A022998 -id:A022998 - cp's OEIS frontend

A143051 Smallest number not occurring earlier and smaller than the largest square so far, the next square if no such number exists.

0, 1, 4, 2, 3, 9, 5, 6, 7, 8, 16, 10, 11, 12, 13, 14, 15, 25, 17, 18, 19, 20, 21, 22, 23, 24, 36, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 49, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 64, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 81, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Reinhard Zumkeller, Jul 20 2008

nonn

Permutation of the natural numbers, inverse: A143052;
A143053(n) = a(a(n));
a(A000290(n)) = A005563(n-1);
a(A002522(n)) = A000290(n+1);
for n>1: GCD(a(n^2),a(n^2+1)) = A050873(A000290(n),A002522(n)) =
A022998(n+1).

a(n) = if n=0 then 0 else if n=1+k^2 then n+2*k else n-1.

A225975 Square root of A226008(n).

Original entry on oeis.org

0, 2, 2, 6, 1, 10, 6, 14, 4, 18, 10, 22, 3, 26, 14, 30, 8, 34, 18, 38, 5, 42, 22, 46, 12, 50, 26, 54, 7, 58, 30, 62, 16, 66, 34, 70, 9, 74, 38, 78, 20, 82, 42, 86, 11, 90, 46, 94, 24, 98, 50, 102, 13, 106, 54, 110, 28, 114, 58

Offset: 0

Paul Curtz, May 22 2013

Repeated terms of A016825 are in the positions 1,2,3,6,5,10,... (A043547).
From Wolfdieter Lang, Dec 04 2013: (Start)
This sequence a(n), n>=1, appears in the formula 2*sin(2*Pi/n) = R(p(n), x) modulo C(a(n), x), with x = rho(a(n)) = 2*cos(Pi/a(n)), the R-polynomials given in A127672 and the minimal C-polynomials of rho given in A187360. This follows from the identity 2*sin(2*Pi/n) = 2*cos(Pi*p(n)/a(n)) with gcd(p(n), a(n)) = 1. For p(n) see a comment on A106609,
Because R is an integer polynomial it shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(a(n))) of degree delta(a(n)) (the degree of C(a(n), x)), with delta(k) = A055034(k). This degree is given in A093819. For the coefficients of 2*sin(2*Pi/n) in the power basis of Q(rho(a(n))) see A231189 . (End)

			For the first formula: a(0)=-1+1=0, a(1)=-3+5=2, a(2)=-1+3=2, a(3)=-1+7=6, a(4)=0+1=1.

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

Cf. A016825, A017089, A017137, A022998, A106609, A145979, A226008.

a[0]=0; a[n_] := Sqrt[Denominator[1/4 - 4/n^2]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, May 30 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,2,2,6,1,10,6,14,4,18,10,22,3,26,14,30},60] (* Harvey P. Dale, Nov 21 2019 *)

a(n) = A106609(n-4) + A106609(n+4) with A106609(-4)=-1, A106609(-3)=-3, A106609(-2)=-1, A106609(-1)=-1.
a(n) = 2*a(n-8) -a(n-16).
a(2n+1) = A016825(n), a(2n) = A145979(n-2) for n>1, a(0)=0, a(2)=2.
a(4n)   = A022998(n).
a(4n+1) = A017089(n).
a(4n+2) = A016825(n).
a(4n+3) = A017137(n).
G.f.: x*(2 +2*x +6*x^2 +x^3 +10*x^4 +6*x^5 +14*x^6 +4*x^7 +14*x^8 +6*x^9 +10*x^10 +x^11 +6*x^12 +2*x^13 +2*x^14)/((1-x)^2*(1+x)^2*(1+x^2)^2*(1+x^4)^2). [Bruno Berselli, May 23 2013]
From Wolfdieter Lang, Dec 04 2013: (Start)
a(n) = 2*n if n is odd; if n is even then a(n) is n if n/2 == 1, 3, 5, 7 (mod 8), it is n/2 if n/2 == 0, 4 (mod 8) and it is n/4 if n/2 == 2, 6 (mod 8). This leads to the given G.f..
With c(n) = A178182(n), n>=1, a(n) = c(n)/2 if c(n) is even and c(n) if c(n) is odd. This leads to the preceding formula. (End)

Edited by Bruno Berselli, May 24 2013

A317312 Multiples of 12 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 12, 3, 24, 5, 36, 7, 48, 9, 60, 11, 72, 13, 84, 15, 96, 17, 108, 19, 120, 21, 132, 23, 144, 25, 156, 27, 168, 29, 180, 31, 192, 33, 204, 35, 216, 37, 228, 39, 240, 41, 252, 43, 264, 45, 276, 47, 288, 49, 300, 51, 312, 53, 324, 55, 336, 57, 348, 59, 360, 61, 372, 63, 384, 65, 396, 67, 408, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 16-gonal numbers (A274978).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 16-gonal numbers.

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

Cf. A008594 and A005408 interleaved.
Column 12 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).
Cf. A274978.

{0}~Join~Riffle[2 Range@ # - 1, 12 Range@ #] &@ 35 (* or *)
CoefficientList[Series[x (1 + 12 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 12, 3}, 70] (* Michael De Vlieger, Jul 26 2018 *)

a(2n) = 12*n, a(2n+1) = 2*n + 1.
From Michael De Vlieger, Jul 26 2018: (Start)
G.f.: x*(1 + 12*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) = 3*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 + 5*2^(1-s)). - Amiram Eldar, Oct 25 2023
a(n) = (7 + 5*(-1)^n)*n/2. - Aaron J Grech, Aug 20 2024

A317313 Multiples of 13 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 13, 3, 26, 5, 39, 7, 52, 9, 65, 11, 78, 13, 91, 15, 104, 17, 117, 19, 130, 21, 143, 23, 156, 25, 169, 27, 182, 29, 195, 31, 208, 33, 221, 35, 234, 37, 247, 39, 260, 41, 273, 43, 286, 45, 299, 47, 312, 49, 325, 51, 338, 53, 351, 55, 364, 57, 377, 59, 390, 61, 403, 63, 416, 65, 429, 67, 442, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 17-gonal numbers (A303305).
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.
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 17-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. A008595 and A005408 interleaved.
Column 13 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).
Cf. A303305.

Table[{13n, 2n + 1}, {n, 0, 35}] // Flatten (* or *)
CoefficientList[Series[(x^3 + 13 x^2 + x)/(x^2 - 1)^2, {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 13, 3}, 70] (* Robert G. Wilson v, Jul 26 2018 *)

a(n) = if(n%2==0, return((n/2)*13), return(n)) \\ Felix Fröhlich, Jul 26 2018

concat(0, Vec(x*(1 + 13*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 13*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 13*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) = 13*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 + 11/2^s). - Amiram Eldar, Oct 25 2023
a(n) = (15 + 11*(-1)^n)*n/4. - Aaron J Grech, Aug 20 2024

A118411 Numerator of sum of reciprocals of first n pentatope numbers A000332.

Original entry on oeis.org

1, 6, 19, 136, 83, 119, 656, 73, 190, 121, 1816, 559, 679, 815, 3872, 1139, 886, 513, 2360, 2023, 2299, 2599, 11696, 3275, 7306, 1353, 5992, 1653, 5455, 5983, 26176, 7139, 15538, 8435, 12184, 3293, 3553, 11479, 49360

Offset: 1

Jonathan Vos Post, Apr 27 2006

Denominators are A118412. Fractions are: 1/1, 6/5, 19/15, 136/105, 83/63, 119/90, 656/495, 73/55, 190/143, 121/91, 1816/1365, 559/420, 679/510, 815/612, 3872/2907, 1139/855, 886/665, 513/385, 2360/1771, 2023/1518, 2299/1725, 2599/1950, 11696/8775, 3275/2457, 7306/5481, 1353/1015, 5992/4495, 1653/1240, 5455/4092, 5983/4488, 26176/19635, 7139/5355, 15538/11655, 8435/6327, 12184/9139, 3293/2470, 3553/2665, 11479/8610, 49360/37023. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392.

			a(1) = 1 = numerator of 1/1.
a(2) = 6 = numerator of 6/5 = 1/1 + 1/5.
a(3) = 19 = numerator of 19/15 = 1/1 + 1/5 + 1/15.
a(4) = 136 = numerator of 136/105 = 1/1 + 1/5 + 1/15 + 1/35.
a(5) = 55 = numerator of 55/42 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70.
a(10) = 190 = numerator of 190/143 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715.
a(20) = 2360 = numerator of 2360/1771 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715 + 1/1001 + 1/1365 + 1/1820 + 1/2380 + 1/3060 + 1/3876 + 1/4845 + 1/5985 + 1/7315 + 1/8855.

Cf. A000332, A022998, A026741, A118391, A118391, A118412.

s=0;for(i=4,50,s+=1/binomial(i,4);print(numerator(s))) /* Phil Carmody, Mar 27 2012 */

A118411(n)/A118412(n) = SUM[i=1..n] (1/A000332(n)). A118411(n)/A118412(n) = SUM[i=1..n] (1/C(n+2,4)). A118411(n)/A118412(n) = SUM[i=1..n] (1/(n*(n+1)*(n+2)*(n+3)/24)).

A118412 Denominator of sum of reciprocals of first n pentatope numbers A000332.

Original entry on oeis.org

1, 5, 15, 105, 42, 63, 90, 495, 55, 143, 91, 1365, 420, 510, 612, 2907, 855, 665, 385, 1771, 1518, 1725, 1950, 8775, 2457, 5481, 1015, 4495, 1240, 4092, 4488, 19635, 5355, 11655, 6327, 9139, 2470, 2665, 8610, 37023

Offset: 1

Jonathan Vos Post, Apr 27 2006

Numerators are A118411. Fractions are: 1/1, 6/5, 19/15, 136/105, 83/63, 119/90, 656/495, 73/55, 190/143, 121/91, 1816/1365, 559/420, 679/510, 815/612, 3872/2907, 1139/855, 886/665, 513/385, 2360/1771, 2023/1518, 2299/1725, 2599/1950, 11696/8775, 3275/2457, 7306/5481, 1353/1015, 5992/4495, 1653/1240, 5455/4092, 5983/4488, 26176/19635, 7139/5355, 15538/11655, 8435/6327, 12184/9139, 3293/2470, 3553/2665, 11479/8610, 49360/37023. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392.

			a(1) = 1 = denominator of 1/1.
a(2) = 5 = denominator of 6/5 = 1/1 + 1/5.
a(3) = 15 = denominator of 19/15 = 1/1 + 1/5 + 1/15.
a(4) = 105 = denominator of 136/105 = 1/1 + 1/5 + 1/15 + 1/35.
a(5) = 42 = denominator of 55/42 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70.
a(10) = 143 = denominator of 190/143 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715.
a(20) = 1771 = denominator of 2360/1771 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715 + 1/1001 + 1/1365 + 1/1820 + 1/2380 + 1/3060 + 1/3876 + 1/4845 + 1/5985 + 1/7315 + 1/8855.

Cf. A000332, A022998, A026741, A118391, A118391, A118411.

s=0;for(i=4,50,s+=1/binomial(i,4);print(denominator(s))) /* Phil Carmody, Mar 27 2012 */

A118411(n)/A118412(n) = SUM[i=1..n] (1/A000332(n)). A118411(n)/A118412(n) = SUM[i=1..n] (1/C(n+2,4)). A118411(n)/A118412(n) = SUM[i=1..n] (1/(n*(n+1)*(n+2)*(n+3)/24)).

A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

Original entry on oeis.org

1, 3, 9, 7, 17, 11, 25, 15, 33, 19, 41, 23, 49, 27, 57, 31, 65, 35, 73, 39, 81, 43, 89, 47, 97, 51, 105, 55, 113, 59, 121, 63, 129, 67, 137, 71, 145, 75, 153, 79, 161, 83, 169, 87, 177, 91, 185, 95, 193, 99, 201, 103, 209, 107, 217, 111, 225, 115, 233, 119, 241, 123, 249

Offset: 1

T. D. Noe, May 28 2010

The odd non-multiples of 3 are 1, 5, 7, 11,... (A007310). The odd multiples of 3 have no odd numbers their Collatz pre-image. The next odd number in the Collatz iteration of a(2n) is 6n-1. The next odd number in the Collatz iteration of a(2n+1) is 6n+1. For each non-multiple of 3, there are an infinite number of odd numbers in its Collatz pre-image. For example:
Odd pre-images of 1: 1, 5, 21, 85, 341,... (A002450)
Odd pre-images of 5: 3, 13, 53, 213, 853,... (A072197)
Odd pre-images of 7: 9, 37, 149, 597, 2389,...
Odd pre-images of 11: 7, 29, 117, 469, 1877,...(A072261)
In each case, the pre-image sequence is t(k+1) = 4*t(k) + 1 with t(0)=a(n). The array of pre-images is in A178415.
a(n) = A047529(P(n)), with the permutation P(n) = A006368(n-1) + 1, for n >= 1. This shows that this sequence gives the numbers {1, 3, 7} (mod 8) uniquely. - Wolfdieter Lang, Sep 21 2021

Matthew House, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A002450, A006368, A007310, A022998, A047529, A072197, A072261, A114752.

Riffle[1+8*Range[0,50], 3+4*Range[0,50]]

a(n) = (n - 1)*(3 - (-1)^n) + 1. [Bogart B. Strauss, Sep 20 2013, adapted to the offset by Matthew House, Feb 14 2017]
From Matthew House, Feb 14 2017: (Start)
G.f.: x*(1 + 3*x + 7*x^2 + x^3)/((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4). (End)
From Philippe Deléham, Nov 06 2023: (Start)
a(2*n) = 4*n-1, a(2*n+1) = 8*n+1.
a(n) = 2*A022998(n-1)+1.
a(n) = 2*A114752(n)-1. (End)

A317314 Multiples of 14 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 14, 3, 28, 5, 42, 7, 56, 9, 70, 11, 84, 13, 98, 15, 112, 17, 126, 19, 140, 21, 154, 23, 168, 25, 182, 27, 196, 29, 210, 31, 224, 33, 238, 35, 252, 37, 266, 39, 280, 41, 294, 43, 308, 45, 322, 47, 336, 49, 350, 51, 364, 53, 378, 55, 392, 57, 406, 59, 420, 61, 434, 63, 448, 65, 462, 67, 476, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 18-gonal numbers (A274979).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 18-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. A008596 and A005408 interleaved.
Column 14 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).
Cf. A274979.

Table[4 n + 3 n (-1)^n, {n, 0, 80}] (* Wesley Ivan Hurt, Nov 25 2021 *)

a(n) = if(n%2==0, return(14*n/2), return(n)) \\ Felix Fröhlich, Jul 26 2018

concat(0, Vec(x*(1 + 14*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 14*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 14*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
a(n) = 4*n + 3*n*(-1)^n. - Wesley Ivan Hurt, Nov 25 2021
Multiplicative with a(2^e) = 7*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^(2-s)). - Amiram Eldar, Oct 25 2023

A317315 Multiples of 15 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 15, 3, 30, 5, 45, 7, 60, 9, 75, 11, 90, 13, 105, 15, 120, 17, 135, 19, 150, 21, 165, 23, 180, 25, 195, 27, 210, 29, 225, 31, 240, 33, 255, 35, 270, 37, 285, 39, 300, 41, 315, 43, 330, 45, 345, 47, 360, 49, 375, 51, 390, 53, 405, 55, 420, 57, 435, 59, 450, 61, 465, 63, 480, 65, 495, 67, 510, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 19-gonal numbers (A303813).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 19-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. A008597 and A005408 interleaved.
Column 15 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. A303813.

a[n_] := If[OddQ[n], n, 15*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

concat(0, Vec(x*(1 + 15*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 15*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 15*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) = 15*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 + 13/2^s). - Amiram Eldar, Oct 25 2023

A317316 Multiples of 16 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 16, 3, 32, 5, 48, 7, 64, 9, 80, 11, 96, 13, 112, 15, 128, 17, 144, 19, 160, 21, 176, 23, 192, 25, 208, 27, 224, 29, 240, 31, 256, 33, 272, 35, 288, 37, 304, 39, 320, 41, 336, 43, 352, 45, 368, 47, 384, 49, 400, 51, 416, 53, 432, 55, 448, 57, 464, 59, 480, 61, 496, 63, 512, 65, 528, 67, 544, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 20-gonal numbers (A218864).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 20-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. A008598 and A005408 interleaved.
Column 16 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. A218864.

a[n_] := If[OddQ[n], n, 8*n]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

concat(0, Vec(x*(1 + 16*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 16*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 16*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) = 2^(e+3), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 7*2^(1-s)). - Amiram Eldar, Oct 25 2023

cp's OEIS Frontend

A143051 Smallest number not occurring earlier and smaller than the largest square so far, the next square if no such number exists.

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A225975 Square root of A226008(n).

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A317312 Multiples of 12 and odd numbers interleaved.

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A317313 Multiples of 13 and odd numbers interleaved.

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A118411 Numerator of sum of reciprocals of first n pentatope numbers A000332.

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PARI

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A118412 Denominator of sum of reciprocals of first n pentatope numbers A000332.

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A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

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A317314 Multiples of 14 and odd numbers interleaved.

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