Sergio Falcon - cp's OEIS frontend

A336890 Numbers that eventually reach the fixed point 8208 under "x --> sum of the fourth powers of digits of x".

12, 17, 21, 46, 64, 71, 102, 107, 120, 137, 145, 154, 170, 173, 201, 210, 224, 242, 279, 288, 297, 317, 349, 357, 371, 375, 379, 394, 397, 406, 415, 422, 439, 451, 460, 493, 514, 537, 541, 573, 599, 604, 640, 701, 710, 713, 729, 731, 735, 739, 753, 792, 793, 828, 882, 927, 934, 937, 943, 959, 972, 973, 995

Offset: 1

Sergio Falcon, Aug 07 2020

			12 --> 1^4+2^4 = 17 --> 1^4+7^4 = 2402 --> 2^4+4^4+0^4+2^4 = 288 --> 2^4+8^4+8^4 = 8208.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A219111, A031185, A031184.

V:= Vector(32805): V[8208]:= true:
g:= proc(n) local L, t;
  add(t^4, t = convert(n,base,10))
end proc:
f:= proc(n) local x,S; global V;
  if n <= 32805 then
     if V[n] <> 0 then return V[n]
     else S:= [n]
     fi
  else S:= []
  fi;
  x:= n;
  do
    x:= g(x);
    if V[x] <> 0 then
       V[S]:= V[x];
       return V[x]
    elif member(x,S) then
       V[S]:= false;
       return false
    fi;
    if x <= 32805 then S:= [op(S), x] fi;
  od;
end proc;
select(f, [$1..10000]); # Robert Israel, Sep 03 2020

okQ[n] := MemberQ[NestList[Total[IntegerDigits[#]^4]&, n, 30], 8208]; Select[Range[1000], okQ]

A282756 Let F(k,n) = k*F(k,n-1) + F(k,n-2) with initial conditions F(k,0) = 0, F(k,1) = 1. Sequence lists the minimum 'n' such that F(k,n) > k^n.

Original entry on oeis.org

3, 6, 14, 27, 45, 70, 101, 139, 184, 236, 296, 364, 440, 524, 616, 716, 826, 943, 1070, 1205, 1350, 1503, 1666, 1838, 2019, 2210, 2410, 2620, 2839, 3069, 3308, 3557, 3815, 4084, 4363, 4652, 4951, 5261, 5580, 5910, 6251, 6601, 6963, 7334, 7717, 8110, 8513, 8928, 9353, 9788

Offset: 1

Sergio Falcon, Feb 21 2017

nonn

			F(1,3) = 2 > 1^1;
F(2,6) = 70 > 2^6 = 64;
F(3,14) = 5097243 > 3^14 = 4782969;
...

Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons and Fractals, Elsevier, 32 (5), 1615 - 1624, 2007.

Cf. A000045, A000129, A006190.

f[k_, n_] := Fibonacci[n, k]
Do[Do[If[f[k, n] > k^n, {Print[{k, n}], Break[]}], {n, 0, 10000}], {k, 50}]

A238391 Expansion of (1+x)/(1-x^2-3*x^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 4, 7, 7, 10, 19, 22, 40, 43, 70, 100, 136, 220, 265, 430, 565, 838, 1225, 1633, 2515, 3328, 5029, 7003, 9928, 14548, 19912, 29635, 40921, 59419, 84565, 119155, 173470, 241918, 351727, 495613, 709192, 1016023, 1434946, 2071204, 2921785, 4198780, 5969854, 8503618, 12183466, 17268973, 24779806

Offset: 0

Sergio Falcon, Feb 26 2014

			a(5) = 3*a(0)+a(3)=4; a(6) = 3*a(1)+a(4)=4; a(7) = 3*a(2)+a(5)=7.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 [Terms 0 through 500 were computed by G. C. Greubel]
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,3).

For[j = 0, j < 5, j++, a[j] = 1]; For[j = 5, j < 51, j++, a[j] = 3 a[j - 5] + a[j - 2]]; Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 3 x^5), {x, 0, 50}], x] (* Michael De Vlieger, Jan 27 2016 *)

Vec((1+x)/(1-x^2-3*x^5) + O(x^50)) \\ Michel Marcus, Jan 27 2016

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1; a(n) = 3*a(n-5)+a(n-2) for n>4.
a(2n) = Sum_{j=0..n/5} binomial(n-3j,2j)*3^(2j) + Sum_{j=0..(n-3)/5} binomial(n-2-3j,2j+1)*3^(2j+1).
a(2n+1) = Sum_{j=0..n/5} binomial(n-3j,2j)*3^{2j} + Sum_{j=0..(n-2)/5} binomial(n-1-3j,2j+1)*3^(2j+1).

A237714 Expansion of (1 + x)/(1 - x^2 - 2*x^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 11, 13, 21, 23, 35, 45, 61, 87, 107, 157, 197, 279, 371, 493, 685, 887, 1243, 1629, 2229, 2999, 4003, 5485, 7261, 9943, 13259, 17949, 24229, 32471, 44115, 58989, 80013, 107447, 144955, 195677, 262933, 355703, 477827, 645613, 869181, 1171479, 1580587

Offset: 0

Sergio Falcon, Feb 12 2014

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

Cf. A000129, A122522, A159284.

For[j = 0, j < 5, j++, a[j] = 1]
For[j = 5, j < 51, j++, a[j] = 2 a[j - 5] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{0,1,0,0,2},{1,1,1,1,1},70] (* Harvey P. Dale, Nov 24 2024 *)

Vec( (1 + x)/(1 - x^2 - 2*x^5) + O(x^66) ) \\ Joerg Arndt, Feb 24 2014

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(n) = 2*a(n-5) + a(n-2) for n>=5.
a(2n) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-3)/5} C(n-2-3j,2j+1)*2^(2j+1).
a(2n+1) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-2)/5} C(n-1-3j,2j+1)*2^(2j+1).

A238389 Expansion of (1+x)/(1-x^2-3*x^3).

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 16, 19, 37, 67, 94, 178, 295, 460, 829, 1345, 2209, 3832, 6244, 10459, 17740, 29191, 49117, 82411, 136690, 229762, 383923, 639832, 1073209, 1791601, 2992705, 5011228, 8367508, 13989343, 23401192, 39091867, 65369221, 109295443

Offset: 0

Sergio Falcon, Feb 26 2014

			a(3) = 3*a(0)+a(1) = 4; a(4) = 3*a(1)+a(2) = 4; a(5) = 3*a(2)+a(3) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,3).

Cf. A006190, A134816, A159284, A213713.

[n le 3 select 1 else Self(n-2) +3*Self(n-3): n in [1..41]]; // G. C. Greubel, May 09 2021

a:= n-> (<<0|1|0>, <0|0|1>, <3|1|0>>^n.<<(1$3)>>)[(1$2)]:
seq(a(n), n=0..44);  # Alois P. Heinz, May 09 2021

(* First program *)
For[j=0, j<3, j++, a[j] = 1]
For[j=3, j<51, j++, a[j] = 3a[j-3] + a[j-2]]
Table[a[j], {j, 0, 50}]
(* Second program *)
CoefficientList[Series[(1+x)/(1-x^2-3x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)
LinearRecurrence[{0,1,3},{1,1,1},40] (* Harvey P. Dale, Feb 28 2023 *)

Vec((1+x)/(1-x^2-3*x^3)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

def A238389_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/(1-x^2-3*x^3) ).list()
A238389_list(40) # G. C. Greubel, May 09 2021

a(0)=1, a(1)=1, a(2)=1; for n>2, a(n) = a(n-2) + 3*a(n-3).
a(2n) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-2)/3} binomial(n-1-j,2j+1)*3^(2j+1).
a(2n+1) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-1)/3} binomial(n-j,2j+1)*3^(2j+1).
a(n) = |A106855(n)| + |A106855(n-1)| . - R. J. Mathar, Mar 13 2014

Terms corrected by Charles R Greathouse IV, Mar 06 2014

A237718 9-distance Pell numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 15, 17, 25, 27, 39, 41, 57, 59, 79, 89, 113, 139, 167, 217, 249, 331, 367, 489, 545, 715, 823, 1049, 1257, 1547, 1919, 2281, 2897, 3371, 4327, 5017, 6425, 7531, 9519

Offset: 0

Sergio Falcon, Feb 12 2014

			a(9)=2a(0)+a(7)=3; a(10)=2a(1)+a(8)=3; a(11)=2a(2)+a(9)=5.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,0,0,2).

Cf. A000129, A122522, A159284, A237714, A237716, A237717.

For[j = 0, j < 9, j++, a[j] = 1]
For[j = 9, j < 51, j++, a[j] = 2 a[j - 9] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^9), {x,0,50}], x] (* G. C. Greubel, May 01 2017 *)

Vec((1+x)/(1-x^2-2*x^9)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=1, a(8)=1; a(n) = 2*a(n-9) + a(n-2) for n>=9.
G.f. (1+x)/(1-x^2-2x^9).
a(2*n) = Sum_{j=0..n/9} Binomial[n-7j, 2j]*2^{2j} + Sum_{j=0..(n-5)/9} Binomial[n-4-7j, 2j+1]*2^{2j+1}.
a(2*n+1) = Sum_{j=0..n/9} Binomial[n-7j, 2j]*2^{2j} + Sum_{j=0..(n-4)/9} Binomial[n-3-7j, 2j+1]*2^{2j+1}.

A237716 7-distance Pell sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 13, 15, 23, 25, 37, 39, 55, 65, 85, 111, 135, 185, 213, 295, 343, 465, 565, 735, 935, 1161, 1525, 1847, 2455, 2977, 3925, 4847, 6247, 7897, 9941, 12807, 15895, 20657, 25589, 33151, 41383, 53033, 66997

Offset: 0

Sergio Falcon, Feb 12 2014

			a(7)=2a(0)+a(5)=3; a(8)=2a(1)+a(6)=3; a(9)=2a(2)+a(7)=5.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,2).

Cf. A000129, A122522, A159284, A237714.

For[j = 0, j < 7, j++, a[j] = 1]
For[j = 7, j < 51, j++, a[j] = 2 a[j - 7] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^7), {x,0,50}], x] (* G. C. Greubel, May 01 2017 *)

Vec((1+x)/(1-x^2-2*x^7)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1; a(n) = 2*a(n-7) + a(n-2) for n>=7.
G.f.: (1 + x)/(1 - x^2 - 2*x^7).
a(2*n) = Sum_{j=0..n/7} binomial(n-5*j, 2*j)*2^(2*j) + Sum_{j=0..(n-4)/7} binomial(n-3-5*j, 2*j+1)*2^(2*j+1).
a(2*n+1) = Sum_{j=0..n/7} binomial(n-5*j, 2*j)*2^(2*j) + Sum_{j=0..(n-3)/7} binomial(n-2-5*j, 2*j+1)*2^(2*j+1).

A143646 Catalan transform of the 3-Fibonacci sequence A006190.

Original entry on oeis.org

0, 1, 4, 18, 83, 387, 1815, 8541, 40276, 190182, 898844, 4250780, 20111394, 95181166, 450565602, 2133227418, 10101126723, 47834649675, 226540406571, 1072931019393, 5081776592061, 24069823974879, 114009427284309

Offset: 0

Sergio Falcon, Oct 27 2008

nonn

Paul Barry, A Catalan Transform and Related Transformations of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.

Cf. A006190, A143464.

q=50 #change q for more terms
[0]+[sum((k/n)*binomial(2*n-k-1,n-k)*lucas_number1(k,3,-1) for k in [0..n]) for n in [1..q]] # Tom Edgar, Mar 09 2014

a(n) = Sum_{k=0..n} A039599(n,k)*A006130(k-1) with A006130(-1) = 0. - Philippe Deléham, Nov 01 2008

For n>0, a(n) = sum_{k=0..n} (k/n)*C(2n-k-1,n-k)*A006190(k). - Tom Edgar, Mar 09 2014

A143464 Catalan transform of the Pell sequence.

Original entry on oeis.org

0, 1, 3, 11, 42, 164, 649, 2591, 10408, 41998, 170050, 690370, 2808714, 11446642, 46715469, 190876527, 780679200, 3195628806, 13090353594, 53655587034, 220045073988, 902842397664, 3705876933930, 15216954519222, 62503485455208

Offset: 0

Sergio Falcon, Oct 24 2008

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
Sergio Falcón, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals 2007; 33(1): 38-49.
Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
Merve Taştan and Engin Özkan, Catalan transform of the k-Pell, k-Pell-Lucas and modified k-Pell sequence, Notes on Num. Theory and Disc. Math. (2021) Vol. 27, No. 1, 198-207.

Cf. A000035, A000129, A016116, A039599, A106566, A109262.

a[n_]:= a[n]= If[n==0, 0, Sum[i*Binomial[2n-i,n-i]*Fibonacci[i,2]/(2n-i), {i,n}]];
Table[a[n], {n,0,30}] (* modified by G. C. Greubel, May 31 2022 *)

my(x='x+O('x^66)); concat([0],Vec((1-5*x-(1+x)*sqrt(1-4*x))/(2*x^2+16*x-4))) \\ Joerg Arndt, May 01 2013

def Pell(n): return round( ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2)) )
[0]+[(1/n)*sum(k*binomial(2*n-k-1, n-1)*Pell(k) for k in (1..n)) for n in (1..30)] # G. C. Greubel, May 31 2022

a(n) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n)*Pell(j), with a(0) = 0.
From Philippe Deléham, Oct 28 2008: (Start)
a(n) = Sum_{k=0..n} A106566(n,k)*A000129(k).
a(n) = Sum_{k=0..n} A039599(n,k)*A000035(k)*A016116(k). (End)
G.f.: ((1+x)*sqrt(1-4*x) - (1-5*x))/(2*(2 - 8*x - x^2)). - Mark van Hoeij, May 01 2013
a(n) = (1/(2*sqrt(2)))*Catalan(n-1)*Sum_{j=0..1} ((-1)^j + sqrt(2)) * Hypergeometric2F1([2,1-n], [2*(1-n)], 1+(-1)^j*sqrt(2)) - [n=0]/2. - G. C. Greubel, May 31 2022
a(n) ~ (1 + sqrt(2))^(2*n - 1) / 2^(2 + n/2). - Vaclav Kotesovec, May 31 2022

Offset corrected by Philippe Deléham, Oct 28 2008

A139798 Coefficient of x^5 in (1-x-x^2)^(-n).

Original entry on oeis.org

8, 38, 111, 256, 511, 924, 1554, 2472, 3762, 5522, 7865, 10920, 14833, 19768, 25908, 33456, 42636, 53694, 66899, 82544, 100947, 122452, 147430, 176280, 209430, 247338, 290493, 339416, 394661, 456816, 526504, 604384, 691152, 787542

Offset: 1

Sergio Falcon, May 22 2008

The coefficient of x^5 in (1-x-x^2)^(-n) is the coefficient of x^5 in (1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5)^n. Using the multinomial theorem one then finds that a(n) = n(n+1)(n+2)(n^2 + 27n + 132)/5!

The inverse binomial transform yields 8,30,43,29,9,1,0,0,... (0 continued) - R. J. Mathar, May 23 2008

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000096, A006503, A006504.

a[n_] := n(n + 1)(n + 2)(n^2 + 27n + 132)/5! Do[Print[n, " ", a[n]], {n, 1, 25}]
LinearRecurrence[{6,-15,20,-15,6,-1},{8,38,111,256,511,924},40] (* Harvey P. Dale, Oct 13 2015 *)

a(n)=binomial(n+2,3)*(n^2+27*n+132)/20 \\ Charles R Greathouse IV, Jul 29 2011

a(n) = n(n+1)(n+2)(n^2 + 27n + 132)/5!

O.g.f.: x(3x-4)(x-2)/(1-x)^6. - R. J. Mathar, May 23 2008

More terms from R. J. Mathar, May 23 2008

cp's OEIS Frontend

User: Sergio Falcon

A336890 Numbers that eventually reach the fixed point 8208 under "x --> sum of the fourth powers of digits of x".

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A282756 Let F(k,n) = k*F(k,n-1) + F(k,n-2) with initial conditions F(k,0) = 0, F(k,1) = 1. Sequence lists the minimum 'n' such that F(k,n) > k^n.

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A238391 Expansion of (1+x)/(1-x^2-3*x^5).

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A237714 Expansion of (1 + x)/(1 - x^2 - 2*x^5).

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A238389 Expansion of (1+x)/(1-x^2-3*x^3).

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Magma

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A237718 9-distance Pell numbers.

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A237716 7-distance Pell sequence.

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A143646 Catalan transform of the 3-Fibonacci sequence A006190.

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