A019523 -id:A019523 - cp's OEIS frontend

A038399 Concatenate first n nonzero Fibonacci numbers in reverse order.

1, 11, 211, 3211, 53211, 853211, 13853211, 2113853211, 342113853211, 55342113853211, 8955342113853211, 1448955342113853211, 2331448955342113853211, 3772331448955342113853211, 6103772331448955342113853211, 9876103772331448955342113853211

Offset: 1

M. I. Petrescu (mipetrescu(AT)yahoo.com)

Mihaly Bencze [Beneze], L. Tutescu, Some Notions and Questions in Number Theory, Sequence 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..96

Cf. A000045, A000422, A019523.

a038399 n = a038399_list !! (n-1)
a038399_list = h "" $ tail a000045_list where
   h xs (f:fs) = (read ys :: Integer) : h ys fs
     where ys = show f ++ xs
-- Reinhard Zumkeller, Mar 01 2014

Module[{nn=20,fibs},fibs=Fibonacci[Range[nn]];Table[FromDigits[ Flatten[ IntegerDigits/@ Reverse[Take[fibs,n]]]],{n,nn}]] (* Harvey P. Dale, Aug 30 2016 *)

a(n) = my(t=fibonacci(n)); forstep(k=n-1, 1, -1, t=t*10^#Str(fibonacci(k))+fibonacci(k)); t; \\ Michel Marcus, Apr 06 2024

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000
Offset changed by Reinhard Zumkeller, Mar 01 2014
More terms from Harvey P. Dale, Aug 30 2016

A094704 Convolution of Fibonacci(n) and 10^n.

Original entry on oeis.org

0, 1, 11, 112, 1123, 11235, 112358, 1123593, 11235951, 112359544, 1123595495, 11235955039, 112359550534, 1123595505573, 11235955056107, 112359550561680, 1123595505617787, 11235955056179467, 112359550561797254

Offset: 0

Paul Barry, May 21 2004

The convolution of Fibonacci(n) and k^n for k > 1 has a(n) = (1/k^2 - k -1))*(2*k^(n+1) - k*Lucas(n) - (k+2)*Fibonacci(n)).

G. C. Greubel, Table of n, a(n) for n = 0..990
Index entries for linear recurrences with constant coefficients, signature (11,-9,-10).

Cf. A000032, A000045, A019523.

[(10^(n+1) -Fibonacci(n+3) -8*Fibonacci(n+1))/89: n in [0..30]]; // G. C. Greubel, Feb 09 2023

CoefficientList[Series[x/((1-x-x^2)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[ {11,-9,-10},{0,1,11},20] (* Harvey P. Dale, Mar 18 2013 *)

def A094704(n): return (10^(n+1) -fibonacci(n+3) -8*fibonacci(n+1))/89
[A094704(n) for n in range(31)] # G. C. Greubel, Feb 09 2023

G.f. : x/((1-10*x)*(1-x-x^2)).
a(n) = (1/89)*( 10^(n+1) - 5*( ((1 + sqrt(5))/2)^n + ((1 - sqrt(5))/2)^n ) - (6/sqrt(5))*( ((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n ) ).
a(n) = 10*a(n-1) + Fibonacci(n) for n >= 1. - Mark Dols, Aug 31 2009
a(n) = 11*a(n-1) - 9*a(n-2) - 10*a(n-3), n > 2. - Harvey P. Dale, Mar 18 2013
a(n) = (1/89)*( 10^(n+1) - Fibonacci(n+3) - 8*Fibonacci(n+1) ). - G. C. Greubel, Feb 09 2023

A134069 Primes in A038399.

Original entry on oeis.org

11, 211, 853211

Offset: 1

Alexander Adamchuk, Oct 06 2007

A038399 = {1, 11, 211, 3211, 53211, 853211, 13853211, 2113853211, ...} = concatenation of first n nonzero Fibonacci numbers in reverse order.
From Robert Israel, Sep 14 2016: (Start)
No more terms < A038399(500), which has 26252 digits.
The probability of a number of the order of magnitude of A038399(n) being prime is approximately constant/n^2. Since Sum_n 1/n^2 converges, we should expect this sequence to be finite. (End)

Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).

Cf. A019523 (concatenation of Fibonacci numbers).

count:= 0:
A038399[1]:= 1:
for n from 2 to 100 do
A038399[n]:= combinat:-fibonacci(n)*10^(1+ilog10(A038399[n-1]))+A038399[n-1];
   if isprime(A038399[n]) then count:= count+1; A[count]:= A038399[n] fi
od:
seq(A[i],i=1..count); # Robert Israel, Sep 14 2016

Module[{nn=10,bif},bif=Fibonacci[Range[nn]];Select[Table[FromDigits[ Flatten[ IntegerDigits/@Reverse[Take[bif,n]]]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Sep 27 2019 *)

A134072 Concatenation of A000204 Lucas numbers (beginning at 1) in reverse order.

Original entry on oeis.org

1, 31, 431, 7431, 117431, 18117431, 2918117431, 472918117431, 76472918117431, 12376472918117431, 19912376472918117431, 32219912376472918117431, 52132219912376472918117431, 84352132219912376472918117431

Offset: 1

Alexander Adamchuk, Oct 06 2007

Indices of prime terms are {2, 3, 5, 11, ...}. Primes are listed in A134071 = {31, 431, 117431, 19912376472918117431, ...}.

Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A000204 (Lucas numbers).
Cf. A130774 (concatenation of Lucas numbers).
Cf. A019523 (concatenation of Fibonacci numbers).
Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).
Cf. A134069 (primes in A038399).
Cf. A134070 (primes in A130774).
Cf. A134071 (primes in A134072).

Module[{nn=20,lnos},lnos=LucasL[Range[nn]];Table[FromDigits[Flatten[ IntegerDigits/@ Reverse[Take[lnos,n]]]],{n,nn}]] (* Harvey P. Dale, Jul 27 2015 *)

Edited by Charles R Greathouse IV, Apr 26 2010

A134732 Concatenation of first n members of the Padovan sequence A000931, starting at (1, 1, 1, 2,).

Original entry on oeis.org

1, 11, 111, 1112, 11122, 111223, 1112234, 11122345, 111223457, 1112234579, 111223457912, 11122345791216, 1112234579121621, 111223457912162128, 11122345791216212837, 1112234579121621283749, 111223457912162128374965

Offset: 1

Omar E. Pol, Nov 10 2007

Harvey P. Dale, Table of n, a(n) for n = 1..125

Cf. A000931, A007908, A019523, A102397. See A132347 for another version.

Module[{nn=20,padseq},padseq=LinearRecurrence[{0,1,1},{1,1,1,2},nn];Table[ FromDigits[ Flatten[IntegerDigits/@Take[padseq,n]]],{n,nn}]] (* Harvey P. Dale, Feb 18 2023 *)

A134070 Primes in A130774.

Original entry on oeis.org

13, 13471118294776123, 134711182947761231993225218431364220735715778934915127244763960364079103682167761271443439204710647114985118604983010349

Offset: 1

Alexander Adamchuk, Oct 06 2007

A130774 = {1, 13, 134, 1347, 134711, 13471118, 1347111829, 134711182947, 13471118294776, 13471118294776123, ...} = Concatenate Lucas numbers. Indices of prime terms in A130774 are {2, 10, 31, ...}.

a(4) > A130774(500) = 1.347...*10^26425, if it exists. - Amiram Eldar, Jul 17 2025

			a(1) = A130774(2) = 13.
a(2) = A130774(10) = 13471118294776123.
a(3) = A130774(31) = 13471118294776123199322521843136422073571577893491512724476396036407910368216776127144343920471064711498511860498301034.

Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Smarandache Sequences.

Cf. A000204 (Lucas numbers).
Cf. A130774 (concatenation of Lucas numbers).
Cf. A019523 (concatenation of Fibonacci numbers).
Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).
Cf. A134072 (concatenation of A000204 Lucas numbers (beginning at 1) in reverse order).
Cf. A134069 (primes in A038399).
Cf. A134071 (primes in A134072).

Select[Module[{nn=40,ll},ll=LucasL[Range[nn]];Table[FromDigits[Flatten[IntegerDigits/@Take[ll,n]]],{n,nn}]],PrimeQ] (* Harvey P. Dale, May 07 2023 *)

Edited by Charles R Greathouse IV, Apr 24 2010

A134071 Primes in A134072.

Original entry on oeis.org

31, 431, 117431, 19912376472918117431

Offset: 1

Alexander Adamchuk, Oct 06 2007

A134072 = {1, 31, 431, 7431, 117431, 18117431, 2918117431, 472918117431, 76472918117431, 12376472918117431, 19912376472918117431, ...} = concatenation of Lucas numbers in reverse order. Indices of prime terms in A134072 are {2, 3, 5, 11, ...}.

No further terms through 500 Lucas numbers. - Harvey P. Dale, Aug 17 2013

			a(1) = A134072(2) = 31.
a(2) = A134072(3) = 431.
a(3) = A134072(5) = 117431.

Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A000204 (Lucas numbers).
Cf. A130774 (concatenation of Lucas numbers).
Cf. A019523 (concatenation of Fibonacci numbers).
Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).
Cf. A134072 (concatenation of A000204 Lucas numbers (beginning at 1) in reverse order).
Cf. A134069 (primes in A038399).
Cf. A134070 (primes in A130774).

nn=500;With[{lucs=LucasL[Range[nn]]},Select[Table[FromDigits[ Flatten[ IntegerDigits/@ Reverse[ Take[lucs,n]]]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Aug 17 2013 *)

Edited by Charles R Greathouse IV, Apr 24 2010

A132347 Concatenation of first n members of the Padovan sequence A000931.

Original entry on oeis.org

1, 10, 100, 1001, 10010, 100101, 1001011, 10010111, 100101112, 1001011122, 10010111223, 100101112234, 1001011122345, 10010111223457, 100101112234579, 10010111223457912, 1001011122345791216, 100101112234579121621

Offset: 1

Omar E. Pol, Nov 10 2007

Cf. A000931, A007908, A019523, A102397. See A134732 for another version.

Module[{nn=20,padseq},padseq=LinearRecurrence[{0,1,1},{1,0,0},nn];Table[FromDigits[ Flatten[ IntegerDigits/@Take[padseq,n]]],{n,nn}]] (* Harvey P. Dale, Feb 18 2024 *)

A329719 Numbers whose digits can be partitioned into at least 3 segments (not beginning with 0) where each segment is the sum of the previous two segments.

Original entry on oeis.org

112, 123, 134, 145, 156, 167, 178, 189, 213, 224, 235, 246, 257, 268, 279, 314, 325, 336, 347, 358, 369, 415, 426, 437, 448, 459, 516, 527, 538, 549, 617, 628, 639, 718, 729, 819, 1123, 1235, 1347, 1459, 1910, 2134, 2246, 2358, 2810, 2911, 3145, 3257, 3369

Offset: 1

Bi Cheng Wu, Nov 19 2019

			112 -> 1+1=2;
1235 -> 1+2=3 and 2+3=5;
224610 -> 2+2=4 and 2+4=6 and 4+6=10.

Bi Cheng Wu, Table of n, a(n) for n = 1..4958 (a(n) < 1000000)

Shares subsequences with A108203, A308104, A214527.

Cf. A019523.

part[n_] := part[n] = Select[Flatten[ Permutations /@ Reverse /@ IntegerPartitions[n, {3,n}], 1], 0 <= #[[3]] - Max[#[[1]], #[[2]]] <= 1 && AllTrue[Rest@ Rest@ Differences@ #, 0 <= # <= 1 &] &]; spl[x_, L_] := Map[ FromDigits@ Take[x, #] &, Transpose[{Most@ #, Rest[#]-1}& [ FoldList[ Plus, 1, L]]]]; sumQ[w_] := AllTrue[Range[3, Length@w], w[[#]] == w[[#-1]] + w[[#-2]] &]; ok[n_] := Block[{m = IntegerLength@ n}, AnyTrue[ spl[ IntegerDigits[n], #] & /@ part[m], sumQ[#] && Total[ IntegerLength /@ #] == m &]]; Select[ Range[100, 6000], ok] (* Giovanni Resta, Dec 03 2019 *)

def isSegmentSum(digits,segment1=None,segment2=None):
    digits = str(digits)
    N = len(digits)
    if N == 0:
        return True
    else:
        if (segment1 is None) and (segment2 is None):
            for i in range(N):
                try:
                    slice1 = digits[:i+1]
                    for j in range(N-(i+1)):
                        slice2 = digits[i+1:i+1+j+1]
                        slice3 = digits[i+1+j+1:]
                        if (isSegmentSum(slice3,slice1,slice2) and
                         len(slice3)>0 and not (slice1.startswith("0") or
                         slice2.startswith("0") or
                         (slice3.startswith("0")))):
                            return True
                except:
                    return False
        else:
            sumOfDigits = str(int(segment1)+int(segment2))
            nS = len(sumOfDigits)
            try:
                if digits[:nS] == sumOfDigits:
                    return isSegmentSum(digits[nS:],segment2,digits[:nS])
                else:
                    return False
            except:
                return False
    return False
def findSegmentSum(lower,upper):
    for i in range(lower,upper+1):
        if isSegmentSum(i):
            print(str(i))
findSegmentSum(1, 5200)

cp's OEIS Frontend

A038399 Concatenate first n nonzero Fibonacci numbers in reverse order.

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A094704 Convolution of Fibonacci(n) and 10^n.

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A134069 Primes in A038399.

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A134072 Concatenation of A000204 Lucas numbers (beginning at 1) in reverse order.

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A134732 Concatenation of first n members of the Padovan sequence A000931, starting at (1, 1, 1, 2,).

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A134070 Primes in A130774.

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A134071 Primes in A134072.

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A132347 Concatenation of first n members of the Padovan sequence A000931.

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