A099837 -id:A099837 - cp's OEIS frontend

A036605 a(n) = a(n-2) + 2*a(n-3) + a(n-4).

1, 4, 4, 7, 13, 19, 31, 52, 82, 133, 217, 349, 565, 916, 1480, 2395, 3877, 6271, 10147, 16420, 26566, 42985, 69553, 112537, 182089, 294628, 476716, 771343, 1248061, 2019403, 3267463, 5286868, 8554330, 13841197, 22395529, 36236725

Offset: 0

N. J. A. Sloane

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.2, Eq. (25).

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

Cf. A004695.

A036605 := proc(n) option remember; if n <= 0 then 1 else A036605(n-2)+2*A036605(n-3)+A036605(n-4); fi; end;

LinearRecurrence[{0, 1, 2, 1}, {1, 4, 4, 7}, 36] (* Jean-François Alcover, Apr 01 2020 *)

3 * [Fibonacci(n+2)/2] + 1. - Ralf Stephan, Dec 02 2004

a(n) = (A099837(n+2)+A022086(n+2))/2. G.f. ( -1-4*x-3*x^2-x^3 ) / ( (1+x+x^2)*(x^2+x-1) ). - R. J. Mathar, Mar 21 2011

A107751 a(n) = A107750(n+1) - A107750(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Reinhard Zumkeller, May 23 2005

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A100063, A099837, A100051, A107688.

A107751[ nmax_ ] := Length /@ Split[ (Total[ 1 - IntegerDigits[ #1, 2 ] ] &) /@ Range[ 0, nmax ] ]; A107751[ 200 ] (* Peter Pein (petsie(AT)dordos.net), Oct 12 2007 *)

up_to = 65537;
A107751list(up_to) = { my(v=vector(up_to)); v[1]=v[2]=v[3]=v[4]=1; for(n=5,up_to,v[n] = (0^(v[n-1]-1) + 0^(v[n-2]-1))); (v); };
v107751 = A107751list(up_to);
A107751(n) = if(!n,1,v107751[n]); \\ Antti Karttunen, Dec 23 2018

a(n) = if n<=4 then 1 else 0^(a(n-1)-1) + 0^(a(n-2)-1).

A144429 Starts 1 2 3 then successive terms have differences 1 2 3.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 27, 28, 30, 33, 34, 36, 39, 40, 42, 45, 46, 48, 51, 52, 54, 57, 58, 60, 63, 64, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 87, 88, 90, 93, 94, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 117, 118, 120, 123, 124, 126, 129

Offset: 1

Milton L. Brown (miltbrown(AT)earthlink.net), Oct 13 2008

Essentially the same sequence as A047231. - Stefan Steinerberger, Oct 17 2008

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

Cf. A144430, A047231.

Join[{1,2},LinearRecurrence[{1,0,1,-1},{3,4,6,9},100]] (* or *) nxt[ {a_,b_,c_}]:= {c+1,c+3,c+6}; Flatten[NestList[nxt,{1,2,3},30]] (* Harvey P. Dale, Nov 23 2012 *)

G.f.: x(1+x)(1+x^2-x^3+2x^4)/((1+x+x^2)(1-x)^2). a(n)=a(n-3)+6, n>5. a(n) = 2n -11/3 +A099837(n+3)/3, n>2. [From R. J. Mathar, Oct 15 2008]

Extended by R. J. Mathar, Oct 15 2008

A153349 Period 6: repeat [1, 7, 4, 4, 7, 1].

Original entry on oeis.org

1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4, 4, 7, 1, 1, 7, 4

Offset: 0

Paul Curtz, Dec 24 2008

Also: the decimal expansion of 5287/30303. [R. J. Mathar, Jan 03 2009]

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

Cf. A010892, A099837.

&cat [[1, 7, 4, 4, 7, 1]^^20]; // Wesley Ivan Hurt, Jun 23 2016

A153349:=n->[1, 7, 4, 4, 7, 1][(n mod 6)+1]: seq(A153349(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

PadRight[{}, 100, {1, 7, 4, 4, 7, 1}] (* Wesley Ivan Hurt, Jun 23 2016 *)

G.f.: (x^4+6*x^3-2*x^2+6*x+1)/((1-x)*(x^2-x+1)*(1+x+x^2)). a(n) = 4 + 3*A099837(n+2)/2 + 3*A010892(n+4)/2. [R. J. Mathar, Jan 03 2009]
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = (8 - 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) + sqrt(3)*sin(n*Pi/3) + 3*sqrt(3)*sin(2*n*Pi/3))/2. (End)

Extended by R. J. Mathar, Jan 03 2009

A318624 Number of 3-member subsets of [3*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 10, 30, 55, 91, 138, 190, 253, 327, 406, 496, 597, 703, 820, 948, 1081, 1225, 1380, 1540, 1711, 1893, 2080, 2278, 2487, 2701, 2926, 3162, 3403, 3655, 3918, 4186, 4465, 4755, 5050, 5356, 5673, 5995, 6328, 6672, 7021, 7381, 7752, 8128, 8515, 8913, 9316

Offset: 0

Alois P. Heinz, Aug 30 2018

			a(1) = 1: {1,2,3}.
a(2) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}.
a(3) = 30: {1,2,3}, {1,2,6}, {1,2,9}, {1,3,5}, {1,3,8}, {1,4,7}, {1,5,6}, {1,5,9}, {1,6,8}, {1,8,9}, {2,3,4}, {2,3,7}, {2,4,6}, {2,4,9}, {2,5,8}, {2,6,7}, {2,7,9}, {3,4,5}, {3,4,8}, {3,5,7}, {3,6,9}, {3,7,8}, {4,5,6}, {4,5,9}, {4,6,8}, {4,8,9}, {5,6,7}, {5,7,9}, {6,7,8}, {7,8,9}.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Row n=3 of A318557.

LinearRecurrence[{2,-1,1,-2,1},{0,1,10,30,55,91},50] (* Harvey P. Dale, Mar 27 2019 *)

G.f.: -x*(3*x^4+4*x^3+11*x^2+8*x+1)/((x^2+x+1)*(x-1)^3).
a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5) for n>5.
3*a(n) = 5+2*A099837(n)+27*n*(n-1)/2 for n>0. - R. J. Mathar, Sep 02 2018

A014679 G.f.: (1+x^3)^2/((1-x^2)(1-x^3)(1-x^4)).

Original entry on oeis.org

1, 0, 1, 3, 2, 3, 6, 6, 7, 10, 11, 13, 16, 17, 20, 24, 25, 28, 33, 35, 38, 43, 46, 50, 55, 58, 63, 69, 72, 77, 84, 88, 93, 100, 105, 111, 118, 123, 130, 138, 143, 150, 159, 165, 172, 181, 188, 196, 205, 212, 221, 231, 238, 247, 258, 266, 275, 286, 295, 305

Offset: 0

N. J. A. Sloane

Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of M_12.

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 255, Theorem 3.20, where the series is given in the form GF_2 (see formula line).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Adem, Recent developments in the cohomology of finite groups, Notices Amer. Math. Soc., 44 (1997), 806-812.
Alejandro Adem; John Maginnis; James R. Milgram, The geometry and cohomology of the Mathieu group M_12, J. Algebra 139 (1991), no. 1, 90-133.
Index entries for linear recurrences with constant coefficients, signature (2,-2,3,-3,2,-2,1).

I:=[1,0,1,3,2,3,6]; [n le 7 select I[n] else 2*Self(n-1)- 2*Self(n-2)+3*Self(n-3)-3*Self(n-4)+2*Self(n-5)-2*Self(n-6)+Self(n-7): n in [1..60]]; // Vincenzo Librandi, Jul 19 2015

Maple
```
(1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4));
```

CoefficientList[Series[(1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4)), {x,0,60}],x] (* Harvey P. Dale, Mar 17 2011 *)
LinearRecurrence[{2,-2,3,-3,2,-2,1},{1,0,1,3,2,3,6},60] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,-2,2,-3,3,-2,2]^n*[1;0;1;3;2;3;6])[1,1] \\ Charles R Greathouse IV, Feb 10 2017

Can also be written as GF_2 = (1 + x^2 + 3*x^3 + x^4 + 3*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + x^10 + 3*x^11 + x^12 + x^14 ) / ( (1-x^4)*(1-x^6)*(1-x^7)).
G.f.: (1-x+x^2)^2/((1-x)^3*(1+x^2)(1+x+x^2)). a(n)=n^2/12+n/4+13/36-A057077(n)/4+4*A099837(n+3)/9. - R. J. Mathar, Jan 11 2009
a(0)=1, a(1)=0, a(2)=1, a(3)=3, a(4)=2, a(5)=3, a(6)=6, a(n)= 2*a(n-1)- 2*a(n-2)+3*a(n-3)-3*a(n-4)+2*a(n-5)-2*a(n-6)+a(n-7). - Harvey P. Dale, Apr 10 2012

A025771 Expansion of 1/((1-x)(1-x^3)(1-x^11)).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 40, 42, 43, 45, 47, 48, 50, 52, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 82, 84, 86, 89

Offset: 0

N. J. A. Sloane

a(n) is the number of partitions of n into parts 1, 3, and 11. - Joerg Arndt, Aug 20 2025

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,0,0,0,0,0,1,-1,0,-1,1).

A025771 := proc(n)
        round(n^2/66 +5*n/22 +68/99 + A099837(n+3)/9) ;
end proc: # R. J. Mathar, Aug 11 2012

A025771[n_] := Quotient[n*(n + 15) + 78, 66];
Array[A025771, 100, 0] (* Paolo Xausa, Aug 20 2025 *)

Vec(1/((1-x)*(1-x^3)*(1-x^11))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = +a(n-1) +a(n-3) -a(n-4) +a(n-11) -a(n-12) -a(n-14) +a(n-15). - R. J. Mathar, Aug 21 2014

a(n) = floor((n^2 + 15*n + 78)/66). - Hoang Xuan Thanh, Aug 18 2025

A089108 Convoluted convolved Fibonacci numbers G_4^(r).

Original entry on oeis.org

3, 5, 7, 10, 13, 16, 20, 24, 28, 33, 38, 43, 49, 55, 61, 68, 75, 82, 90, 98, 106, 115, 124, 133, 143, 153, 163, 174, 185, 196, 208, 220, 232, 245, 258, 271, 285, 299, 313, 328, 343, 358, 374, 390, 406, 423, 440, 457, 475, 493, 511, 530, 549, 568, 588, 608, 628

Offset: 1

N. J. A. Sloane, Dec 05 2003

P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

with(numtheory): f := z->1/(1-z-z^2): m := proc(r,j) d := divisors(r): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]),i=1..nops(d)): Wser := simplify(series(W,z=0,80)): coeff(Wser,z^j) end: seq(m(r,4),r=1..60);

LinearRecurrence[{2, -1, 1, -2, 1}, {3, 5, 7, 10, 13}, 60] (* Jean-François Alcover, Nov 28 2017 *)

G.f.: x*(3 - x - 2*x^3 + x^4)/((1 - x^3)*(1 - x)^2).

9*a(n) = 11 +27*n/2 +3*n^2/2 -A099837(n+3). - R. J. Mathar, Jan 09 2024

Edited by Emeric Deutsch, Mar 06 2004

A103819 Whitney transform of Jacobsthal numbers.

Original entry on oeis.org

0, 1, 3, 8, 23, 63, 172, 471, 1287, 3516, 9607, 26247, 71708, 195911, 535239, 1462300, 3995079, 10914759, 29819676, 81468871, 222577095, 608091932, 1661338055, 4538859975, 12400396060, 33878512071, 92557816263, 252872656668

Offset: 0

Paul Barry, Feb 16 2005

The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))*g(x(1+x)).

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

Cf. A004070, A001045, A002605, A099837.

LinearRecurrence[{2,2,1,-2,-2},{0,1,3,8,23},30] (* Harvey P. Dale, Nov 02 2024 *)

G.f.: x(1+x)/((1-x)(1+x+x^2)(1-2x-2x^2)).
a(n) = 2a(n-1)+2a(n-2)+a(n-3)-2a(n-4)-2a(n-5).
a(n) = Sum_{k=0..n} Sum_{i=0..n} C(k, i-k)*A001045(k).
9*a(n) = -2 +2*(A002605(n)+2*A002605(n+1))-A099837(n+3). - R. J. Mathar, Oct 23 2011

A173714 Floor(Lucas(n+1)/2), Lucas(n) = A000032(n).

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 14, 23, 38, 61, 99, 161, 260, 421, 682, 1103, 1785, 2889, 4674, 7563, 12238, 19801, 32039, 51841, 83880, 135721, 219602, 355323, 574925, 930249, 1505174

Offset: 0

Gary Detlefs, Nov 25 2010

Sequences of the form a(0)=1, a(1)=b,
a(n) = a(n-1) + a(n-2) + 1 if n mod 3 =2, else
a(n) = a(n-1) + a(n-2) have a closed form of
a(n) = F(n-1)*a + F(n)*b + floor(F(n+1)/2),
where F(n)= Fibonacci(n) = A000045(n), floor(F(n+1)/2) = A004695(n+1).
We can generalize the definition of this sequence by changing the added 1 to any value of k and changing the last term of the formula to floor(F(n+1)/2)*k.
Two variants: if we add the constant at n mod 3 = 0, then a(n)=F(n-1)*a + F(n)*b + floor(F(n)/2), and if for n mod 3 =1, then a(n)=F(n-1)*a + F(n)*b + floor(F(n-1)/2).

			a(5) = a(4) + a(3) + 1 = 5 +3 +1 =9 because 5 mod 3 = 2.
a(6) = a(5) + a(4) = 9 +5 =14 because 6 mod 3 <>2.

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

[Floor(Lucas(n+1)/2): n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

with(combinat):
g:=(a,b,n)->fibonacci(n-1)*a+fibonacci(n)*b + floor(fibonacci(n+1)/2):
seq(g(0,1,n),n=0..30)

Table[Floor[LucasL[n + 1]/2], {n,0,50}] (* G. C. Greubel, Nov 24 2016 *)

a(0)= 0, a(1)=1, a(n)=a(n-1)+a(n-2)+1 if n mod 3 =2, else a(n)=a(n-1)+a(n-2).
G.f.: x*(1+x-x^3)/[(1-x-x^2)*(1-x^3)].
a(n) = a(n-1) +a(n-2) +(1+(-1)^Fib(n+1))/2.
a(n) = A000204(n+1)/2 + A099837(n+1)/6 - 1/3. - R. J. Mathar, Nov 26 2010
a(n) = Fibonacci(n) + floor(Fibonacci(n+1)/2). - Gary Detlefs, Dec 10 2010

cp's OEIS Frontend

A036605 a(n) = a(n-2) + 2*a(n-3) + a(n-4).

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A107751 a(n) = A107750(n+1) - A107750(n).

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A144429 Starts 1 2 3 then successive terms have differences 1 2 3.

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A153349 Period 6: repeat [1, 7, 4, 4, 7, 1].

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Magma

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A318624 Number of 3-member subsets of [3*n] whose elements sum to a multiple of n.

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A014679 G.f.: (1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4)).

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Magma

Maple

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Formula

A025771 Expansion of 1/((1-x)*(1-x^3)*(1-x^11)).

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Maple

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A014679 G.f.: (1+x^3)^2/((1-x^2)(1-x^3)(1-x^4)).

A025771 Expansion of 1/((1-x)(1-x^3)(1-x^11)).