A037481 -id:A037481 - cp's OEIS frontend

A140796 a(n)=a(n-1)+6a(n-2), n>2.

1, 5, 14, 44, 128, 392, 1160, 3512, 10472, 31544, 94376, 283640, 849896, 2551736, 7651112, 22961528, 68868200, 206637368, 619846568, 1859670776, 5578750184, 16736774840, 50209275944, 150629924984, 451885580648, 1355665130552

Offset: 0

Paul Curtz, Jul 15 2008

nonn

The binomial transform is A037481.

The recurrence of the definition is also satisfied by A087451, A102901 and A140725.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 6).

Cf. A140725

Join[{1},LinearRecurrence[{1,6},{5,14},30]] (* Harvey P. Dale, Nov 20 2011 *)

a(n+1)-3a(n) = (-1)^n*A000079(n-1), n>0.
d(n+1)-3d(n) = (-1)^(n+1)*A000079(n-1), n>0, where d(n) is the sequence of pair sums d(n)= a(n)+a(n+1)=6, 19, 58, 172,...
O.g.f.: (1+x)(3x+1)/((2x+1)(1-3x)). - R. J. Mathar, Jul 29 2008
a(n) = (-1)^(n+1)*2^n/10+8*3^n/5, n>0. - R. J. Mathar, Jul 29 2008
a(n) = A140725(n)+A140725(n+1). - Philippe Deléham, Nov 17 2013

Edited and extended by R. J. Mathar, Jul 29 2008

A341910 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of runs in the binary expansion of n equals the number of ones in the binary expansion of a(n).

Original entry on oeis.org

0, 1, 3, 2, 5, 7, 6, 4, 9, 11, 15, 13, 10, 14, 12, 8, 17, 19, 23, 21, 27, 31, 29, 22, 18, 25, 30, 26, 20, 28, 24, 16, 33, 35, 39, 37, 43, 47, 45, 38, 46, 55, 63, 59, 51, 61, 53, 41, 34, 42, 54, 44, 57, 62, 58, 49, 36, 50, 60, 52, 40, 56, 48, 32, 65, 67, 71, 69

Offset: 0

Rémy Sigrist, Feb 23 2021

This sequence is a permutation of the nonnegative integers with inverse A341911.

Apparently, A037481 corresponds to the fixed points of this sequence.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)   bin(a(n))
  --  ----  -------  ---------
   0     0        0          0
   1     1        1          1
   2     3       10         11
   3     2       11         10
   4     5      100        101
   5     7      101        111
   6     6      110        110
   7     4      111        100
   8     9     1000       1001
   9    11     1001       1011
  10    15     1010       1111

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored scatterplot of the first 2^16 terms (where the color is function of A005811(n))
Rémy Sigrist, PARI program for A341910
Rémy Sigrist, PARI program for A341910 (faster)
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A005811, A037481, A298847, A341911 (inverse).

Block[{a = {0}, k}, Do[k = 1; While[Nand[FreeQ[a, k], DigitCount[k, 2, 1] == #], k++] &@ Length[Split@ IntegerDigits[i, 2]]; AppendTo[a, k], {i, 67}]; a] (* Michael De Vlieger, Feb 24 2021 *)

PARI
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See Links section.
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A005811(n) = A000120(a(n)).

a(n) < 2^k for any n < 2^k.

A362364 a(n) is the product of the first n primes that are coprime to a(n-1); a(0) = 1.

Original entry on oeis.org

1, 2, 15, 154, 3315, 67298, 2980185, 102091066, 6022953885, 319238763382, 24615812527995, 1654614510608906, 161405882746063215, 14284287070086685498, 1679105398207295625645, 166597640098421012963174, 24096841569672899523631395, 2989927846846361919650083778, 499069685749495422033929821845

Offset: 0

Robert Israel, Apr 18 2023

nonn

Lexicographically first sequence of squarefree numbers such that A001222(a(n)) = n and each term is coprime to the next.

			a(0) = 1.
a(1) = 2 is the least prime coprime to a(0).
a(2) = 3*5 is the product of the two least primes coprime to a(1).
a(3) = 2*7*11 is the product of the three least primes coprime to a(2).
a(4) = 3*5*13*17 = 3315 is the product of the four least primes coprime to a(3).

Robert Israel, Table of n, a(n) for n = 0..315

Cf. A001222.

See the formula section for the relationships with A002110, A019565, A037481, A100112, A117214.

f:= proc(n) local i;
   if n::odd then 2 * mul(ithprime(4*i)*ithprime(4*i+1),i=1..(n-1)/2)
   else mul(ithprime(4*i-2)*ithprime(4*i-1),i=1..(n/2))
   fi
end proc:
map(f, [$0..20]);

from math import prod
from sympy import prime
def A362364(n): return prod(prime(i)*prime(i+1) for i in range(2+((n&1)<<1),(n<<1)-1,4))<<(n&1) # Chai Wah Wu, Apr 20 2023

If n is even, a(n) = Product_{i=1..n/2} prime(4*i-2)*prime(4*i-1).
If n is odd, a(n) = 2 * Product_{i=1..(n-1)/2} prime(4*i)*prime(4*i+1).
From Peter Munn, Apr 21 2023: (Start)
a(0) = 1, for n >= 1, a(n) = A002110(2n-1)/a(n-1).
a(n) = A019565(A037481(n)).
For n >= 1, a(n-1) = A117214(A100112(a(n))).
(End)

A363773 a(n) = (4^(n+1) + (-1)^n + 5)/10.

Original entry on oeis.org

1, 2, 7, 26, 103, 410, 1639, 6554, 26215, 104858, 419431, 1677722, 6710887, 26843546, 107374183, 429496730, 1717986919, 6871947674, 27487790695, 109951162778, 439804651111, 1759218604442, 7036874417767, 28147497671066, 112589990684263, 450359962737050

Offset: 0

Evangelos G. Filothodoros, Jun 21 2023

a(n) is a part of the numerator of the approximate solutions x(n) = (Pi/2)*(1+5/((4^(n+1)-(-1)^(n+1)))) = a(n)*Pi/A015521(n+1) of D_d(exp(-i*x(n))) = Cl_d(x(n)+Pi) = 0, where D_d(exp(-i*x(n))) is the Bloch-Wigner-Ramakrishnan polylogarithm function and Cl_d(x(n)+Pi) is the Clausen function for odd d >= 3 and n >= 0.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Evangelos G. Filothodoros, Anastasios C. Petkou, and Nicholas D. Vlachos, The fermion-boson map for large d, Nuclear Physics B, Volume 941, 2019, pp. 195-224.
Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Cf. A015521, A037481.

A363773list[nmax_]:=LinearRecurrence[{4,1,-4},{1,2,7},nmax+1];A363773list[50] (* Paolo Xausa, Jun 29 2023 *)

def A363773(n): return (1<<(n<<1|1))//5+1 # Chai Wah Wu, Jun 28 2023

a(n) = 1 + A037481(n).
G.f.: (1-2*x-2*x^2)/((x-1)*(4*x-1)*(x+1)).
E.g.f.: (4*e^(4*x) + e^-x + 5*e^x)/10.

cp's OEIS Frontend

A140796 a(n)=a(n-1)+6a(n-2), n>2.

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A341910 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of runs in the binary expansion of n equals the number of ones in the binary expansion of a(n).

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A362364 a(n) is the product of the first n primes that are coprime to a(n-1); a(0) = 1.

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A363773 a(n) = (4^(n+1) + (-1)^n + 5)/10.

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Programs

Mathematica

Python

Formula