A103609 -id:A103609 - cp's OEIS frontend

A101706 Numbers n such that reversal(n)=(7/3)*n.

0, 3267, 32967, 329967, 3299967, 32673267, 32999967, 326703267, 329999967, 3267003267, 3296732967, 3299999967, 32670003267, 32967032967, 32999999967, 326700003267, 326732673267, 329670032967, 329967329967, 329999999967, 3267000003267, 3267329673267, 3296700032967, 3299670329967, 3299999999967

Offset: 1

Farideh Firoozbakht, Jan 01 2005

If m is in the sequence then all numbers of the form g(m,s,t) for nonnagative integers s and t are in the sequence (the function g has been defined in the sequence A101704), for example g(3267,1,1)= 326703267 is in the sequence. If n=0 or n>1 then 33*(10^n-1) is in the sequence.

There are Fibonacci(floor((n-2)/2)) terms with n digits, n>1 (this is essentially A103609). - Ray Chandler, Oct 12 2017

			g(3267,10,2) = 32670000000000326700000000003267 is in the sequence
because reversal(32670000000000326700000000003267) =
76230000000000762300000000007623 =
(7/3)*32670000000000326700000000003267, g(3267,0,4) =
32673267326732673267 is in the sequence because
reversal(32673267326732673267) = 76237623762376237623 =
(7/3)*32673267326732673267.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A101704, A101705, A001232, A008918.

Do[If[FromDigits[Reverse[IntegerDigits[n]]] == (7/3)*n, Print[n]], {n, 100000000}]

Terms a(8) onward from Max Alekseyev, Aug 18 2013

A094707 Partial sums of repeated Fibonacci sequence.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 32, 40, 53, 66, 87, 108, 142, 176, 231, 286, 375, 464, 608, 752, 985, 1218, 1595, 1972, 2582, 3192, 4179, 5166, 6763, 8360, 10944, 13528, 17709, 21890, 28655, 35420, 46366, 57312, 75023, 92734, 121391, 150048, 196416

Offset: 0

Paul Barry, May 21 2004

Equals row sums of triangle A139147 starting with "1". - Gary W. Adamson, Apr 11 2008

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

Cf. A000045, A103609, A131524, A139147.

[Fibonacci(Floor((n+6)/2))*((n+1) mod 2) + 2*Fibonacci(Floor((n+3)/2))*(n mod 2) - 2: n in [0..60]]; // G. C. Greubel, Feb 12 2023

LinearRecurrence[{1,1,-1,1,-1}, {0,0,1,2,3}, 50] (* Jean-François Alcover, Nov 18 2017 *)

def A094707(n): return fibonacci((n+6)//2) - 2 if (n%2==0) else 2*fibonacci((n+3)//2) - 2
[A094707(n) for n in range(61)] # G. C. Greubel, Feb 12 2023

G.f. : x^2*(1+x)/((1-x)*(1-x^2-x^4)).
a(n) = a(n-1) + a(n+2) - a(n-3) + a(n-4) - a(n-5).
a(n) = Sum_{k=0..n} Fibonacci(floor(k/2)).
a(n) = -2 - (sqrt(5)/2 - 1/2)^(n/2)*((2*sqrt(5)/5 - 1)*cos(Pi*n/2) + sqrt(4*sqrt(5)/5 - 8/5)*sin(Pi*n/2)) - (sqrt(5)/2 + 1/2)^(n/2)*((sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5 - 1/2)*(-1)^n - sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5-1/2).
a(n) = A131524(n) + A131524(n+1). - R. J. Mathar, Jul 07 2011
a(n) = Fibonacci(n/2 +3) - 2 if n even, otherwise a(n) =  2*Fibonacci((n-1)/2 + 2) - 2. - G. C. Greubel, Feb 12 2023

A099574 Diagonal sums of triangle A099573.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 9, 11, 18, 23, 37, 48, 74, 97, 147, 195, 290, 387, 568, 763, 1108, 1495, 2152, 2915, 4167, 5662, 8047, 10962, 15506, 21168, 29825, 40787, 57280, 78448, 109870, 150657, 210521, 288969, 403020, 553677, 770963, 1059932, 1473898

Offset: 0

Paul Barry, Oct 23 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A003269, A079977, A099573, A099577, A103609.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^4)/((1-x^2-x^4)*(1-x-x^4)) )); // G. C. Greubel, Jul 25 2022

a[n_]:= a[n]= Sum[Binomial[n-k-j, j], {k,0,Floor[n/2]}, {j,0,Floor[k/2]}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jul 25 2022 *)

@CachedFunction
def A099574(n): return sum(sum(binomial(n-k-j, j) for j in (0..(k//2))) for k in (0..(n//2)))
[A099574(n) for n in (0..40)] # G. C. Greubel, Jul 25 2022

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor(k/2)} binomial(n-k-j, j).
G.f.: (1-x)*(1+x)*(1+x^2) / ( (1-x-x^4)*(1-x^2-x^4) ). - R. J. Mathar, Nov 11 2014
From G. C. Greubel, Jul 25 2022: (Start)
a(n) = A003269(n+5) - A079977(n+3) - A079977(n+2).
a(n) = A003269(n+5) - A103609(n+5). (End)

A173951 Positive integers with the property that if the base-3 representation is reversed the result is twice the original number.

Original entry on oeis.org

32, 104, 320, 968, 2624, 2912, 7808, 8744, 23360, 25376, 26240, 70016, 75920, 78728, 209984, 212576, 227552, 233600, 236192, 629888, 638312, 682448, 700160, 708584, 1889600, 1897376, 1915520, 2047136, 2054912, 2099840, 2117984, 2125760

Offset: 1

John W. Layman, Mar 03 2010

The number of terms of this sequence containing n ternary digits is given by {d(n)}={0,0,0,1,1,1,1,2,2,3,3,5,5,8,8,13,13,21,...} for n=1,2,3,... and thus appears to be essentially the doubling-up of the Fibonacci numbers A103609. For example, 2624 = 10121012(base-3) and 2912 = 10222212(base-3) are the only two terms that have 8 digits when written in base 3, so d(8)=2.
(This conjecture is correct - see A223077. - N. J. A. Sloane, Mar 19 2013)
All terms of sequence A173952, defined by b(1)=32 and, for n>1, b(n)=9*b(n-1)+32, appear to be terms of the above sequence {a(n)}; in fact each term b(n) appears to be the largest term of {a(k)} that has 2n+2 digits when written in base 3.

Cf. A103609, A173952, A223077, A214927.

A129361 a(n) = Sum_{k=floor((n+1)/2)..n} Fibonacci(k+1).

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 29, 47, 81, 131, 220, 356, 589, 953, 1563, 2529, 4126, 6676, 10857, 17567, 28513, 46135, 74792, 121016, 196041, 317201, 513619, 831053, 1345282, 2176712, 3522981, 5700303, 9224881, 14926171, 24153636, 39081404, 63239221, 102323209

Offset: 0

Paul Barry, Apr 11 2007

			  1 =  1.
  1 =  1.
  1 +  2 =  3.
  2 +  3 =  5.
  2 +  3 +  5 = 10.
  3 +  5 +  8 = 16.
  3 +  5 +  8 + 13 = 29.
  5 +  8 + 13 + 21 = 47.
  5 +  8 + 13 + 21 + 34 =  81.
  8 + 13 + 21 + 34 + 55 = 131.
  8 + 13 + 21 + 34 + 55 +  89 = 220.

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

Cf. A000045, A103609, A129362.

I:=[1,1,3,5,10,16]; [n le 6 select I[n] else Self(n-1) +2*Self(n-2)-Self(n-3)-Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Mar 01 2014

a[n_]:= Sum[Fibonacci@k, {k, Floor[(n + 3)/2], n + 1}]; Array[a, 33, 0] (* Robert G. Wilson v, Mar 15 2011 *)
Table[Sum[Fibonacci[n - i + 2], {i, Floor[(n + 2)/2]}], {n, 0, 50}] (* Wesley Ivan Hurt, Feb 25 2014 *)
LinearRecurrence[{1,2,-1,0,-1,-1},{1,1,3,5,10,16},40] (* Harvey P. Dale, Feb 02 2019 *)

Vec( (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x^2-x^4)) +O(x^66) ) \\ Joerg Arndt, Mar 01 2014

[sum(fibonacci(n-j+2) for j in range(1,2+(n//2))) for n in range(51)] # G. C. Greubel, Jan 31 2024

G.f.: (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x^2-x^4)).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6).
a(n) = Sum_{k=0..n} ( F(k+1) - F((k+1)/2)*(1-(-1)^k)/2 ).
a(n) = A000045(n+3) - A103609(n+5). - R. J. Mathar, Mar 15 2011

More terms from Vincenzo Librandi, Mar 01 2014

A223077 Positive integers, written in base 3, with the property that if the base-3 representation is reversed the result is twice the original number.

Original entry on oeis.org

1012, 10212, 102212, 1022212, 10121012, 10222212, 101201012, 102222212, 1012001012, 1021210212, 1022222212, 10120001012, 10212010212, 10222222212, 101200001012, 101210121012, 102120010212, 102212102212, 102222222212, 1012000001012, 1012102121012, 1021200010212, 1022120102212, 1022222222212

Offset: 1

N. J. A. Sloane, Mar 14 2013

For the decimal representations of these same numbers see A173951.

Theorem: The number of terms of length n is equal to A103609(n-2).

N. J. A. Sloane, Table of n, a(n) for n = 1..108

Cf. A173951, A214927.

b3rQ[n_]:=FromDigits[Reverse[IntegerDigits[n]],3]/FromDigits[ IntegerDigits[ n],3] ==2; Select[FromDigits/@Tuples[{0,1,2},13],b3rQ]//Quiet (* Harvey P. Dale, Jun 10 2018 *)

A071685 Non-palindromic numbers n, not divisible by 10, such that either n divides R(n) or R(n) divides n, where R(n) is the digit-reversal of n.

Original entry on oeis.org

1089, 2178, 8712, 9801, 10989, 21978, 87912, 98901, 109989, 219978, 879912, 989901, 1099989, 2199978, 8799912, 9899901, 10891089, 10999989, 21782178, 21999978, 87128712, 87999912, 98019801, 98999901, 108901089, 109999989

Offset: 1

Labos Elemer, Jun 03 2002

The quotient R(n)/n or n/R(n) is always 4 or 9.
This is the union of the four sequence A001232, A222814, A008918, A222815. Equivalently, the union of A008919 and A031877.
There are 4*Fibonacci(floor((n-2)/2)) terms with n digits (this is 2*A214927 or essentially 4*A103609). - Ray Chandler, Oct 12 2017
Conjecture: every term mod 100 is equal to 1, 12, 78, or 89. - Harvey P. Dale, Dec 13 2017

			Palindromic solutions like 12021 or also solutions divisible by 10 were filtered out like {8380,838; q=10} or {8400,48; q=175}. In case of m>R(m), q=m/R(m)=4 or 9.

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, 2178 And All That, arXiv:1307.0453 [math.NT], 2013; Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Cf. A001232, A222814, A008918, A222815, A008919, A031877.

Cf. A004086, A055483, A069554, A103609, A214927.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[IntegerDigits[x]] Do[s=Mod[Max[{n, tn[red[n]]}], Min[{n, r=tn[red[n]]}]]; If[Equal[s, 0]&&!Equal[Mod[n, 10], 0] &&!Equal[n, r], Print[{n, r/n}]], {n, 1, 1000000}]
npnQ[n_]:=Module[{r=IntegerReverse[n]},!PalindromeQ[n]&&!Divisible[ n,10] &&(Mod[n,r]==0||Mod[r,n]==0)]; Select[Range[11*10^7],npnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2017 *)

x = q*R(x), q is an integer q<>1, q<>10^j and neither of x or R(x) is divisible by 10.

Corrected and extended by Harvey P. Dale, Jul 01 2013
Edited by N. J. A. Sloane, Jul 02 2013
Missing terms inserted by Ray Chandler, Oct 09 2017
Incorrect comment removed by Ray Chandler, Oct 12 2017

A114199 Row sums of a Pascal-Fibonacci triangle.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 87, 200, 458, 1044, 2373, 5388, 12233, 27782, 63112, 143392, 325805, 740266, 1681935, 3821412, 8682310, 19726316, 44818473, 101828344, 231355953, 525645354, 1194276812, 2713420728, 6164945513, 14006877390

Offset: 0

Paul Barry, Nov 16 2005

Binomial transform of double Fibonacci sequence A103609(n+2). Row sums of A114197.

Michael De Vlieger, Table of n, a(n) for n = 0..2806
Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,1).

Cf. A000045, A103609, A114197.

[n le 4 select 2^(n-1) else 4*Self(n-1) -5*Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..30]]; // G. C. Greubel, Oct 23 2024

LinearRecurrence[{4,-5,2,1},{1,2,4,8},30] (* Harvey P. Dale, Dec 07 2015 *)

@CachedFunction # a = A114199
def a(n): return 2^n if n<4 else 4*a(n-1) -5*a(n-2) +2*a(n-3) +a(n-4)
[a(n) for n in range(71)] # G. C. Greubel, Oct 23 2024

G.f.: (1-x)^2/(1-4*x+5*x^2-2*x^3-x^4).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n-k, j)*C(k, j)*Fibonacci(j).
a(n) = Sum_{k=0..n} C(n, k)*Fibonacci(floor((k+2)/2)).

A174618 For n odd a(n) = a(n-2) + a(n-3), for n even a(n) = a(n-2) + a(n-5); with a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 10, 11, 17, 17, 28, 27, 45, 44, 72, 72, 116, 117, 188, 189, 305, 305, 494, 493, 799, 798, 1292, 1292, 2090, 2091, 3382, 3383, 5473, 5473, 8856, 8855, 14329, 14328, 23184, 23184, 37512, 37513, 60696

Offset: 1

Mark Dols, Mar 23 2010

Combination a(2n)=A005252(n-1) and a(2n+1)=A024490(n). Consecutive pairs add up to A000045 and subtract to A010892. If a(1)= 1 formula gives: A103609.

			As consecutive pairs: (0,1),(0,1),(1,1),(2,1),(3,2),(4,4),...

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

Cf. A000045, A005252, A010892, A024490, A079977, A110161.

R:=PowerSeriesRing(Integers(), 70);
[0] cat Coefficients(R!( x^2*(1-x^2+x^3)/((1-x^2+x^4)*(1-x^2-x^4)) )); // G. C. Greubel, Oct 23 2024

nxt[{n_,a_,b_,c_,d_,e_}]:={n+1,b,c,d,e,If[EvenQ[n],d+c,d+a]}; NestList[nxt,{5,0,1,0,1,1},50][[All,2]] (* or *) LinearRecurrence[ {0,2,0,-1,0,0,0,1},{0,1,0,1,1,1,2,1},60] (* Harvey P. Dale, Nov 15 2019 *)

def A174618(n): return (kronecker(12,n-3) - kronecker(12,n-2) + ((n+1)%2)*fibonacci(n//2) + (n%2)*fibonacci((n+1)//2))//2
[A174618(n) for n in range(1,71)] # G. C. Greubel, Oct 23 2024

G.f.: x^2*(1-x^2+x^3) / ( (1-x^2+x^4)*(1-x^2-x^4) ). - R. J. Mathar, Jan 27 2011

a(n) = (1/2)*(A110161(n-3) - A110161(n-2) + A079977(n-2) + A079977(n-1)). - G. C. Greubel, Oct 23 2024

A169592 Expansion of 1/((1-x)(1-x^2-x^4)) + x/(1-3x^3).

Original entry on oeis.org

1, 2, 2, 2, 7, 4, 7, 16, 12, 12, 47, 20, 33, 114, 54, 54, 331, 88, 143, 872, 232, 232, 2563, 376, 609, 7170, 986, 986, 21279, 1596, 2583, 61632, 4180, 4180, 183911, 6764, 10945, 542386, 17710, 17710, 1622979, 28656, 46367, 4829336, 75024, 75024, 14470299, 121392

Offset: 0

Roger L. Bagula, Dec 02 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 303.
Index entries for linear recurrences with constant coefficients, signature (1,1,2,-2,-4,3,-3,3).

Cf. A103609.

R:=PowerSeriesRing(Integers(), 60);
Coefficients(R!( 1/((1-x)*(1-x^2-x^4)) +x/(1-3*x^3) )); // G. C. Greubel, Oct 23 2024

p[t_]= 1/((1-t)*(1-t^2-t^4)) + t/(1-3*t^3);
CoefficientList[ Series[p[t], {t, 0, 60}], t]

def A169592_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^2-x^4)) +x/(1-3*x^3) ).list()
A169592_list(60) # G. C. Greubel, Oct 23 2024

G.f.: (1+x+x^4-x^5+x^6-x^2-4*x^3)/((1-x)*(1-3*x^3)*(1-x^2-x^4)).
a(n) = +a(n-1) +a(n-2) +2*a(n-3) -2*a(n-4) -4*a(n-5) +3*a(n-6) -3*a(n-7) 3*a(n-8).
a(n) = A103609(n+6) - 1 + 3^((n-1)/3) if n == 1 mod 3.
a(n) = A103609(n+6) - 1 if n == 0 or 2 mod 3.

Notation adapted to OEIS standards, sequence extended, formulas added by the Assoc. Editors of the OEIS [Dec 05 2009]

cp's OEIS Frontend

A101706 Numbers n such that reversal(n)=(7/3)*n.

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A094707 Partial sums of repeated Fibonacci sequence.

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A099574 Diagonal sums of triangle A099573.

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A173951 Positive integers with the property that if the base-3 representation is reversed the result is twice the original number.

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A129361 a(n) = Sum_{k=floor((n+1)/2)..n} Fibonacci(k+1).

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A223077 Positive integers, written in base 3, with the property that if the base-3 representation is reversed the result is twice the original number.

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A071685 Non-palindromic numbers n, not divisible by 10, such that either n divides R(n) or R(n) divides n, where R(n) is the digit-reversal of n.

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A114199 Row sums of a Pascal-Fibonacci triangle.

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A169592 Expansion of 1/((1-x)(1-x^2-x^4)) + x/(1-3x^3).