A080512 -id:A080512 - cp's OEIS frontend

A159634 Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

1, 2, 4, 4, 6, 8, 8, 8, 12, 12, 12, 16, 14, 16, 24, 16, 18, 24, 20, 24, 32, 24, 24, 32, 30, 28, 36, 32, 30, 48, 32, 32, 48, 36, 48, 48, 38, 40, 56, 48, 42, 64, 44, 48, 72, 48, 48, 64, 56, 60, 72, 56, 54, 72, 72, 64, 80, 60, 60, 96, 62, 64, 96, 64, 84, 96, 68, 72, 96, 96

Offset: 1

Steven Finch, Apr 17 2009

Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).
We have
m(7/2,N)+s(5/2,N) = m(5/2,N)+s(7/2,N) =
(m(11/2,N)+s(9/2,N))/2 = (m(9/2,N)+s(11/2,N))/2 =
(m(15/2,N)+s(13/2,N))/3 = (m(13/2,N)+s(15/2,N))/3 = ...
(m((4j+3)/2,N)+s((4j+1)/2,N))/j = (m((4j+1)/2,N)+s((4j+3)/2,N))/j = ...
where N is any positive multiple of 4 and j>=1.
Multiplicative because A001615 is multiplicative and a(1) = A001615(2)/3 = 1. - Andrew Howroyd, Aug 08 2018

Ken Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004, (p. 16, theorem 1.56).

Peter Luschny, Table of n, a(n) for n = 1..1000
H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78; Scanned copy.
S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]
Peter Humphries, Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula", MathOverflow, 2014.
Jon Maiga, Computer-generated formulas for A159634, Sequence Machine.
Wikipedia, Cusp Form.

Cf. A159635, A159636. - Steven Finch, Apr 22 2009

Cf. A000010, A000082, A001221, A001615, A008683, A059956, A080512, A159630.

[[4*n,(Dimension(HalfIntegralWeightForms(4*n,7/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2))))/2] : n in [1..70]]; [[4*n,(Dimension(HalfIntegralWeightForms(4*n,5/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,7/2))))/2] : n in [1..70]];

(* per Enrique Pérez Herrero's conjecture proved by P. Humphries, see link *)
dedekindPsi[n_Integer]:=n Apply[Times,1+1/Map[First,FactorInteger[n]]];
1/3 dedekindPsi /@ (2 Range[70]) (* Wouter Meeussen, Apr 06 2014 *)

a(n) = 2*n*sumdiv( 2*n, d, moebius(d)^2 / d)/3; \\ Andrew Howroyd, Aug 08 2018

a(n) = A159636(n) + A159630(n). - Enrique Pérez Herrero, Apr 15 2014
a(n) = A001615(2*n)/3. - Enrique Pérez Herrero, Jan 31 2014
From Peter Bala, Mar 19 2019: (Start)
a(n)= n*Product_{p|n, p odd prime} (1 + 1/p).
a(n) = Sum_{d|n, d odd} mu(d)^2*n/d, where mu(n) = A008683(n) is the Möbius function.
If n = m*2^k , where 2^k is the largest power of 2 dividing n, then
a(n) = (2^k)*a(m) = 2^k * Sum_{d^2|m} mu(d)*sigma(m/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also
a(n) = 2^k * Sum_{d|m} 2^omega(d)*phi(m/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.
O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^(2^n-1)/(1 - x^(2*n-1))^2. (End)
a(n) = A000082(n)/A080512(n). [obvious by prime products, discovered by Sequence Machine]. - R. J. Mathar, Jun 24 2021
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(2^e) = 2^e, and a(p^e) = (p+1)*p^(e-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 6/Pi^2 = 0.607927... (A059956). (End)

A296955 Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 3, 4, 3, 1, 10, 1, 3, 9, 7, 1, 12, 1, 12, 11, 3, 1, 24, 6, 3, 13, 14, 1, 27, 1, 15, 15, 3, 13, 37, 1, 3, 17, 30, 1, 33, 1, 18, 33, 3, 1, 52, 8, 18, 21, 20, 1, 39, 17, 36, 23, 3, 1, 78, 1, 3, 41, 31, 19, 45, 1, 24, 27, 39, 1, 87, 1, 3, 49, 26, 19, 51, 1, 66, 40, 3, 1, 98, 23, 3, 33, 48, 1, 99, 21, 30, 35, 3, 25, 108, 1

Offset: 1

Wesley Ivan Hurt, Dec 22 2017

The number of partitions of n into 3 parts whose "middle" part divides n. - Wesley Ivan Hurt, Oct 21 2021

			a(12) = 10; the partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). 1 divides 11, 2 divides 10, 3 divides 9 and 4 divides 8, so the sum of the smaller parts gives 1 + 2 + 3 + 4 = 10.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Index entries for sequences related to partitions

Cf. A297024, A001065, A023645, A000203, A080512.

with(numtheory):
a := n -> add( d, d = divisors(n) minus {floor((n+1)/2), n} ):
seq(a(n), n = 1..100); # Peter Bala, Jan 13 2021

Table[Sum[i (Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
f[n_] := Plus @@ Select[Divisors@n, 2 # < n &]; Array[f, 75] (* Robert G. Wilson v, Dec 23 2017 *)

A296955(n) = sumdiv(n,d,(d<(n/2))*d); \\ Antti Karttunen, Sep 25 2018

a(n) = Sum_{i=1..floor((n-1)/2)} i * (floor(n/i) - floor((n-1)/i)).
a(n) = the sum of the divisors < n/2. - Robert G. Wilson v, Dec 23 2017
a(n) = 1 iff n is an odd prime or n=4. - Robert G. Wilson v, Dec 23 2017
G.f.: Sum_{k>=1} k * x^(3*k) / (1 - x^k). - Ilya Gutkovskiy, May 30 2020
G.f.: Sum_{k >= 3} x^k/(1 - x^k)^2. Cf. A023645. - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+2))*( n*q^(3*n+4) - (n + 1)*q^(2*n+2) - (n - 1)*q^(n+2) + n )/( (1 - q^n )*(1 - q^(n+2))^2 ). (In equation 1 in Arndt, after combining the two n = 0 summands to get t/(1 - t), apply the operator t*d/dt and then set t = q^2 and x = 1. Cf. A001065.) - Peter Bala, Jan 22 2021
a(n) = A000203(n) - A080512(n). - Ridouane Oudra, Aug 15 2024

More terms from Antti Karttunen, Sep 25 2018

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

A080511 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose arithmetic mean is an integer.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 1, 2, 3, 6, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 9, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 21

Offset: 1

Amarnath Murthy, Mar 20 2003

The n-th row is {1,2,...,n-1,x}, where x=n if n is odd, x=3n/2 if n is even.

			Triangle starts:
1;
1, 3;
1, 2, 3;
1, 2, 3, 6;
1, 2, 3, 4, 5;
1, 2, 3, 4, 5, 9;
1, 2, 3, 4, 5, 6, 7;
1, 2, 3, 4, 5, 6, 7, 12;
...

Cf. A080512, A008619, A080504, A080508.

T:= proc(n) $1..n-1, `if`(irem(n, 2)=1, n, 3*n/2) end:
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 29 2013

row[n_] := Append[Range[n - 1], If[OddQ[n], n, 3 n/2]];
Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, May 21 2016 *)

A111710 Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.

Original entry on oeis.org

1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112, 133, 148, 172, 189, 216, 235, 265, 286, 319, 342, 378, 403, 442, 469, 511, 540, 585, 616, 664, 697, 748, 783, 837, 874, 931, 970, 1030, 1071, 1134, 1177, 1243, 1288, 1357, 1404, 1476, 1525, 1600, 1651, 1729

Offset: 1

Amarnath Murthy, Aug 24 2005

			The fourth row is 8,9,10 and 13,(8+9+10 +13)/4 = 10.
Triangle begins:
1
2 4
5 6 7
8 9 10 13
14 15 16 17 18
19 20 21 22 23 27
28 29 30 31 32 33 34

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

Cf. A111711, A111712.

Cf. A085787. - R. J. Mathar, Aug 15 2008

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 7, 13, 18}, 100] (* Paolo Xausa, Feb 09 2024 *)

Vec(x*(1+3*x+x^2)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

a(1)=1, a(2n) = a(2n-1)+3n, a(2n+1)=a(2n)+2n+1. - Franklin T. Adams-Watters, May 01 2006
G.f.: -x*(1+3*x+x^2) / ( (1+x)^2*(x-1)^3 ). a(n+1)-a(n) = A080512(n+1). - R. J. Mathar, May 02 2013
From Colin Barker, Jan 26 2016: (Start)
a(n) = (10*n^2+2*(-1)^n*n+10*n+(-1)^n-1)/16.
a(n) = (5*n^2+6*n)/8 for n even.
a(n) = (5*n^2+4*n-1)/8 for n odd. (End)

More terms from Franklin T. Adams-Watters, May 01 2006

A111711 Leading column of triangle mentioned in A111710.

Original entry on oeis.org

1, 2, 5, 8, 14, 19, 28, 35, 47, 56, 71, 82, 100, 113, 134, 149, 173, 190, 217, 236, 266, 287, 320, 343, 379, 404, 443, 470, 512, 541, 586, 617, 665, 698, 749, 784, 838, 875, 932, 971, 1031, 1072, 1135, 1178, 1244, 1289, 1358, 1405, 1477, 1526, 1601, 1652

Offset: 1

Amarnath Murthy, Aug 24 2005

Also partial sums of A257143. - Reinhard Zumkeller, Apr 17 2015

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

Cf. A111710, A111712, A080512, A257143.

a111711 n = a111711_list !! (n-1)
a111711_list = 1 : zipWith (+) a111711_list a080512_list
-- Reinhard Zumkeller, Apr 17 2015

LinearRecurrence[{1,2,-2,-1,1},{1,2,5,8,14},60] (* Harvey P. Dale, Jun 21 2023 *)

a(1)=1, a(2n) = a(2n-1)+2n-1, a(2n+1)=a(2n)+3n; a(n) = A111710(n-1)+1. - Franklin T. Adams-Watters, May 01 2006
From Chai Wah Wu, Mar 05 2021: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
G.f.: x*(-x^4 - x^3 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2). (End)
a(n) = (10*n*(n-1) + (-1)^n*(1-2*n)+15)/16.  - Eric Simon Jacob, Jun 11 2022

More terms from Franklin T. Adams-Watters, May 01 2006

A257143 a(2n) = 3n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

Original entry on oeis.org

1, 1, 3, 3, 6, 5, 9, 7, 12, 9, 15, 11, 18, 13, 21, 15, 24, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 29, 45, 31, 48, 33, 51, 35, 54, 37, 57, 39, 60, 41, 63, 43, 66, 45, 69, 47, 72, 49, 75, 51, 78, 53, 81, 55, 84, 57, 87, 59, 90, 61, 93, 63, 96, 65, 99

Offset: 0

Michael Somos, Apr 16 2015

			G.f. = 1 + x + 3*x^2 + 3*x^3 + 6*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 12*x^8 + ...

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

Cf. A080512, A111711 (partial sums), A188626.

Cf. A005408, A008486.

import Data.List (transpose)
a257143 n = a257143_list !! n
a257143_list = concat $ transpose [a008486_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], n, True, 3 n/2];
a[ n_] := SeriesCoefficient[ (1 + x + x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4), {x, 0, n}];

{a(n) = if( n<1, n==0, n%2, n, 3*n/2)};

{a(n) = if( n<0, 0, polcoeff( (1 - x^5) / ((1 - x) * (1 - x^2)^2) + x * O(x^n), n))};

a(n) is multiplicative with a(2^e) = 3 * 2^(e-1) if e>0, a(p^e) = p^e otherwise and a(0) = 1.
Euler transform of length 5 sequence [ 1, 2, 0, 0, -1].
G.f.: (1 - x^5) / ((1 - x) * (1 - x^2)^2).
G.f.: (1 + x + x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4).
a(n) = A080512(n) if n>0.
First difference of A111711.
A188626(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.
From Amiram Eldar, Jan 03 2023: (Start)
Dirichlet g.f.: zeta(s-1)*(1+1/2^s).
Sum_{k=1..n} a(k) ~ (5/8) * n^2. (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

cp's OEIS Frontend

A159634 Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

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A296955 Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger.

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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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A080511 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose arithmetic mean is an integer.

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A111710 Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.

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A111711 Leading column of triangle mentioned in A111710.

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A257143 a(2n) = 3n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.