..385
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Lucas Number

a(0, j, k) = A000045(k).
a(1, 2, k+1) - a(1, 2, k) = A099036(k).
a(3, 2, k+1) - a(3, 2, k) = A104004(k).
a(4, 2, k+1) - a(4, 2, k) = A027973(k).
a(1, 3, k+1) - a(1, 3, k) = A099159(k).
a(i, 0, k) = A109754(i, k).
a(i, i+1, 3) = A002522(i+1).
a(i, i+1, 4) = A071568(i+1).
a(2^i-2, 0, k+1) = A118654(i, k), for i > 0.
Sequences of the form a(n, 0, k): A000045(k+1) (n=1), A000032(k) (n=2), A000285(k-1) (n=3), A022095(k-1) (n=4), A022096(k-1) (n=5), A022097(k-1) (n=6), A022098(k-1) (n=7), A022099(k-1) (n=8), A022100(k-1) (n=9), A022101(k-1) (n=10), A022102(k-1) (n=11), A022103(k-1) (n=12), A022104(k-1) (n=13), A022105(k-1) (n=14), A022106(k-1) (n=15), A022107(k-1) (n=16), A022108(k-1) (n=17), A022109(k-1) (n=18), A022110(k-1) (n=19), A088209(k-2) (n=k-2), A007502(k) (n=k-1), A094588(k) (n=k).
Sequences of the form a(1, n, k): A000071(k+2) (n=1), A027934(k-1) (n=2), A098703(k) (n=3).
Sequences of the form a(2, n, k): A001911(k) (n=1), A008466(k+1) (n=2), A106517(k-1) (n=3).
Sequences of the form a(3, n, k): A027961(k) (n=1), A094688(k) (n=3).
Sequences of the form a(4, n, k): A053311(k-1) (n=1), A027974(k-1) (n=2).
Cf. A001622, A001891, A001924, A023537, A104161, A106517.

A101220:= func< n | (&+[n^k*Fibonacci(n-k): k in [0..n]]) >;
[A101220(n): n in [0..30]]; // G. C. Greubel, Jun 01 2025

Join[{0}, Table[Sum[Fibonacci[n-k]*n^k, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 03 2021 *)

a(n)=sum(k=0,n,fibonacci(n-k)*n^k) \\ Joerg Arndt, Jan 03 2021

def A101220(n): return sum(n^k*fibonacci(n-k) for k in range(n+1))
print([A101220(n) for n in range(31)]) # G. C. Greubel, Jun 01 2025

a(i, j, 0) = 0, a(i, j, 1) = 1, a(i, j, 2) = i+1; a(i, j, k) = ((j+1)*a(i, j, k-1)) - ((j-1)*a(i, j, k-2)) - (j*a(i, j, k-3)), for k > 2.
a(i, j, k) = Fibonacci(k) + i*a(j, j, k-1), for i, k > 0.
a(i, j, k) = (Phi^k - (-Phi)^-k + i(((j^k - Phi^k) / (j - Phi)) - ((j^k - (-Phi)^-k) / (j - (-Phi)^-1)))) / sqrt(5), where Phi denotes the golden mean/ratio (A001622).
i^k = a(i-1, i, k) + a(i-2, i, k+1).
A104161(k) = Sum_{m=0..k} a(k-m, 0, m).
a(i, j, 0) = 0, a(i, j, 1) = 1, a(i, j, 2) = i+1, a(i, j, 3) = i*(j+1) + 2; a(i, j, k) = (j+2)*a(i, j, k-1) - 2*j*a(i, j, k-2) - a(i, j, k-3) + j*a(i, j, k-4), for k > 3. a(i, j, 0) = 0, a(i, j, 1) = 1; a(i, j, k) = a(i, j, k-1) + a(i, j, k-2) + i * j^(k-2), for k > 1.
G.f.: x*(1 + (i-j)*x)/((1-j*x)*(1-x-x^2)).
a(n, n, n) = Sum_{k=0..n} Fibonacci(n-k) * n^k. - Ross La Haye, Jan 14 2006
Sum_{m=0..k} binomial(k,m)*(i-1)^m = a(i-1,i,k) + a(i-2,i,k+1), for i > 1. - Ross La Haye, May 29 2006
From Ross La Haye, Jun 03 2006: (Start)
a(3, 3, k+1) - a(3, 3, k) = A106517(k).
a(1, 1, k) = A001924(k) - A001924(k-1), for k > 0.
a(2, 1, k) = A001891(k) - A001891(k-1), for k > 0.
a(3, 1, k) = A023537(k) - A023537(k-1), for k > 0.
Sum_{j=0..i+1} a(i-j+1, 0, j) - Sum_{j=0..i} a(i-j, 0, j) = A001595(i). (End)
a(i,j,k) = a(j,j,k) + (i-j)*a(j,j,k-1), for k > 0.
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jan 03 2021

New name from Joerg Arndt, Jan 03 2021

A001595 a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049, 242785, 392835, 635621, 1028457, 1664079, 2692537, 4356617, 7049155, 11405773, 18454929, 29860703, 48315633, 78176337

Offset: 0

N. J. A. Sloane

2-ranks of difference sets constructed from Segre hyperovals.
Sometimes called Leonardo numbers. - George Pollard, Jan 02 2008
a(n) is the number of nodes in the Fibonacci tree of order n. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node (see the Knuth reference, p. 417). - Emeric Deutsch, Jun 14 2010
Also odd numbers whose index is a Fibonacci number: odd(Fib(k)). - Carmine Suriano, Oct 21 2010
This is the sequence A(1,1;1,1;1) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 17 2010
In general, adding a constant to each successive term of a Horadam sequence with signature (c,d) will result in a third-order recurrence with signature (c+1, d-c,-d). - Gary Detlefs, Feb 01 2023

			a(7) = odd(F(7)) = odd(8) = 15. - _Carmine Suriano_, Oct 21 2010

E. W. Dijkstra, 'Fibonacci numbers and Leonardo numbers', circulated privately, July 1981.
E. W. Dijkstra, 'Smoothsort, an alternative for sorting in situ', Science of Computer Programming, 1(3): 223-233, 1982.
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Ziegenbalg, Algorithmen, Spektrum Akademischer Verlag, 1996, p. 172.

T. D. Noe, Table of n, a(n) for n = 0..500
Paula M. M. C. Catarino and Anabela Borges, On Leonardo numbers, Acta Mathematica Universitatis Comenianae (2019), 1-12.
P. Catarino and A. Borges, A Note on Incomplete Leonardo Numbers, INTEGERS 20A (2020) A43.
E. W. Dijkstra, Smoothsort, an alternative for sorting in situ (EWD796a).
E. W. Dijkstra, Fibonacci numbers and Leonardo numbers (EWD797).
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Supplement to "Gauss Sums, Jacobi Sums and p-Ranks ...".
Sergio Falcon, Sum of the Squares of the Extended (k, t)-Fibonacci Numbers, Preprints 2024, 2024081150. See p. 2.
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 2.
Yasuichi Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1019
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Kálmán Liptai, László Németh, Tamás Szakács, and László Szalay, On certain Fibonacci representations, arXiv:2403.15053 [math.NT], 2024. See p. 8.
Thor Martinsen, Non-Fisherian generalized Fibonacci numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 2, 370-389. See p. 388.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Diana Savin and Elif Tan, On Companion sequences associated with Leonardo quaternions: Applications over finite fields, arXiv:2403.01592 [math.CO], 2024. See p. 2.
David Singmaster, Some counterexamples and problems on linear recurrences and part 2, Fib. Quart. 8 (1970), 264-267.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 24.
Hunar Sherzad Taher and Saroj Kumar Dash, Fibonacci Numbers that Are eta-concatenations of Leonardo and Lucas Numbers, Proc. Bulgar. Acad. Sci. (2025) Vol. 78, No. 2, 171-180.
Elif Tan and Ho-Hon Leung, On Leonardo p-Numbers, Integers (2023) Vol. 23, #A7.
Bibhu Prasad Tripathy and Bijan Kumar Patel, Common Values of Generalized Fibonacci and Leonardo Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.6.2.
Qing Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.
Tülay Yaǧmur, On Gaussian Leonardo Hybrid Polynomials, Symmetry (2023) Vol. 15, No. 7, 1422.
Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers (2024) Vol. 24, Art. No. A82.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071, A006355, A033538, A049112, A049114, A101220, A109754, A128587.

List([0..40], n-> 2*Fibonacci(n+1) -1); # G. C. Greubel, Jul 10 2019

a001595 n = a001595_list !! n
a001595_list =
   1 : 1 : (map (+ 1) $ zipWith (+) a001595_list $ tail a001595_list)
-- Reinhard Zumkeller, Aug 14 2011

[2*Fibonacci(n+1)-1: n in [0..40]]; // G. C. Greubel, Jul 10 2019

L := 1,3: for i from 3 to 40 do l := nops([ L ]): L := L,op(l,[ L ])+op(l-1,[ L ])+1: od: [ L ];
A001595:=(1-z+z**2)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(n-1)+fibonacci(n+2)-1, n=0..40); # Zerinvary Lajos, Jan 31 2008

Join[{1, 3}, Table[a[1]=1; a[2]=3; a[i]=a[i-1]+a[i-2]+1, {i, 3, 40} ] ]
Table[2*Fibonacci[n+1]-1, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009; modified by G. C. Greubel, Jul 10 2019 *)
RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+a[n-2]+1},a,{n,40}] (* or *) LinearRecurrence[{2,0,-1},{1,1,3},40] (* Harvey P. Dale, Aug 07 2012 *)

a(n) = 2*fibonacci(n+1)-1 \\ Franklin T. Adams-Watters, Sep 30 2009

from sympy import fibonacci
def A001595(n): return (fibonacci(n+1)<<1)-1 # Chai Wah Wu, Sep 10 2024

[2*fibonacci(n+1)-1 for n in (0..40)] # G. C. Greubel, Jul 10 2019

a(n) = 2*Fibonacci(n+1) - 1 = A006355(n+2) - 1. - Richard L. Ollerton, Mar 22 2002
G.f.: (1-x+x^2)/(1-2x+x^3) = 2/(1-x-x^2) - 1/(1-x). [Conjectured by Simon Plouffe in his 1992 dissertation; this is readily verified.]
a(n) = (2/sqrt(5))*((1+sqrt(5))/2)^(n+1) - 2/sqrt(5)*((1-sqrt(5))/2)^(n+1) - 1.
a(n+1)/a(n) is asymptotic to Phi = (1+sqrt(5))/2. - Jonathan Vos Post, May 26 2005
For n >= 2, a(n+1) = ceiling(Phi*a(n)). - Franklin T. Adams-Watters, Sep 30 2009
a(n) = Sum_{k=0..n+1} A109754(n-k+1,k) - Sum_{k=0..n} A109754(n-k,k) = Sum_{k=0..n+1} A101220(n-k+1,0,k) - Sum_{k=0..n} A101220(n-k,0,k). - Ross La Haye, May 31 2006
a(n) = A000071(n+3) - A000045(n). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009
a(n) = Fibonacci(n-1) + Fibonacci(n+2) - 1. - Zerinvary Lajos, Jan 31 2008, corrected by R. J. Mathar, Dec 17 2010
a(n) = 2*a(n-1) - a(n-3); a(0)=1, a(1)=1, a(2)=3. - Harvey P. Dale, Aug 07 2012
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x). - Stefano Spezia, Jan 23 2024

Additional comments from Christian Krattenthaler (kratt(AT)ap.univie.ac.at)

Further edits from Franklin T. Adams-Watters, Sep 30 2009, and N. J. A. Sloane, Oct 03 2009

A027926 Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 2, 3, 5, 8, 13, 20, 26, 25, 16, 6, 1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 46, 51, 41, 22, 7, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 79, 97, 92, 63, 29, 8, 1

Offset: 0

Clark Kimberling

T(n,k) = number of strings s(0),...,s(n) such that s(0)=0, s(n)=n-k and for 1<=i<=n, s(i)=s(i-1)+d, with d in {0,1,2} if i=0, 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.

Can be seen as concatenation of triangles A104763 and A105809, with identifying column of Fibonacci numbers, see example. - Reinhard Zumkeller, Aug 15 2013

			.   0:                           1
.   1:                        1  1   1
.   2:                     1  1  2   2   1
.   3:                  1  1  2  3   4   3   1
.   4:               1  1  2  3  5   7   7   4   1
.   5:            1  1  2  3  5  8  12  14  11   5   1
.   6:          1 1  2  3  5  8 13  20  26  25  16   6   1
.   7:        1 1 2  3  5  8 13 21  33  46  51  41  22   7   1
.   8:      1 1 2 3  5  8 13 21 34  54  79  97  92  63  29   8  1
.   9:    1 1 2 3 5  8 13 21 34 55  88 133 176 189 155  92  37  9  1
.  10:  1 1 2 3 5 8 13 21 34 55 89 143 221 309 365 344 247 129 46 10  1
.
.   1:                           1
.   2:                        1  1
.   3:                     1  1  2
.   4:                  1  1  2  3
.   5:               1  1  2  3  5      columns = A000045, > 0
.   6:            1  1  2  3  5  8     +---------+
.   7:          1 1  2  3  5  8 13     | A104763 |
.   8:        1 1 2  3  5  8 13 21     +---------+
.   9:      1 1 2 3  5  8 13 21 34
.  10:    1 1 2 3 5  8 13 21 34 55
.  11:  1 1 2 3 5 8 13 21 34 55 89
.
.   0:                           1
.   1:                           1   1                +---------+
.   2:                           2   2   1            | A105809 |
.   3:                           3   4   3   1        +---------+
.   4:                           5   7   7   4   1
.   5:                           8  12  14  11   5   1
.   6:                          13  20  26  25  16   6   1
.   7:                          21  33  46  51  41  22   7   1
.   8:                          34  54  79  97  92  63  29   8  1
.   9:                          55  88 133 176 189 155  92  37  9  1
.  10:                          89 143 221 309 365 344 247 129 46 10  1

Reinhard Zumkeller, Rows n = 0..100 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Many columns of T are A000045 (Fibonacci sequence), also in T: A001924, A004006, A000071, A000124, A014162, A014166, A027927-A027933.

Some other Fibonacci-Pascal triangles: A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

Flat(List([0..10], n-> List([0..2*n], k-> Sum([0..Int((2*n-k+1)/2) ], j-> Binomial(n-j, 2*n-k-2*j) )))); # G. C. Greubel, Sep 05 2019

a027926 n k = a027926_tabf !! n !! k
a027926_row n = a027926_tabf !! n
a027926_tabf = iterate (\xs -> zipWith (+)
                               ([0] ++ xs ++ [0]) ([1,0] ++ xs)) [1]
-- Variant, cf. example:
a027926_tabf' = zipWith (++) a104763_tabl (map tail a105809_tabl)
-- Reinhard Zumkeller, Aug 15 2013

[&+[Binomial(n-j, 2*n-k-2*j): j in [0..Floor((2*n-k+1)/2)]]: k in [0..2*n], n in [0..10]]; // G. C. Greubel, Sep 05 2019

A027926 := proc(n,k)
    add(binomial(n-j,2*n-k-2*j),j=0..(2*n-k+1)/2) ;
end proc: # R. J. Mathar, Apr 11 2016

z = 15; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := 1;
t[n_, k_] := t[n, k] = t[n - 1, k - 2] + t[n - 1, k - 1];
u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u] (* A027926 array *)
v = Flatten[u] (* A027926 sequence *)
(* Clark Kimberling, Aug 31 2014 *)
Table[Sum[Binomial[n-j, 2*n-k-2*j], {j, 0, Floor[(2*n-k+1)/2]}], {n, 0, 10}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Sep 05 2019 *)

{T(n, k) = if( k<0 || k>2*n, 0, if( k<=1 || k==2*n, 1, T(n-1, k-2) + T(n-1, k-1)))}; /* _Michael Somos, Feb 26 1999 */

{T(n, k) = if( k<0 || k>2*n, 0, sum( j=max(0, k-n), k\2, binomial(k-j, j)))}; /* Michael Somos */

[[sum(binomial(n-j, 2*n-k-2*j) for j in (0..floor((2*n-k+1)/2))) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Sep 05 2019

T(n, k) = Sum_{j=0..floor((2*n-k+1)/2)} binomial(n-j, 2*n-k-2*j). - Len Smiley, Oct 21 2001

Incorporates comments from Michael Somos.

Example extended by Reinhard Zumkeller, Aug 15 2013

A000126 A nonlinear binomial sum.

Original entry on oeis.org

1, 2, 4, 8, 15, 27, 47, 80, 134, 222, 365, 597, 973, 1582, 2568, 4164, 6747, 10927, 17691, 28636, 46346, 75002, 121369, 196393, 317785, 514202, 832012, 1346240, 2178279, 3524547, 5702855, 9227432, 14930318, 24157782, 39088133, 63245949

Offset: 1

N. J. A. Sloane

a(n)-1 counts ternary numbers with no 0 digit (A007931) and at least one 2 digit, where the total of ternary digits is <= n. E.g., a(4)-1 = 7: 2 12 21 22 112 121 211. - Frank Ellermann, Dec 02 2001
A107909(a(n-1)) = A000079(n-1) = 2^(n-1). - Reinhard Zumkeller, May 28 2005
a(n) is the permanent of the n X n 0-1 matrix whose (i,j) entry is 1 iff i=1 or j=n or |i-j|<=1. For example, a(5)=15 is per([[1, 1, 1, 1, 1], [1, 1, 1, 0, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]]). - David Callan, Jun 07 2006
Conjecture. Let S(1)={1} and, for n>1, let S(n) be the smallest set containing x+1 and 2x+1 for each element x in S(n-1). Then a(n) is the sum of the elements in S(n). (See A122554 for a sequence defined in this way.) - John W. Layman, Nov 21 2007
a(n+1) indexes the corner blocks on the Fibonacci spiral built from blocks of unit area (using F(1) and F(2) as the sides of the first block). - Paul Barry, Mar 06 2008
The number of length n binary words with fewer than 2 0-digits between any pair of consecutive 1-digits. - Jeffrey Liese, Dec 23 2010
If b(n) = a(n+1) then b(0) = 1 and 2*b(n) >= b(n+1) for all n > 1 which is sufficient for b(n) to be a complete sequence. - Frank M Jackson, Mar 17 2013
From Gus Wiseman, Feb 10 2019: (Start)
Also the number of non-singleton subsets of {1, ..., n + 1} with no successive elements. For example, the a(5) = 15 subsets are:
  {},
  {1,3}, {1,4}, {1,5}, {1,6}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},
  {1,3,5}, {1,3,6}, {1,4,6}, {2,4,6}.
Also the number of binary sequences with all zeros or at least 2 ones and no adjacent ones. For example, the a(1) = 1 through a(4) = 8 sequences are:
  (00)  (000)  (0000)  (00000)
        (101)  (0101)  (00101)
               (1001)  (01001)
               (1010)  (01010)
                       (10001)
                       (10010)
                       (10100)
                       (10101)
(End)

Ralph P. Grimaldi, A generalization of the Fibonacci sequence. Proceedings of the seventeenth Southeastern international conference on combinatorics, graph theory, and computing (Boca Raton, Fla., 1986). Congr. Numer. 54 (1986), 123--128. MR0885268 (89f:11030). - N. J. A. Sloane, Apr 08 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..201
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See p. 6.
Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, Permutations of point sets in R^d, arXiv:2106.14140 [math.CO], 2021.
Tamsin Forbes and Tony Forbes, Hanoi revisited, Math. Gaz. 100, No. 549, 435-441 (2016).
Thomas Langley, Jeffrey Liese, and Jeffrey Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298, doi: 10.1080/00150517.1965.12431407.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Heap-transform of A000071. - John W. Layman
Cf. A066067, A001924, A065220.
Cf. A007931: binary strings with leading 0's, or ternary strings without 0's.
Differences are A000071.
Cf. A122554.
Cf. A005251, A066982, A169985.
Cf. A000045.

List([1..40], n-> Fibonacci(n+3)-(n+1)); # G. C. Greubel, Jul 09 2019

[Fibonacci(n+3)-(n+1): n in [1..40]]; // G. C. Greubel, Jul 09 2019

a:= n-> (Matrix([[1,1,1,2]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [3,-2,-1,1][i] else 0 fi)^n)[1,2]; seq(a(n), n=1..36); # Alois P. Heinz, Aug 26 2008
# alternative
A000126 := proc(n)
    combinat[fibonacci](n+3)-n-1 ;
end proc:
seq(A000126(n),n=1..40) ; # R. J. Mathar, Aug 05 2022

LinearRecurrence[{3,-2,-1,1},{1,2,4,8},40] (* or *) CoefficientList[ Series[-(1-x+x^3)/((x^2+x-1)(x-1)^2),{x,0,40}],x]  (* Harvey P. Dale, Apr 24 2011 *)
Table[Length[Select[Subsets[Range[n]],Min@@Abs[Subtract@@@Partition[#,2,1,1]]>1&]],{n,15}] (* Gus Wiseman, Feb 10 2019 *)

Vec((1-x+x^3)/(1-x-x^2)/(1-x)^2+O(x^40)) \\ Charles R Greathouse IV, Oct 06 2011

vector(40, n, fibonacci(n+3) -(n+1)) \\ G. C. Greubel, Jul 09 2019

def seq(n):
    if n < 0:
        return 1
    a, b = 1, 1
    for i in range(n + 1):
        a, b = b, a + b + i
    return a
[seq(i) for i in range(n)] # Reza K Ghazi, Mar 03 2019

[fibonacci(n+3)-(n+1) for n in (1..40)] # G. C. Greubel, Jul 09 2019

G.f.: (1 - x + x^3 ) / (( 1 - x - x^2 )*( 1 - x )^2). - Simon Plouffe in his 1992 dissertation.
From Henry Bottomley, Oct 22 2001: (Start)
a(n) = Fibonacci(n+3) - (n+1) = a(n-1) + a(n-2) + n - 2
a(n) = A001924(n-1) + 1 = A065220(n+3) + 2. (End)
a(n) = 2*a(n-1) - a(n-3) + 1. - Franklin T. Adams-Watters, Jan 13 2006
a(n+1) = 1 + Sum_{k=0..n} (Fibonacci(k+2) - 1) = Sum_{k=0..n} Fibonacci(k+2) - n. - Paul Barry, Mar 06 2008
a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Harvey P. Dale, May 05 2011
Closed-form without extra leading 1: ((15+7*sqrt(5))*((1+sqrt(5))/2)^n+(15-7*sqrt(5))*((1-sqrt(5))/2)^n-10*n-20)/10; closed-form with extra leading 1: ((20+8*sqrt(5))*((1+sqrt(5))/2)^n+(20-8*sqrt(5))*((1-sqrt(5))/2)^n-20*n-20)/20. - Tim Monahan, Jul 16 2011
G.f. for closed-form with extra leading 1: (1-2*x+x^2+x^3)/((1-x-x^2)*(x-1)^2). - Tim Monahan, Jul 17 2011

A008937 a(n) = Sum_{k=0..n} T(k) where T(n) are the tribonacci numbers A000073.

Original entry on oeis.org

0, 1, 2, 4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760, 117897840, 216847936, 398845537, 733591314, 1349284788

Offset: 0

N. J. A. Sloane, Alejandro Teruel (teruel(AT)usb.ve)

a(n+1) is the number of n-bit sequences that avoid 1100. - David Callan, Jul 19 2004 [corrected by Kent E. Morrison, Jan 08 2019]. Also the number of n-bit sequences avoiding one of the patterns 1000, 0011, 1110, ... or any binary string of length 4 without overlap at beginning and end. Strings where it is not true are: 1111, 1010, 1001, ... and their bitwise complements. - Alois P. Heinz, Jan 09 2019
Row sums of Riordan array (1/(1-x), x(1+x+x^2)). - Paul Barry, Feb 16 2005
Diagonal sums of Riordan array (1/(1-x)^2, x(1+x)/(1-x)), A104698.
A shifted version of this sequence can be found in Eqs. (4) and (3) on p. 356 of Dunkel (1925) with r = 3. (Equation (3) follows equation (4) in the paper!) The whole paper is a study of the properties of this and other similar sequences indexed by the parameter r. For r = 2, we get a shifted version of A000071. For r = 4, we get a shifted version of A107066. For r = 5, we get a shifted version of A001949. For r = 6, we get a shifted version of A172316. See also the table in A172119. - Petros Hadjicostas, Jun 14 2019
Officially, to match A000073, this should start with a(0)=a(1)=0, a(2)=1. - N. J. A. Sloane, Sep 12 2020
Numbers with tribonacci representation that is a prefix of 100100100100... . - Jeffrey Shallit, Jul 10 2024

			G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 28*x^6 + 52*x^7 + 96*x^8 + 177*x^9 + ... [edited by _Petros Hadjicostas_, Jun 12 2019]

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 41.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Isha Agarwal, Matvey Borodin, Aidan Duncan, Kaylee Ji, Tanya Khovanova, Shane Lee, Boyan Litchev, Anshul Rastogi, Garima Rastogi, and Andrew Zhao, From Unequal Chance to a Coin Game Dance: Variants of Penney's Game, arXiv:2006.13002 [math.HO], 2020.
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See pp. 6, 9.
Erik Bates, Blan Morrison, Mason Rogers, Arianna Serafini, and Anav Sood, A new combinatorial interpretation of partial sums of m-step Fibonacci numbers, arXiv:2503.11055 [math.CO], 2025. See p. 3.
Otto Dunkel, Solutions of a probability difference equation, Amer. Math. Monthly, 32 (1925), 354-370; see pp. 356 and 369.
Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See p. 7.
Thomas Langley, Jeffrey Liese, and Jeffrey Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011), Article # 11.4.2.
William Verreault, MacMahon Partition Analysis: a discrete approach to broken stick problems, arXiv:2107.10318 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Partial sums of A000073. Cf. A000213, A018921, A027084, A077908, A209972.
Row sums of A055216.
Column k = 1 of A140997 and second main diagonal of A140994.

a:=[0,1,1];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Sep 13 2019

a008937 n = a008937_list !! n
a008937_list = tail $ scanl1 (+) a000073_list
-- Reinhard Zumkeller, Apr 07 2012

[ n eq 1 select 0 else n eq 2 select 1 else n eq 3 select 2 else n eq 4 select 4 else 2*Self(n-1)-Self(n-4): n in [1..40] ]; // Vincenzo Librandi, Aug 21 2011

A008937 := proc(n) option remember; if n <= 3 then 2^n else 2*procname(n-1)-procname(n-4) fi; end;
a:= n-> (Matrix([[1,1,0,0], [1,0,1,0], [1,0,0,0], [1,0,0,1]])^n)[4,1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 24 2008

CoefficientList[Series[x/(1-2x+x^4), {x, 0, 40}], x]
Accumulate[LinearRecurrence[{1,1,1},{0,1,1},40]] (* Harvey P. Dale, Dec 04 2017 *)
LinearRecurrence[{2, 0, 0, -1},{0, 1, 2, 4},40] (* Ray Chandler, Mar 01 2024 *)

{a(n) = if( n<0, polcoeff( - x^3 / (1 - 2*x^3 + x^4) + x * O(x^-n), -n), polcoeff( x / (1 - 2*x + x^4) + x * O(x^n), n))}; /* Michael Somos, Aug 23 2014 */

a(n) = sum(j=0, n\2, sum(k=0, j, binomial(n-2*j,k+1)*binomial(j,k)*2^k)); \\ Michel Marcus, Sep 08 2017

def A008937_list(prec):
    P = PowerSeriesRing(ZZ, 'x', prec)
    x = P.gen().O(prec)
    return (x/(1-2*x+x^4)).list()
A008937_list(40) # G. C. Greubel, Sep 13 2019

a(n) = A018921(n-2) = A027084(n+1) + 1.
a(n) = (A000073(n+2) + A000073(n+4) - 1)/2.
From Mario Catalani (mario.catalani(AT)unito.it), Aug 09 2002: (Start)
G.f.: x/((1-x)*(1-x-x^2-x^3)).
a(n) = 2*a(n-1) - a(n-4), a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 4. (End)
a(n) = 1 + a(n-1) + a(n-2) + a(n-3). E.g., a(11) = 1 + 600 + 326 + 177 = 1104. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 29 2007
a(n) = term (4,1) in the 4 X 4 matrix [1,1,0,0; 1,0,1,0; 1,0,0,0; 1,0,0,1]^n. - Alois P. Heinz, Jul 24 2008
a(n) = -A077908(-n-3). - Alois P. Heinz, Jul 24 2008
a(n) = (A000213(n+2) - 1) / 2. - Reinhard Zumkeller, Apr 07 2012
a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} binomial(n-2j,k+1) *binomial(j,k)*2^k. - Tony Foster III, Sep 08 2017
a(n) = Sum_{k=0..floor(n/2)} (n-2*k)*hypergeom([-k,-n+2*k+1], [2], 2). - Peter Luschny, Nov 09 2017
a(n) = 2^(n-1)*hypergeom([1-n/4, 1/4-n/4, 3/4-n/4, 1/2-n/4], [1-n/3, 1/3-n/3, 2/3-n/3], 16/27) for n > 0. - Peter Luschny, Aug 20 2020
a(n-1) = T(n) + T(n-3) + T(n-6) + ... + T(2+((n-2) mod 3)), for n >= 4, where T is A000073(n+1). - Jeffrey Shallit, Dec 24 2020

A228074 A Fibonacci-Pascal triangle read by rows: T(n,0) = Fibonacci(n), T(n,n) = n and for n > 0: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 4, 3, 3, 5, 7, 7, 4, 5, 8, 12, 14, 11, 5, 8, 13, 20, 26, 25, 16, 6, 13, 21, 33, 46, 51, 41, 22, 7, 21, 34, 54, 79, 97, 92, 63, 29, 8, 34, 55, 88, 133, 176, 189, 155, 92, 37, 9, 55, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10

Offset: 0

Reinhard Zumkeller, Aug 15 2013

Sum of n-th row is 2^(n+1) - F(n+1) - 1 = A228078(n+1). - Greg Dresden and Sadek Mohammed, Aug 30 2022

			.    0:                                 0
.    1:                               1   1
.    2:                             1   2   2
.    3:                          2    3    4   3
.    4:                       3    5    7    7   4
.    5:                     5    8   12   14   11   5
.    6:                  8   13   20   26   25   16   6
.    7:               13   21   33   46   51   41   22   7
.    8:            21   34   54   79   97   92   63   29   8
.    9:          34   55   88  133  176  189  155   92   37   9
.   10:       55   89  143  221  309  365  344  247  129   46  10
.   11:     89  144  232  364  530  674  709  591  376  175  56   11
.   12:  144 233  376  596  894 1204 1383 1300  967  551  231  67   12 .

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045 (left edge), A001477 (right edge), A228078 (row sums), A027988 (maxima per row);
diagonals T(*,k): A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309;
diagonals T(k,*): A001477, A001245, A004006, A027927, A027928, A027929, A027930, A027931, A027932, A027933;
some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741.

T:= function(n,k)
    if k=0 then return Fibonacci(n);
    elif k=n then return n;
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Sep 05 2019

a228074 n k = a228074_tabl !! n !! k
a228074_row n = a228074_tabl !! n
a228074_tabl = map fst $ iterate
   (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [1]))) ([0], [1,1])

with(combinat);
T:= proc (n, k) option remember;
if k = 0 then fibonacci(n)
elif k = n then n
else T(n-1, k-1) + T(n-1, k)
end if
end proc;
seq(seq(T(n, k), k = 0..n), n = 0..12); # G. C. Greubel, Sep 05 2019

T[n_, k_]:= T[n, k]= If[k==0, Fibonacci[n], If[k==n, n, T[n-1, k-1] + T[n -1, k]]]; Table[T[n, k], {n,0,12}, {k,0,n}] (* G. C. Greubel, Sep 05 2019 *)

T(n,k) = if(k==0, fibonacci(n), if(k==n, n, T(n-1, k-1) + T(n-1, k)));
for(n=0, 12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 05 2019

def T(n, k):
    if (k==0): return fibonacci(n)
    elif (k==n): return n
    else: return T(n-1, k) + T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 05 2019

cp's OEIS Frontend

A001924 Apply partial sum operator twice to Fibonacci numbers.

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A003754 Numbers with no adjacent 0's in binary expansion.

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A001610 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 2.

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A008949 Triangle read by rows of partial sums of binomial coefficients: T(n,k) = Sum_{i=0..k} binomial(n,i) (0 <= k <= n); also dimensions of Reed-Muller codes.

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A101220 a(n) = Sum_{k=0..n} Fibonacci(n-k)*n^k.

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