Oboifeng Dira - cp's OEIS frontend

A337129 Triangular array read by rows: T(n,0) = 2^n, T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j) for k > 0.

1, 2, 3, 4, 6, 16, 8, 12, 32, 84, 16, 24, 64, 168, 440, 32, 48, 128, 336, 880, 2304, 64, 96, 256, 672, 1760, 4608, 12064, 128, 192, 512, 1344, 3520, 9216, 24128, 63168, 256, 384, 1024, 2688, 7040, 18432, 48256, 126336, 330752, 512, 768, 2048, 5376, 14080, 36864, 96512, 252672, 661504, 1731840

Offset: 0

Oboifeng Dira, Sep 14 2020

			The triangle  T(n,k) begins:
   n\k  0    1    2    3    4    5
   0:   1
   1:   2    3
   2:   4    6    16
   3:   8    12   32  84
   4:   16   24   64  168  440
   5:   32   48   128 336  880  2304
   ...
T(3,2) = ((3+sqrt(5))^3-(3-sqrt(5))^3)*(2)/(4*sqrt(5)) = (64*sqrt(5))/(2*sqrt(5)) = 32.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Cf. A000079 (1st column), A069429 (diagonal), A018903 (row sums), A001906, A004171.

T := proc (n, k) if k = 0 and 0 <= n then 2^n elif 1 <= k and k <= n then round((((3+sqrt(5))^(k+1)-(3-sqrt(5))^(k+1))*(2^(n-k))/(4*sqrt(5)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);

T[n_, 0] := 2^n;
T[n_, n_] := 2^(n-1) Fibonacci[2n+2];
T[n_, k_] /; 0Jean-François Alcover, Nov 13 2020 *)

T(n,k) = if (k == 0, 2^n, my(w=quadgen(5, 'w)); ((2*w+2)^(k+1)-(4-2*w)^(k+1))*(2^(n-k))/(4*(2*w-1))); \\ Michel Marcus, Sep 14 2020

Row(n)={Vecrev(polcoef((1-x*y)*(1-2*x*y)/((1-6*x*y+4*x^2*y^2)*(1-2*x)) + O(x*x^n), n))} \\ Andrew Howroyd, Sep 23 2020

T(n,0) = 2^n.
T(n,k) = ((3+sqrt(5))^(k+1)-(3-sqrt(5))^(k+1))*(2^(n-k))/(4*sqrt(5)) for 1<=k<=n.
T(n+1,n) = 2*T(n,n).
T(n+m,n) = 2^m*T(n,n), for m>=1.
T(n,n) = A069429(n) = 2^(n-1)*A001906(n+1) for n>=1.
T(2*n,n) = (1/2)*A099157(n+1) = A004171(n-1)*A001906(n+1) for n>=1.
G.f.: (1 - x*y)*(1 - 2*x*y)/((1 - 6*x*y + 4*x^2*y^2)*(1 - 2*x)). - Andrew Howroyd, Sep 23 2020

A335087 Row sums of A335436.

Original entry on oeis.org

1, 7, 34, 150, 628, 2540, 10024, 38840, 148368, 560368, 2096928, 7786592, 28726592, 105390272, 384788096, 1398978432, 5067403520, 18294707968, 65854095872, 236421150208, 846732997632, 3025927678976, 10792083499008, 38420157773824, 136547503083520, 484546494459904, 1716976084393984

Offset: 0

Oboifeng Dira, Sep 11 2020

nonn

This sequence is also a composition of generating functions H(x) = G(F(x)), where G(x) = x/(1-4*x)^2 is the generating function of A002697 and F(x) = x*(1-x)/(1-2*x^2) is the generating function of 0, A016116*(-1)^n.

			For n = 4, a(4) = 8*a(3)-20*a(2)+16*a(1)-4*a(0) = 8*150-20*34+16*7-4*1 = 628.

Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Index entries for linear recurrences with constant coefficients, signature (8,-20,16,-4).

Composition of g.fs of A002697 and A016116.

Cf. A335436.

f:=x->x*(1-x)/(1-2*x^2):g:=x->(x)/(1-4*x)^2:
C:=n->coeff(series(g(f(x))/x,x,n+1),x,n): seq(C(n),n=0..30);

a(n) = 8*a(n-1)-20*a(n-2)+16*a(n-3)-4*a(n-4), a(0)=1, a(1)=7, a(2)=34, a(3)=150 for  n>=4.
G.f.: (1-x)*(1-2*x^2)/(1-4*x+2*x^2)^2.
a(0)=1; a(n) = 2*n+1+Sum_{k=1..n}[(2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1)]*(2n-k+1)/(4*sqrt(2)), n>=1.
G.f.: G(F(x))/x where G(x) is g.f of A002697 and F(x) is g.f of 0,A016116*(-1)^n.

A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

Original entry on oeis.org

1, 3, 4, 5, 8, 21, 7, 12, 35, 96, 9, 16, 49, 144, 410, 11, 20, 63, 192, 574, 1680, 13, 24, 77, 240, 738, 2240, 6692, 15, 28, 91, 288, 902, 2800, 8604, 26112, 17, 32, 105, 336, 1066, 3360, 10516, 32640, 100296, 19, 36, 119, 384, 1230, 3920, 12428, 39168, 122584, 380480

Offset: 0

Oboifeng Dira, Jul 14 2020

			Triangle begins:
  1;
  3,  4;
  5,  8, 21;
  7, 12, 35,  96;
  9, 16, 49, 144, 410;
  ...
T(3,2) = ((2+sqrt(2))^3-(2-sqrt(2))^3)*(6-2+1)/(4*sqrt(2)) = (28*sqrt(2))*(5)/(4*sqrt(2)) = 35.

Oboifeng Dira, Table of n, a(n) for n = 0..54

Cf. A053199, A294285, A064339.

T := proc (n, k) if k = 0 and 0 <= n then 2*n+1 elif 1 <= k and k <= n then round((((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);

T(n,k) = if (k==0, 2*n+1, if (k<=n, sum(i=n-k, n, sum(j=0, i-n+k, if ((i==n) && (j==k), 0, T(i,j)), 0))));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 08 2020

T(n, k) = if (k==0, 2*n+1, if (k>n, 0, my(w=quadgen(8, 'w)); ((2+w)^(k+1)-(2-w)^(k+1))*(2*n-k+1)/(4*w)));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 10 2020

T(n,0) = 2*n+1 for k=0;

T(n,k) = ((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)) for 1<=k<=n.

A334293 First quadrisection of Padovan sequence.

Original entry on oeis.org

1, 0, 2, 5, 16, 49, 151, 465, 1432, 4410, 13581, 41824, 128801, 396655, 1221537, 3761840, 11584946, 35676949, 109870576, 338356945, 1042002567, 3208946545, 9882257736, 30433357674, 93722435101, 288627200960, 888855064897, 2737314167775, 8429820731201, 25960439030624

Offset: 0

Oboifeng Dira, Apr 21 2020

			For n=3, a(3) = 2*a(2) + 3*a(1) + a(0) = 2*2 + 3*0 + 1 = 5.

Colin Barker, Table of n, a(n) for n = 0..1000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (2,3,1).

Quadrisection of A000931.
Bisection (even part) of A099529.
Cf. A099098, A091140.

Vec((1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3) + O(x^30)) \\ Colin Barker, Apr 27 2020

a(n) = A000931(4n).
a(n) = A099529(2n).
a(n) = Sum_{k=0..n} binomial(2*n-k-1, 2*k-1).
a(n) = 2*a(n-1)+3*a(n-2)+a(n-3), a(0)=1, a(1)=0, a(2)=2 for n>=3.
G.f.: (1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3). - Colin Barker, Apr 27 2020

A322573 G.f. = g(f(x)), where f(x) = g.f. of Fibonacci sequence A000045 and g(x) = g.f. of Jacobsthal sequence A001045.

Original entry on oeis.org

0, 1, 2, 7, 22, 73, 240, 793, 2618, 8647, 28558, 94321, 311520, 1028881, 3398162, 11223367, 37068262, 122428153, 404352720, 1335486313, 4410811658, 14567921287, 48114575518, 158911647841, 524849519040, 1733460204961, 5725230133922, 18909150606727, 62452681954102, 206267196469033

Offset: 0

Oboifeng Dira, Aug 29 2019

nonn

Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Index entries for linear recurrences with constant coefficients, signature (3,2,-3,-1).

Cf. A000045, A001045.

g:=x->x/(1-x-2*x^2):
f:=x->x/(1-x-x^2):
C:=n->coeff(series(g(f(x)),x,n+1),x,n):
seq(C(n),n=0..30);

LinearRecurrence[{3, 2, -3, -1}, {0, 1, 2, 7}, 30] (* Jean-François Alcover, Nov 10 2019 *)

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

a(n) = 3a(n-1)+2a(n-2)-3a(n-3)-a(n-4), a(0)=0, a(1)=1, a(2)=2, a(3)=7.

Edited by N. J. A. Sloane, Sep 23 2019

A286012 A Kedlaya-Wilf matrix for the Fibonacci sequence A000045.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 4, 12, 17, 5, 1, 5, 20, 48, 50, 8, 1, 6, 30, 102, 197, 147, 13, 1, 7, 42, 185, 532, 815, 434, 21, 1, 8, 56, 303, 1165, 2804, 3391, 1282, 34

Offset: 1

Oboifeng Dira, Apr 30 2017

For any power series f(x) starting with the term x the first column of the Kedlaya-Wilf matrix are the coefficients of f(x), the second column are the coefficients of f(f(x)), the third column are the coefficients of f(f(f(x))) and so on. This gives a matrix with first row consisting of ones. The sequence given is the diagonal reading of this matrix from right up to left down.

			f^(3)(x) = x + 3x^2 + 12x^4 + ... as in A283679, so a(4)=1, a(8)=3, a(13)=12.

Kiran S. Kedlaya, Another Combinatorial Determinant, Journal of Combinatorial Theory Series A 90(1), November 1998.

Cf. A000045, A270863, A283679.

h:= x-> x/(1-x-x^2):
h2:= n-> coeff(series(h(h(x))), x, n+1), x, n):
h3:= n -> coeff(series(h(h2(x))),x,n+1), x, n):
etc.
h7:= n -> coeff(series(h(h6(x))),x,n+1), x, n): N7:=array(1..7,1..7,sparse): gg:=array([h1,h2,h3,h4,h5,h6,h7]):for k from 1 to 7 do: for j from 1 to 7 do: N7[k,j]:=coeff(series(gg[j],x,12),x^k): od:od:

As an n X n matrix a(i,j) = coefficient of x^i in f^(j)(x) for i,j=1..n where f^(j) is the j-fold composition of f with itself.

A283679 G.f.: F(F(F(x))) where F(x) = x/(1-x-x^2) is the g.f. for the Fibonacci numbers.

Original entry on oeis.org

0, 1, 3, 12, 48, 197, 815, 3391, 14153, 59185, 247791, 1038186, 4351706, 18245861, 76514483, 320899470, 1345931153, 5645394769, 23679726926, 99326654214, 416638208001, 1747652017025, 7330817809523, 30750407615699, 128988186902345, 541064919671773, 2269598571509748, 9520261251293028

Offset: 0

Oboifeng Dira, Mar 14 2017

f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(f(x))), x, n+1), x, n):
seq(a(n), n=0..23);

Edited by N. J. A. Sloane, Apr 21 2017

A279277 Composition of Lucas numbers A000032 with Fibonacci numbers A000045.

Original entry on oeis.org

1, 4, 12, 37, 110, 327, 968, 2864, 8469, 25040, 74029, 218856, 647008, 1912753, 5654670, 16716883, 49420052, 146100276, 431915561, 1276869920, 3774804441, 11159436284, 32990587972, 97529916957, 288327225550, 852380393407, 2519888066928, 7449533000584, 22023018662909

Offset: 1

Oboifeng Dira, Dec 10 2016

G(F(x)) where F(x) = x+x^2+2x^3+3x^4+... is the generating series of the Fibonacci numbers A000045 and G(x) = x+3x^2+4x^3+7x^4 +... is the generating series of the Lucas numbers A000032.

			(x+x^2)/(1-3x) = x + (3+1)x^2+... so a(1) = 1 and a(2) = 4.

Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).

Cf. A000045, A000032, A270863.

Rest@ CoefficientList[Series[(x + x^2 - x^3)/(1 - 3 x - x^2 + 3 x^3 + x^4), {x, 0, 24}], x] (* Michael De Vlieger, Dec 12 2016 *)

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

a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4), a(1)=1, a(2)=4, a(3)=12, a(4)=46.

A270863 Self-composition of the Fibonacci sequence.

Original entry on oeis.org

0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069

Offset: 0

Oboifeng Dira, Mar 24 2016

This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027.
From Oboifeng Dira, Jun 28 2020: (Start)
This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where
f(x) = x/(1-k*x-x^2) and g(x) = (x+(k-1)*x^2)/(1-(3-2*k)*x-(3*k-k^2-1)*x^2).
Some cases of k values are:
k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571
k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570
k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569
k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568
k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567
k=0,  f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1)
k=1,  f(x) g.f. A000045 and g(x) g.f. A000045
k=2,  f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1)
k=3,  f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n
k=4,  f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n
k=5,  f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n
k=6,  f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2)
k=7,  f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2).
(End)

			a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.

Colin Barker, Table of n, a(n) for n = 0..1000
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683
Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).

Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834.

I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // Marius A. Burtea, Aug 03 2019

f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
seq(a(n), n=0..30);

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-3,1,3]^(n-1)*[1;2;6;17])[1,1] \\ Charles R Greathouse IV, Mar 24 2016

concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ Colin Barker, Mar 24 2016

a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=2, a(3)=6.
G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - Colin Barker, Mar 24 2016
G.f.: B(B(x)) where B(x) is the g.f. of A000045. - Joerg Arndt, Mar 25 2016
a(n) = (phi*((phi^2 + 5^(1/4)*sqrt(3*phi))^n - (phi^2 - 5^(1/4)*sqrt(3*phi))^n) + (psi^2 + 5^(1/4)*sqrt(3*psi))^n - (psi^2 - 5^(1/4)*sqrt(3*psi))^n)/(2^n * 5^(3/4) * sqrt(3*phi)), where phi = (sqrt(5) + 1)/2 is the golden ratio, and psi = 1/phi = (sqrt(5) - 1)/2. - Vladimir Reshetnikov, Aug 01 2019
0 = a(n)*(a(n) +6*a(n+1) -a(n+2)) +a(n+1)*(8*a(n+1) -9*a(n+2) +a(n+3)) +a(n+2)*(-8*a(n+2) +6*a(n+3)) +a(n+3)*(-a(n+3)) if n>=0. - Michael Somos, Feb 05 2022

A242728 Sequence a(n) with all (x,y)=(a(2m),a(2m+-1)) satisfying y|x^2+1 and x|y^2+y+1.

Original entry on oeis.org

1, 2, 7, 25, 93, 346, 1291, 4817, 17977, 67090, 250383, 934441, 3487381, 13015082, 48572947, 181276705, 676533873, 2524858786, 9422901271, 35166746297, 131244083917, 489809589370, 1827994273563, 6822167504881, 25460675745961, 95020535478962

Offset: 0

Oboifeng Dira, May 21 2014

a(n) with a(1)=2, a(2)=7 is that two-way sequence such that (a(n),a(n+1)) and (a(n),a(n-1)) for n even together with the corresponding pairs of A242725 give all solutions of the two congruences x^2+1 mod y = 0 and y^2+y+1 mod x = 0. The negative part b(n) = a(-n) is given in sequence A242725.

			Considering the pair a(1)=2 and a(2)=7, 2 divides 7^2+1 and 7 divides 2^2+2+1.

T. Bier, Classifications of solutions of certain positive biquadratic division system, submitted May 12 2014.
T. Bier and O. Dira, Construction of integer sequences, submitted May 12 2014.

Oboifeng Dira, Sequences solving division systems
Index entries for linear recurrences with constant coefficients, signature (4,0,-4,1).

A101368 gives a similar problem with x^2+x+1 mod y = 0 and y^2+y+1 mod x = 0.

x0:=1: x1:=2: L:=[x0,x1]: for k from 1 to 30 do:if k mod 2 = 1 then z:=4*x1-x0: fi: if k mod 2 = 0 then z:=4*x1-x0-1: fi: L:=[op(L),z]: x0:=x1: x1:=z: od: print(L);

LinearRecurrence[{4,0,-4,1},{1,2,7,25},30] (* Harvey P. Dale, Sep 02 2025 *)

Vec(-x*(x^3-x^2-2*x+1)/((x-1)*(x+1)*(x^2-4*x+1)) + O(x^100)) \\ Colin Barker, May 21 2014

a(n+1) = 4*a(n) - a(n-1) - p_n (n>0), where p_n=0 if n is odd and p_n = 1 if n is even.
a(n) = 4*a(n-1) - 4*a(n-3) + a(n-4). - Colin Barker, May 21 2014
G.f.: -(x^3-x^2-2*x+1) / ((x-1)*(x+1)*(x^2-4*x+1)). - Colin Barker, May 21 2014
a(n) = (1/12) * (2*A077136(n) + (-1)^n + 3). - Ralf Stephan, May 24 2014

cp's OEIS Frontend

User: Oboifeng Dira

A337129 Triangular array read by rows: T(n,0) = 2^n, T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j) for k > 0.

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A335087 Row sums of A335436.

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A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

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A334293 First quadrisection of Padovan sequence.

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A322573 G.f. = g(f(x)), where f(x) = g.f. of Fibonacci sequence A000045 and g(x) = g.f. of Jacobsthal sequence A001045.

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A286012 A Kedlaya-Wilf matrix for the Fibonacci sequence A000045.

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A283679 G.f.: F(F(F(x))) where F(x) = x/(1-x-x^2) is the g.f. for the Fibonacci numbers.

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A279277 Composition of Lucas numbers A000032 with Fibonacci numbers A000045.

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