A165998 -id:A165998 - cp's OEIS frontend

A317323 Multiples of 23 and odd numbers interleaved.

0, 1, 23, 3, 46, 5, 69, 7, 92, 9, 115, 11, 138, 13, 161, 15, 184, 17, 207, 19, 230, 21, 253, 23, 276, 25, 299, 27, 322, 29, 345, 31, 368, 33, 391, 35, 414, 37, 437, 39, 460, 41, 483, 43, 506, 45, 529, 47, 552, 49, 575, 51, 598, 53, 621, 55, 644, 57, 667, 59, 690, 61, 713, 63, 736, 65, 759, 67, 782, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 27-gonal numbers (A316725).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 27-gonal numbers.

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

Cf. A008605 and A005408 interleaved.
Column 23 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A316725.

With[{nn=40},Riffle[23*Range[0,nn],Range[1,2*nn,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,23,3},80] (* Harvey P. Dale, May 05 2019 *)

concat(0, Vec(x*(1 + 23*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 23*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 23*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 23*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 21/2^s). - Amiram Eldar, Oct 26 2023

A317324 Multiples of 24 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 24, 3, 48, 5, 72, 7, 96, 9, 120, 11, 144, 13, 168, 15, 192, 17, 216, 19, 240, 21, 264, 23, 288, 25, 312, 27, 336, 29, 360, 31, 384, 33, 408, 35, 432, 37, 456, 39, 480, 41, 504, 43, 528, 45, 552, 47, 576, 49, 600, 51, 624, 53, 648, 55, 672, 57, 696, 59, 720, 61, 744, 63, 768, 65, 792, 67, 816, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 28-gonal numbers (A303812).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 28-gonal numbers.

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

Cf. A008606 and A005408 interleaved.
Column 24 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303812.

&cat[[24*n, 2*n + 1]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2018

Table[If[EvenQ[n], 24 (n/2), n], {n, 0, 70}] (* Vincenzo Librandi, Jul 28 2018 *)
With[{nn=40},Riffle[24*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,24,3},80] (* Harvey P. Dale, Jul 06 2019 *)

concat(0, Vec(x*(1 + 24*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 24*n, a(2n+1) = 2*n + 1.
G.f.: x*(1 + 24*x + x^2)/((1-x)^2*(1+x)^2). - Vincenzo Librandi, Jul 28 2018
a(n) = 2*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 29 2018
Multiplicative with a(2^e) = 3*2^(e+2), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 11*2^(1-s)). - Amiram Eldar, Oct 26 2023

A317325 Multiples of 25 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 25, 3, 50, 5, 75, 7, 100, 9, 125, 11, 150, 13, 175, 15, 200, 17, 225, 19, 250, 21, 275, 23, 300, 25, 325, 27, 350, 29, 375, 31, 400, 33, 425, 35, 450, 37, 475, 39, 500, 41, 525, 43, 550, 45, 575, 47, 600, 49, 625, 51, 650, 53, 675, 55, 700, 57, 725, 59, 750, 61, 775, 63, 800, 65, 825, 67, 850, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 29-gonal numbers (A303815).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 29-gonal numbers.

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

Cf. A008607 and A005408 interleaved.
Column 25 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303815.

Flat(List([0..40],n->[25*n,2*n+1])); # Muniru A Asiru, Jul 28 2018

&cat[[25*n, 2*n + 1]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2018

seq(op([25*n,2*n+1]),n=0..40); # Muniru A Asiru, Jul 28 2018

With[{nn=30}, Riffle[25 Range[0, nn], 2 Range[0, nn] + 1]] (* Vincenzo Librandi, Jul 28 2018 *)

concat(0, Vec(x*(1 + 25*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 25*n, a(2n+1) = 2*n + 1.
G.f.: x*(1 + 25*x + x^2)/((1 - x)^2*(1 + x)^2). - Vincenzo Librandi, Jul 28 2018
a(n) = 2*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 29 2018
Multiplicative with a(2^e) = 25*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 23/2^s). - Amiram Eldar, Oct 26 2023

A317326 Multiples of 26 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69

Offset: 0

Omar E. Pol, Jul 25 2018

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).
Partial sums give the generalized 30-gonal numbers.
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
From Bruno Berselli, Jul 27 2018: (Start)
Also, this type of sequence is characterized by:
O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;
E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;
a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;
a(n) = (m/2)^((1 + (-1)^n)/2)*n;
a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

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

Cf. A252994 and A005408 interleaved.
Column 26 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).
Cf. A316729.

[13^div(1+(-1)^n,2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018

Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

a(2*n) = 26*n, a(2*n+1) = 2*n + 1.
From Bruno Berselli, Jul 27 2018: (Start)
O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.
E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).
a(n) = -a(-n) = (7 + 6*(-1)^n)*n.
a(n) = 13^((1 + (-1)^n)/2)*n.
a(n) = 2*a(n-2) - a(n-4). (End)
Multiplicative with a(2^e) = 13*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(3-s)). - Amiram Eldar, Oct 26 2023

A059944 Denominators of Maclaurin series coefficients for 2cos(x/sqrt(3) + arctan(-sqrt(3))) = cos(x/sqrt(3)) + sqrt(3)sin(x/sqrt(3)).

Original entry on oeis.org

1, 1, 6, 18, 216, 1080, 19440, 136080, 3265920, 29393280, 881798400, 9699782400, 349192166400, 4539498163200, 190658922854400, 2859883842816000, 137274424455168000, 2333665215737856000, 126017921649844224000

Offset: 0

Eric W. Weisstein

Related to the exact solution of the logistic equation with r = -2.

Denominators of Maclaurin series coefficients of (sqrt(n) + 1)/2*exp(x/sqrt(n)) - (sqrt(n) - 1)/2*exp(-x/sqrt(n)) = 1 + x + x^2/(n*2!) + x^3/(n*3!) + x^4/(n^2*4!) + x^5/(n^2*5!) + ... when n = 3. Cf. A268363 (case n = 2). - Peter Bala, Aug 06 2019

Eric Weisstein's World of Mathematics, Logistic Equation.

Cf. A165998, A268363, A057077.

gf := 2*sin(x/sqrt(3) + Pi/6): ser := series(gf, x, 20):
seq(denom(coeff(ser, x, n)), n=0..18); # Peter Luschny, Aug 07 2019

a(n) = 3^floor(n/2)*n! = Product_{k = 1..n} k*(2 - (-1)^(k-1)) = Product_{k = 0..n-1} A165998(k), with empty products set equal to 1. - Peter Bala, Aug 05 2019

a(n) = denominator([x^n] 2*sin(x/sqrt(3) + Pi/6)). Numerator is A057077. - Peter Luschny, Aug 07 2019

A347365 a(n) = n * (2-(-1)^n), or zero together with first differences of even triangular numbers halved (A074378).

Original entry on oeis.org

0, 3, 2, 9, 4, 15, 6, 21, 8, 27, 10, 33, 12, 39, 14, 45, 16, 51, 18, 57, 20, 63, 22, 69, 24, 75, 26, 81, 28, 87, 30, 93, 32, 99, 34, 105, 36, 111, 38, 117, 40, 123, 42, 129, 44, 135, 46, 141, 48, 147, 50, 153, 52, 159, 54, 165, 56, 171, 58, 177, 60, 183, 62, 189, 64

Offset: 0

Federico Provvedi, Aug 29 2021

This sequence and A165998 form a complementary pair as solutions of alternating sequences a(n) + b(n) = 4*n (A008586), and a(n)*b(n) = 3*n^2 (A033428).
This is the particular case of the two integer sequences x(n)=2n and y(n)=n, where more generally, x(n) + y(n) = 2*a(n) and x(n)*y(n) = (a(n) + b(n)) * (a(n) - b(n)) give the two conjugate binomials a(n) = x(n) + (-1)^n*y(n) and b(n) = x(n) - (-1)^n*y(n) as solutions over the integer domain.
a(n) is also A005843 and A016945 interleaved.
For every integer k: a(n*k) = n*k is multiplicative for nonnegative even integers n and a(n*k) = n*a(k) for nonnegative odd integers n.
For every nonnegative odd integer k, the k-th difference of a(k*n)/k = (2n+1)*(-1)^n + 2 = A166519(n), and 1 for all nonnegative even integers.
a(6n+1)/3 = 6n+1, and a(6n+5)/3 = 6n+5, related to Collatz Conjecture.
Half-periods of a(k) mod n is A026741(n).

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A074378, A165998, A008586, A033428, A237420, A193356, A016921, A016969, A166519, A014682, A005843, A016945, A010684, A008585, A000027, A026741.

Table[n(2-(-1)^n),{n,0,99}] (* or *)
LinearRecurrence[{0,2,0,-1}, {0,3,2,9}, 100] (* or *)
If[EvenQ@#,#,3#]&/@Range[0,99]  (* or *)
Drop[Flatten@Transpose[{2#,6#+3}&@Range[0,Quotient[#,2]]],-Boole@EvenQ@#]&@(10^2)

a(n) = n*(2-(-1)^n); \\ Michel Marcus, Sep 13 2021

(x*(3+2*x+3*x^2)/(1-x^2)^2).series(x,65).coefficients(x,sparse=False) # Stefano Spezia, Aug 30 2021

G.f.: x*(3 + 2*x + 3*x^2)/(1 - x^2)^2.
E.g.f.: x*(3*cosh(x) + sinh(x)).
Dirichlet g.f.: 2^(-s) * (3*2^s - 4) * zeta(s-1) = (3 - 4/2^s) * zeta(s-1) = (3 - 1/2^(s-2)) * zeta(s-1).
a(n) = n*(2-(-1)^n) = 3*n / (2+(-1)^n).
a(n) = 3*n if n odd, a(n) = n if n even, implies a(a(2n)) = 2n, a(a(2n+1)) = 9*a(2n+1).
a(n) = 3*b(n), if n odd and a(n) = b(n)/3, if n even, with b(n) = A165998(n).
a(n) = a(a(2k*n)/(2k)) = a((2k+1)*n) / (2k+1), since a(2*k*n) / (2*k) = n.
a(n) = 4*n - A165998(n).
a(n+1) = a(n) + A086970(n+1)*(-1)^n.
a(n) = 2*A014682(n) - A000035(n).
a(n) = n*A010684(n). - Michel Marcus, Sep 13 2021
For positive integers k and n, a(n) = A(n,1) = n * (A(n,k)/n)^(1/k), where the k-th nesting composition A(n,k) = a(a(...a(a(n))...)) = n * ( a(n) / n )^k, and d.g.f. of A(n,k) = (2^(1-s) + (1-2^(1-s))*3^k) * zeta(s-1). - Federico Provvedi, Sep 18 2021
a(n+1) = A165998(n)*(1 + 1/n). - Federico Provvedi, Sep 19 2021

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A317323 Multiples of 23 and odd numbers interleaved.

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A317324 Multiples of 24 and odd numbers interleaved.

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A317325 Multiples of 25 and odd numbers interleaved.

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A317326 Multiples of 26 and odd numbers interleaved.

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A059944 Denominators of Maclaurin series coefficients for 2*cos(x/sqrt(3) + arctan(-sqrt(3))) = cos(x/sqrt(3)) + sqrt(3)*sin(x/sqrt(3)).

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A347365 a(n) = n * (2-(-1)^n), or zero together with first differences of even triangular numbers halved (A074378).

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A059944 Denominators of Maclaurin series coefficients for 2cos(x/sqrt(3) + arctan(-sqrt(3))) = cos(x/sqrt(3)) + sqrt(3)sin(x/sqrt(3)).