A027941 -id:A027941 - cp's OEIS frontend

A216176 Inner product of the Zeckendorf binary representation of n and its reverse.

1, 0, 0, 2, 0, 2, 0, 0, 2, 0, 1, 3, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 1, 3, 1, 0, 2, 0, 2, 4, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 4, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 1, 3, 1, 1, 3, 0, 2, 0, 0, 2, 2, 4, 2, 0, 2, 0, 2, 4, 0, 2, 0, 1, 3, 1

Offset: 1

Reinhard Zumkeller, Mar 10 2013

nonn

a(n) = sum (A189920(n,k) * A213676(n,k): k = 1..A072649(n));

a(A027941(n)) = n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A159780.

a216176 n = sum $ zipWith (*) zs $ reverse zs
   where zs = a189920_row n

A227351 Permutation of nonnegative integers: map each number by lengths of runs of zeros in its Zeckendorf expansion shifted once left to the number which has the same lengths of runs (in the same order, but alternatively of runs of 0's and 1's) in its binary representation.

Original entry on oeis.org

0, 1, 3, 7, 2, 15, 6, 4, 31, 14, 12, 8, 5, 63, 30, 28, 24, 13, 16, 9, 11, 127, 62, 60, 56, 29, 48, 25, 27, 32, 17, 19, 23, 10, 255, 126, 124, 120, 61, 112, 57, 59, 96, 49, 51, 55, 26, 64, 33, 35, 39, 18, 47, 22, 20, 511, 254, 252, 248, 125, 240, 121, 123, 224

Offset: 0

Antti Karttunen, Jul 08 2013

This permutation is based on the fact that by appending one extra zero to the right of Fibonacci number representation of n (aka "Zeckendorf expansion") and then counting the lengths of blocks of consecutive (nonleading) zeros we get bijective correspondence with compositions, and thus also with the binary representation of a unique n. See the chart below:
   n   A014417(n) A014417(A022342(n+1)) Runs of    Binary number   In dec.
                  [shifted once left]   zeros      with same runs  = a(n)
  ......0              ......0    []         .....0           0
  ......1              .....10    [1]        .....1           1
  .....10              ....100    [2]        ....11           3
  ....100              ...1000    [3]        ...111           7
  ....101              ...1010    [1,1]      ....10           2
  ...1000              ..10000    [4]        ..1111          15
  ...1001              ..10010    [2,1]      ...110           6
  ...1010              ..10100    [1,2]      ...100           4
  ..10000              .100000    [5]        .11111          31
  ..10001              .100010    [3,1]      ..1110          14
  ..10010              .100100    [2,2]      ..1100          12
  ..10100              .101000    [1,3]      ..1000           8
  ..10101              .101010    [1,1,1]    ...101           5
  .100000              1000000    [6]        111111          63
Are there any other fixed points after 0, 1, 6, 803, 407483 ?

Antti Karttunen, Table of n, a(n) for n = 0..10946
OEIS Wiki, Run-length encoding
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A227352. Cf. also A003714, A014417, A006068, A048679.

Could be further composed with A075157 or A075159, also A129594.

(define (A227351 n) (A006068 (A048679 n)))

(define (A227351v2 n) (A006068 (A106151 (* 2 (A003714 n)))))

a(n) = A006068(A048679(n)) = A006068(A106151(2*A003714(n))).
This permutation effects following correspondences:
a(A000045(n)) = A000225(n-1).
a(A027941(n)) = A000975(n).
For n >=3, a(A000204(n)) = A000079(n-2).

A371590 Irregular table T(n, k), n >= 0, k = 1..max(2, 2^n), read by rows; the n-th row lists the nonnegative numbers whose Zeckendorf-binary representation has n nonleading zeros.

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 7, 12, 5, 9, 10, 11, 17, 19, 20, 33, 8, 14, 15, 16, 18, 25, 27, 28, 30, 31, 32, 46, 51, 53, 54, 88, 13, 22, 23, 24, 26, 29, 38, 40, 41, 43, 44, 45, 48, 49, 50, 52, 67, 72, 74, 75, 80, 82, 83, 85, 86, 87, 122, 135, 140, 142, 143, 232

Offset: 0

Rémy Sigrist, Mar 28 2024

As a flat sequence, this is a permutation of the nonnegative integers with inverse A371591.

			Array T(n, k) begins:
    0, 1
    2, 4
    3, 6, 7, 12
    5, 9, 10, 11, 17, 19, 20, 33
    8, 14, 15, 16, 18, 25, 27, 28, 30, 31, 32, 46, 51, 53, 54, 88
    ...

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A371592 for a similar sequence.

Cf. A000045, A027941, A102364, A371591 (inverse).

PARI
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A102364(T(n, k)) = n.
T(n, 1) = A000045(n + 2) for any n > 0.
T(n, max(2, 2^n)) = A027941(n + 1) for any n >= 0.

A077784 Numbers k such that (10^k - 1)/3 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 5, 35, 159, 237, 325, 355, 371, 481, 1649, 3641, 4709, 269623

Offset: 1

Patrick De Geest, Nov 16 2002

Prime versus probable prime status and proofs are given in the author's table.

a(13) > 2*10^5. - Robert Price, Apr 03 2016

			5 is a term because (10^5 - 1)/3 + 2*10^2 = 33533.

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
Makoto Kamada, Prime numbers of the form 33...33533...33
Index entries for primes involving repunits.

Partial sums of S(n, x), for x=1...9: A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420.

Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

Do[ If[ PrimeQ[(10^n + 6*10^Floor[n/2] - 1)/3], Print[n]], {n, 3, 4800, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A183175(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

a(13) from Robert Price, Aug 03 2024

A077828 Expansion of 1/(1-3x-3x^2-3*x^3).

Original entry on oeis.org

1, 3, 12, 48, 189, 747, 2952, 11664, 46089, 182115, 719604, 2843424, 11235429, 44395371, 175422672, 693160416, 2738935377, 10822555395, 42763953564, 168976333008, 667688525901, 2638286437419, 10424853888984, 41192486556912, 162766880649945, 643152663287523

Offset: 0

N. J. A. Sloane, Nov 17 2002

Michael De Vlieger, Table of n, a(n) for n = 0..1675
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
Index entries for linear recurrences with constant coefficients, signature (3,3,3).

Partial sums of S(n, x), for x=1...12, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-7.

Cf. A071675.

CoefficientList[Series[1/(1-3x-3x^2-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {3,3,3},{1,3,12},30] (* Harvey P. Dale, Dec 25 2018 *)

Vec(1/(1-3*x-3*x^2-3*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = sum{k=0..n, T(n-k, k)3^(n-k)}, T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005

a(n) = sum{k=0..n, sum{i=0..floor((n-k)/2), C(n-k-i, i)C(k, n-k-i)}*3^k}. - Paul Barry, Apr 26 2005

A077829 Expansion of 1/(1-3x-3x^2-2*x^3).

Original entry on oeis.org

1, 3, 12, 47, 183, 714, 2785, 10863, 42372, 165275, 644667, 2514570, 9808261, 38257827, 149227404, 582072215, 2270414511, 8855914986, 34543132921, 134737972743, 525555146964, 2049965624963, 7996038261267, 31189121952618, 121655411891581, 474525678055131

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,3,2).

Partial sums of S(n, x), for x=1...14, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139.

CoefficientList[Series[1/(1 - 3*x - 3*x^2 - 2*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 20 2024 *)
LinearRecurrence[{3,3,2},{1,3,12},30] (* Harvey P. Dale, Dec 20 2024 *)

Vec(1/(1-3*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

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

a(n) = 3*a(n-1) + 3*a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Jan 20 2024

A077831 Expansion of 1/(1-3x-2x^2-2*x^3).

Original entry on oeis.org

1, 3, 11, 41, 151, 557, 2055, 7581, 27967, 103173, 380615, 1404125, 5179951, 19109333, 70496151, 260067021, 959412031, 3539362437, 13057045415, 48168685181, 177698871247, 655548074933, 2418379337655, 8921631905325, 32912750541151, 121418274109413

Offset: 0

N. J. A. Sloane, Nov 17 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,2).

Partial sums of S(n, x), for x=1...16, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829, A097830.

CoefficientList[Series[1/(1-3x-2x^2-2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,2},{1,3,11},30] (* Harvey P. Dale, Feb 28 2025 *)

Vec(1/(1-3*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A143952 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 12, 1, 1, 33, 8, 1, 88, 42, 1, 1, 232, 183, 13, 1, 609, 717, 102, 1, 1, 1596, 2622, 624, 19, 1, 4180, 9134, 3275, 205, 1, 1, 10945, 30691, 15473, 1650, 26, 1, 28656, 100284, 67684, 11020, 366, 1, 1, 75024, 320466, 279106, 64553, 3716, 34, 1

Offset: 0

Emeric Deutsch, Oct 10 2008

Row n has 1+floor(n/2) terms.
Row sums are the Catalan numbers (A000108).
T(n,1)=A027941(n-1)=Fibonacci(2n-1)-1.
Sum(k*T(n,k),k=0..floor(n/2))=A079309(n-1).
For the statistic "number of peaks in peak plateaux", see A143953.

			T(3,1)=4 because we have UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D (the peak plateaux are shown between parentheses).
The triangle starts:
1;
1;
1,1;
1,4;
1,12,1;
1,33,8;
1,88,42,1;

David Callan, Kreweras's Narayana Number Identity Has a Simple Dyck Path Interpretation .

Cf. A000045, A000108, A027941, A079309, A143953.

C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G:=(1-z)*C(z*(1-z)^2/(1-z+z^2-t*z^2)^2)/(1-z+z^2-t*z^2): Gser:=simplify(series(G,z= 0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

The g.f. G=G(t,z) satisfies z(1-z)G^2 - (1-z+z^2-tz^2)G+1-z = 0 (for the explicit form of G see the Maple program).
The trivariate g.f. g=g(x,y,z) of Dyck paths with respect to number of peak plateaux, number of peaks in the peak plateaux and semilength, marked, by x, y and z, respectively satisfies g=1+zg[g+xyz/(1-yz)-z/(1-z)].
T(n,k) = Sum_{r=1..n} Narayana(n-r,k)*binomial(2n-r-k,r-k) where Narayana(n,k) := binomial(n,k)*binomial(n,k-1)/n is the Narayana number A001263. - David Callan, Oct 31 2008

A186025 a(n) = 0^n + 1 - F(n-1)^2 - F(n)^2, where F = A000045.

Original entry on oeis.org

1, 0, -1, -4, -12, -33, -88, -232, -609, -1596, -4180, -10945, -28656, -75024, -196417, -514228, -1346268, -3524577, -9227464, -24157816, -63245985, -165580140, -433494436, -1134903169, -2971215072, -7778742048, -20365011073, -53316291172, -139583862444

Offset: 0

Paul Barry, Feb 10 2011

Row sums of number triangle A186024.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A000045, A027941.

[0^n+1-Fibonacci(n-1)^2-Fibonacci(n)^2: n in [0..30]]; // Vincenzo Librandi, Apr 24 2015

Join[{1}, Table[0^n + 1 - Fibonacci[n - 1]^2 - Fibonacci[n]^2, {n, 30}]] (* Vincenzo Librandi, Apr 24 2015 *)
LinearRecurrence[{4,-4,1},{1,0,-1,-4},30] (* Harvey P. Dale, Dec 16 2015 *)

x='x+O('x^50); Vec((1-4*x+3*x^2-x^3)/(1-4*x+4*x^2-x^3)) \\ G. C. Greubel, Jul 24 2017

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

a(n) = -A027941(n-1), n>0. - R. J. Mathar, Mar 21 2013

More terms from Vincenzo Librandi, Apr 24 2015

A190757 Lucas Aurifeuillian primitive part A of Lucas(10*n - 5).

Original entry on oeis.org

1, 1, 101, 71, 181, 39161, 24571, 12301, 1158551, 87382901, 21211, 373270451, 28143378001, 32414581, 1322154751061, 9062194370461, 1550853481, 2819407321151, 265272771839851, 2366632711, 137083914639998701, 85417012034751151, 3455782010101

Offset: 1

Arkadiusz Wesolowski, Dec 29 2012

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..250
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), pp. 251-260, S1.
C. K. Caldwell, "Top Twenty" page, Lucas Aurifeuillian primitive part

Cf. A061447, A190781.

lst = {1}; n = 23; Do[f = LucasL[k]; Do[f = f/GCD[f, lst[[d]]], {d, Most@Divisors[k]}]; AppendTo[lst, f], {k, 2, 10*n - 5}]; Table[GCD[lst[[5*k]], 5*Fibonacci[k]*(Fibonacci[k] - 1) + 1], {k, 1, 2*n - 1, 2}]

a(n) = GCD(A061447(10*n-5), A027941(n-1)*A106729(n-1) + 1).

cp's OEIS Frontend

A216176 Inner product of the Zeckendorf binary representation of n and its reverse.

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A227351 Permutation of nonnegative integers: map each number by lengths of runs of zeros in its Zeckendorf expansion shifted once left to the number which has the same lengths of runs (in the same order, but alternatively of runs of 0's and 1's) in its binary representation.

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A371590 Irregular table T(n, k), n >= 0, k = 1..max(2, 2^n), read by rows; the n-th row lists the nonnegative numbers whose Zeckendorf-binary representation has n nonleading zeros.

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A077784 Numbers k such that (10^k - 1)/3 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A077828 Expansion of 1/(1-3*x-3*x^2-3*x^3).

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A077829 Expansion of 1/(1-3*x-3*x^2-2*x^3).

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A077831 Expansion of 1/(1-3*x-2*x^2-2*x^3).

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A143952 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

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A077828 Expansion of 1/(1-3x-3x^2-3*x^3).

A077829 Expansion of 1/(1-3x-3x^2-2*x^3).

A077831 Expansion of 1/(1-3x-2x^2-2*x^3).