A180662 -id:A180662 - cp's OEIS frontend

A180673 a(n) = Fibonacci(n+8) - Fibonacci(8).

0, 13, 34, 68, 123, 212, 356, 589, 966, 1576, 2563, 4160, 6744, 10925, 17690, 28636, 46347, 75004, 121372, 196397, 317790, 514208, 832019, 1346248, 2178288, 3524557, 5702866, 9227444, 14930331, 24157796, 39088148, 63245965, 102334134

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn17 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..275
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+8)-21); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+8) - Fibonacci(8): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+8)-fibonacci(8) od: seq(a(n),n=0..nmax);

Fibonacci[8 +Range[0, 40]] -21 (* G. C. Greubel, Jul 13 2019 *)

concat(0, Vec(x*(13+8*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n)=fibonacci(n+8)-21 \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+8)-21 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+8) - F(8) with F(n) the Fibonacci numbers A000045.
a(n) = a(n-1) + a(n-2) + 21 for n>1, a(0)=0, a(1)=13, and where 21 = F(8).
G.f.: x*(13 + 8*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
a(n) = 13*A000071(n+2) + 8*A000071(n+1). - Bruno Berselli, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-21 + (2^(-1-n)*((1-sqrt(5))^n*(-47+21*sqrt(5)) + (1+sqrt(5))^n*(47+21*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)

A180674 a(n) = Fibonacci(n+9) - Fibonacci(9).

Original entry on oeis.org

0, 21, 55, 110, 199, 343, 576, 953, 1563, 2550, 4147, 6731, 10912, 17677, 28623, 46334, 74991, 121359, 196384, 317777, 514195, 832006, 1346235, 2178275, 3524544, 5702853, 9227431, 14930318, 24157783, 39088135, 63245952, 102334121

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn18 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..275
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+9)-34); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+9) - Fibonacci(9): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=31: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+9)-fibonacci(9) od: seq(a(n),n=0..nmax);

Fibonacci[9 +Range[0, 40]] -34 (* G. C. Greubel, Jul 13 2019 *)
LinearRecurrence[{2,0,-1},{0,21,55},40] (* Harvey P. Dale, Aug 24 2024 *)

concat(0, Vec(x*(21+13*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n) = fibonacci(n+9) - fibonacci(9) \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+9)-34 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+9) - F(9) with F = A000045.
a(n) = a(n-1) + a(n-2) + 34 for n>1, a(0)=0, a(1)=21, and where 34 = F(9).
G.f.: x*(21 + 13*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
a(n) = 21*A000071(n+2) + 13*A000071(n+1). - Bruno Berselli, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-34 + (2^(-n)*((1-sqrt(5))^n*(-38+17*sqrt(5)) + (1+sqrt(5))^n*(38+17*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)

A181532 a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2; a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 10, 18, 31, 55, 96, 169, 296, 520, 912, 1601, 2809, 4930, 8651, 15182, 26642, 46754, 82047, 143983, 252672, 443409, 778128, 1365520, 2396320, 4205249, 7379697, 12950466, 22726483, 39882198, 69988378, 122821042, 215535903, 378239143, 663763424

Offset: 0

Gary W. Adamson, Oct 28 2010

Essentially the same as A060945: a(0)=0 and a(n)=A060945(n-1) for n>=1.
lim(n->infinity) a(n+1)/a(n) = A109134 = 1.754877666..., the square of the absolute value of one of the complex-valued roots of the characteristic polynomial. [R. J. Mathar, Nov 01 2010]
The Ze4 sums, see A180662 for the definition of these sums, of the ‘Races with Ties’ triangle A035317 lead to this sequence. [Johannes W. Meijer, Jul 20 2011]

			a(7) = 18 = a(6) + a(5) + a(3) = 10 + 6 + 2.
a(7) = 18 = (1 0, 2, 0, 2, 0, 3) dot (10, 6, 3, 2, 1, 1, 1) = (10 + 3 + 2 + 3).

Index entries for linear recurrences with constant coefficients, signature (1,1,0,1). [From _R. J. Mathar_, Oct 29 2010]

All of A060945, A077930, A181532 are variations of the same sequence. - N. J. A. Sloane, Mar 04 2012

LinearRecurrence[{1,1,0,1},{0,1,1,2},40] (* Harvey P. Dale, Jun 20 2015 *)

a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2; a(n) = a(n-1) + a(n-2) + a(n-4).
G.f.: x/(1-x-x^2-x^4). [Franklin T. Adams-Watters, Feb 25 2011]
a(n) = |A077930(n)| = ( |A056016(n+2)|-(-1)^n)/5. [R. J. Mathar, Oct 29 2010]
a(n) = A060945(n-1), n>1. [R. J. Mathar, Nov 03 2010]

Values from a(9) on changed by R. J. Mathar, Oct 29 2010

Edited and a(0) added by Franklin T. Adams-Watters, Feb 25 2011

A191372 The Sierpinski-Stern triangle.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 1, 4, 2, 3, 2, 3, 1, 4, 3, 4, 2, 3, 1, 4, 3, 5, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 3, 5, 2, 5, 3, 4, 1, 6, 3, 6, 4, 5, 2, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 4, 6, 2, 5, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3

Offset: 0

Johannes W. Meijer, Jun 05 2011

The knight sums of the first and second kind Kn1y(n) = Kn2y(n), y >= 1, see A180662 for their definitions, of Sierpinski's triangle A047999 lead to the formula Kn1y(n) = A002487(n+(2*y-1)) - AS2S2S2(n,d) where the AS2S2S2(n,d) is the infinite concatenation of a S2(T, d = y-1) sequence; see for the first ten S2(T, d) and the first four Kn1y(n) the examples.
The A191372 sequence is the concatenation of all S2(T, d) sequences, d >= 0. The lengths of the S2(T, d) sequences are 2^ceiling(log(d)/log(2)) for d >= 1 while the length of S2(T, d=0) is 1.
Both the concatenation of the S2(T, d = 2^p) sequences, p >= 0, and the concatenation of the S2(T, d = 2^p-1) sequences, p >= 0, lead to Stern’s diatomic series A002487(n), n >= 2.
The differences of the sequences (AS2S2S2(T, 2^p-delta) - AS2S2S2(T, 2^(p-1)-delta)), T from 0 to (2^(p-1) -1) and 1 <= delta <= (2^(p-1)-1) (take care that p <= pmax), lead to sequences that are snippets of A002487 and, surprisingly, their reverse; see the examples.
The row sums of the Sierpinski-Stern triangle are given by the terms of A191487.

Sam Northshield, Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,..., Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.
Igor Urbiha, Some properties of a function studied by De Rham, Carlitz and Dijkstra and its relation to the (Eisenstein -) Stern's diatomic sequence, Math. Commun. 6, pp. 181-198, 2002.

Cf. A047999 (Sierpinski), A002487 (Stern).

nmax:=2^5; pmax:=log(nmax)/log(2)-1; A047999:=proc(n,k) option remember; A047999(n,k) :=binomial(n,k) mod 2 end: A002487:=proc(n) option remember; if n<=1 then n elif n mod 2=0 then A002487(n/2); else A002487((n-1)/2)+A002487((n+1)/2); fi; end: d:=0: for n from 0 to nmax-d-1 do Kn1(n,d):= add(A047999(n-k+d, k+d),k=0..floor(n/2)): AS2S2S2(n,d):= A002487(n+1+2*d)-Kn1(n,d): od: for p from 1 to pmax do for d from 2^(p-1) to 2^p do for n from 0 to nmax-d-1 do Kn1(n,d):=add(A047999(n-k+d, k+d),k=0..floor(n/2)): AS2S2S2(n,d):= A002487(n+1+2*d)-Kn1(n,d) od: od: od: S2(0,0):=AS2S2S2(0,0): a(0):=S2(0,0): for d from 1 to 2^pmax do for Tx from 0 to 2^ceil(log(d)/log(2))-1 do S2(Tx,d):=AS2S2S2(Tx,d) od: od: Ty:=0: for d from 1 to 2^pmax do for Tx from 0 to 2^ceil(log(d)/log(2))-1 do Ty:=Ty+1: a(Ty):=S2(Tx,d) od: od: S2(0,0); for d from 1 to 2^pmax do seq(S2(Tx,d), Tx=0..2^ceil(log(d)/ log(2))-1) od; seq(a(n),n=0..Ty);

The first few S2(T, d) rows of the Sierpinski-Stern triangle are:
d=0: [0]
d=1: [1]
d=2: [2, 1]
d=3: [2, 1, 3, 2]
d=4: [3, 2, 3, 1]
d=5: [4, 2, 3, 2, 3, 1, 4, 3]
d=6: [4, 2, 3, 1, 4, 3, 5, 2]
d=7: [3, 1, 4, 3, 5, 2, 5, 3]
d=8: [4, 3, 5, 2, 5, 3, 4, 1]
d=9: [6, 3, 6, 4, 5, 2, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4]
The first four Kn1y(n), y = d+1, sequences:
Kn11(n) = A002487(n+1) - A000004(n)
Kn12(n) = A002487(n+3) - A000012(n)
Kn13(n) = A002487(n+5) - A000034(n+1)
Kn14(n) = A002487(n+7) - A157810(n+1)
Three (AS2S2S2(T, 2^p-delta) - AS2S2S2(T, 2^(p-1)-delta)) sequences for p=6:
delta =  1: [1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4]
delta =  8: [4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1]
delta = 16: [5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0]

A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

Original entry on oeis.org

1, 6, 31, 154, 763, 3808, 19197, 97772, 502749, 2607658, 13630635, 71743478, 379949431, 2023314980, 10828048409, 58206726936, 314157742457, 1701817879214, 9249717805207, 50427858276754, 275695956722547, 1511164724634440, 8302888160922965

Offset: 1

Johannes W. Meijer, Jul 15 2013

The terms of this sequence equal the Kn23 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

Cf. A033877, A006603, A006318, A006319, A227504.

A227505 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227505(n), n = 1..23);
A227505 := proc(n): A006603(n+3) - A006318(n+3) - A006319(n+2) end: A006603 := n ->  add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): A006318 := n -> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): A006319 := proc(n): if n=0 then 1 else A006318(n) - A006318(n-1) fi: end: seq(A227505(n), n=1..23);

a(n) = sum(A033877(n-2*k+2,n-k+3), k=1..floor((n+1)/2)).

a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

A011779 Expansion of 1/((1-x)^3*(1-x^3)^2).

Original entry on oeis.org

1, 3, 6, 12, 21, 33, 51, 75, 105, 145, 195, 255, 330, 420, 525, 651, 798, 966, 1162, 1386, 1638, 1926, 2250, 2610, 3015, 3465, 3960, 4510, 5115, 5775, 6501, 7293, 8151, 9087, 10101, 11193, 12376, 13650

Offset: 0

Emeric Deutsch

The Ca2 and Ze4 triangle sums of A139600 are related to the sequence given above, e.g., Ze4(n) = A011779(n-1) - A011779(n-2) - A011779(n-4) + 3*A011779(n-5), with A011779(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Project Euler, Problem 577. Counting hexagons.
Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).

Cf. A011779, A049347, A099254, A139600, A236770 (first trisection, except 0).

R:=PowerSeriesRing(Integers(), 60);
Coefficients(R!( 1/((1-x)^3*(1-x^3)^2) )); // G. C. Greubel, Oct 22 2024

CoefficientList[Series[1 / ((1 - x)^3 (1 - x^3)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)

Vec(1/((1-x)^3*(1-x^3)^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

a(n)=1/216 * n^4 + 1/12 * n^3 + 37/72 * n^2 + [5/4, 139/108, 131/108][1+n%3] * n + [1, 10/9, 7/9][1+n%3] \\ Yurii Ivanov, Jul 06 2021

def A011779_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^3*(1-x^3)^2) ).list()
A011779_list(60) # G. C. Greubel, Oct 22 2024

a(n) = (1/216)*((208 + 270*n + 111*n^2 + 18*n^3 + n^4) - 8*(-1)^n*(A099254(n) + A099254(n-1)) + 16*(A049347(n) + 2*A049347(n-1)) ). - G. C. Greubel, Oct 22 2024

A049853 a(n) = a(n-1) + Sum_{k=0..n-3} a(k) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 2, 3, 6, 11, 19, 33, 58, 102, 179, 314, 551, 967, 1697, 2978, 5226, 9171, 16094, 28243, 49563, 86977, 152634, 267854, 470051, 824882, 1447567, 2540303, 4457921, 7823106, 13728594, 24092003, 42278518, 74193627

Offset: 0

Clark Kimberling

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

Cf. A070550, A180662 (Ca2).

a049853 n = a049853_list !! n
a049853_list = 1 : 2 : 2 : 3 :
   zipWith (+) a049853_list
               (zipWith (+) (drop 2 a049853_list) (drop 3 a049853_list))
-- Reinhard Zumkeller, Aug 06 2011

a := proc(n) option remember: if n<2 then n+1 else a(n-1) + add(a(k), k=0..n-3) fi end: seq(a(n), n=0..33); # Johannes W. Meijer, Jun 18 2018

LinearRecurrence[{2,-1,1},{1,2,2},40] (* Harvey P. Dale, May 12 2022 *)

Vec((1 - x)*(1 + x) / (1 - 2*x + x^2 - x^3) + O(x^40)) \\ Colin Barker, Jun 17 2018

a(n) = 2*a(n-1) - a(n-2) + a(n-3); 3 initial terms required.
a(n) = a(n-1) + a(n-2) + a(n-4) for n > 3. - Reinhard Zumkeller, Aug 06 2011
Empirical: a(n) = Sum_{k=0..floor(n/3)} A084534(n-2*k, n-3*k). - Johannes W. Meijer, Jun 17 2018
G.f.: (1 - x)*(1 + x) / (1 - 2*x + x^2 - x^3). - Colin Barker, Jun 17 2018

A103632 Expansion of (1 - x + x^2)/(1 - x - x^4).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 76, 105, 145, 200, 276, 381, 526, 726, 1002, 1383, 1909, 2635, 3637, 5020, 6929, 9564, 13201, 18221, 25150, 34714, 47915, 66136, 91286, 126000, 173915, 240051, 331337, 457337, 631252, 871303, 1202640

Offset: 0

Paul Barry, Feb 11 2005

Diagonal sums of A103631.
The Kn11 sums, see A180662, of triangle A065941 equal the terms of this sequence without a(0) and a(1). - Johannes W. Meijer, Aug 11 2011
For n >= 2, a(n) is the number of palindromic compositions of n-2 with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1,2 of the Hoggatt et al. reference. Example: a(9) = 8 because we have 7, 151, 11311, 232, 313, 12121, 21112, and 1111111. - Emeric Deutsch, Aug 17 2016
Essentially the same as A003411. - Georg Fischer, Oct 07 2018

Muniru A Asiru, Table of n, a(n) for n = 0..1000
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Cf. A275446.

a:=[1,0,1,1];;  for n in [5..50] do a[n]:=a[n-1]+a[n-4]; od; a; # Muniru A Asiru, Oct 07 2018

I:=[1,0,1,1]; [n le 4 select I[n] else Self(n-1) + Self(n-4): n in [1..50]]; // G. C. Greubel, Mar 10 2019

A103632 := proc(n): add( binomial(floor((2*n-3*k-1)/2), n-2*k), k=0..floor(n/2)) end: seq(A103632(n), n=0..46); # Johannes W. Meijer, Aug 11 2011

LinearRecurrence[{1,0,0,1}, {1,0,1,1}, 50] (* G. C. Greubel, Mar 10 2019 *)

my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^4)) \\ G. C. Greubel, Mar 10 2019

((1-x+x^2)/(1-x-x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Mar 10 2019

G.f.: (1 - x + x^2)/(1 - x - x^4).
a(n) = a(n-1) + a(n-4) with a(0)=1, a(1)=0, a(2)=1 and a(3)=1.
a(n) = Sum_{k=0..floor(n/2)} binomial(floor((2*n-3*k-1)/2), n-2*k).
a(n) = A003269(n+1) - A003269(n-4), n > 4.

Formula corrected by Johannes W. Meijer, Aug 11 2011

A109222 Row sums of a triangle related to the Fibonacci polynomials.

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 40, 76, 145, 276, 526, 1002, 1909, 3637, 6929, 13201, 25150, 47915, 91286, 173915, 331337, 631252, 1202640, 2291229, 4365172, 8316378, 15844082, 30185609, 57508601, 109563441, 208736561, 397677834, 757642355, 1443434582

Offset: 0

Paul Barry, Jun 22 2005

Row sums of A109221.

The Kn4 sums, see A180662, of triangle A065941 equal the terms of this sequence a(n) while the Kn4 sums of triangle A194005 equal a(n+1)-1. - Johannes W. Meijer, Aug 14 2011

Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

A109222 := proc(n): add(binomial(floor((2*n-k)/2)+n-k, 2*n-2*k), k=0..n) end:  seq(A109222(n), n=0..33); # Johannes W. Meijer, Aug 14 2011

G.f.: (1 + x - x^2 - x^3)/(1 - x - 2x^2 + x^4);
a(n) = a(n-1) + 2a(n-2) - a(n-4);
a(n) = Sum_{k=0..n} binomial(floor((2n-k)/2)+n-k, 2n-2k).

A134393 Row sums of triangle A134392.

Original entry on oeis.org

1, 3, 8, 20, 45, 91, 168, 288, 465, 715, 1056, 1508, 2093, 2835, 3760, 4896, 6273, 7923, 9880, 12180, 14861, 17963, 21528, 25600, 30225, 35451, 41328, 47908, 55245, 63395, 72416, 82368, 93313, 105315, 118440, 132756, 148333, 165243, 183560, 203360, 224721, 247723, 272448

Offset: 1

Gary W. Adamson, Oct 23 2007

Binomial transform of [1, 2, 3, 4, 2, 0, 0, 0, ...].

The Kn4 triangle sums of A139600 are given by this sequence. For the definitions of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

			a(4) = 20 = (1, 3, 3, 1) dot (1, 2, 3, 4) = (1 + 6 + 9 + 4).
a(4) = sum of row 4 terms of triangle A134392: (8 + 7 + 4 + 1).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A014628, A134392.

[Binomial(n+3, 4)-2*Binomial(n+2, 4)+ 3*Binomial(n+1, 4): n in [1..40]]; // Vincenzo Librandi, Feb 04 2013

Table[(n^4 - 2*n^3 + 5*n^2 + 8*n)/12, {n, 1, 40}] (* Vincenzo Librandi, Feb 04 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,3,8,20,45},50] (* Harvey P. Dale, May 28 2018 *)

x='x+O('x^99); Vec(x*(1-2*x+3*x^2)/(1-x)^5) \\ Altug Alkan, Aug 16 2017

From R. J. Mathar, Jun 08 2008: (Start)
O.g.f.: x*(1-2*x+3*x^2)/(1-x)^5.
a(n) = A014628(n+1). (End)
a(n) = binomial(n+3,4) - 2*binomial(n+2,4) + 3*binomial(n+1,4). - Johannes W. Meijer, Apr 29 2011, corrected by Eric Rowland, Aug 16 2017
a(n) = n*(n + 1)*(n^2 - 3*n + 8)/12. - Johannes W. Meijer, Apr 29 2011, corrected by Eric Rowland, Aug 16 2017

cp's OEIS Frontend

A180673 a(n) = Fibonacci(n+8) - Fibonacci(8).

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A180674 a(n) = Fibonacci(n+9) - Fibonacci(9).

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A181532 a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2; a(n) = a(n-1) + a(n-2) + a(n-4).

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A191372 The Sierpinski-Stern triangle.

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A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

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A011779 Expansion of 1/((1-x)^3*(1-x^3)^2).

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A049853 a(n) = a(n-1) + Sum_{k=0..n-3} a(k) for n >= 2, a(0)=1, a(1)=2.

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