A259361 -id:A259361 - cp's OEIS frontend

A264052 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A259361(n)) is the number of integer partitions of n having k distinct parts occurring at least twice.

1, 1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 6, 1, 5, 9, 1, 6, 13, 3, 8, 18, 4, 10, 23, 9, 12, 32, 12, 15, 42, 19, 1, 18, 55, 27, 1, 22, 69, 41, 3, 27, 89, 56, 4, 32, 112, 78, 9, 38, 141, 106, 12, 46, 175, 141, 23, 54, 217, 188, 31, 64, 266, 247, 49, 1, 76, 326, 321, 68, 1

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.
T(n,k) is also the number of integer partitions of n having k parts from which one can subtract 2 and still get an integer partition (mapping a partition to its conjugate sends one statistic to the other).
T(n,k) is also the number of integer partitions of n having k distinct even parts. Example: T(6,2)= 1, counting the partition [2,4]. - Emeric Deutsch, Sep 19 2016

			Triangle begins:
   1,
   1,
   1,  1,
   2,  1,
   2,  3,
   3,  4,
   4,  6, 1,
   5,  9, 1,
   6, 13, 3,
   8, 18, 4,
  10, 23, 9,
  ...
T(6,2)= 1; namely [1,1,2,2]. - _Emeric Deutsch_, Sep 19 2016

Alois P. Heinz, Rows n = 0..1000, flattened
FindStat - Combinatorial Statistic Finder, The number of distinct parts of a partition that occur at least twice, The number of parts from which one can subtract 2 and still get an integer partition.
V. V. Tewari, Kronecker Coefficients For Some Near-Rectangular Partitions, arXiv:1403.5327 [math.CO], 2014, MathSciNet:3320625.

Cf. A000009, A000041, A024786, A090867, A259361.

b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
      `if`(j>1, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..25);  # Alois P. Heinz, Nov 02 2015
# second Maple program:
g := product((1-(1-t)*x^(2*j))/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 25)): for n from 0 to 23 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, i), i = 0 .. degree(P[n])) end do; # yields sequence in triangular form - Emeric Deutsch, Nov 12 2015

T[n_, k_] := SeriesCoefficient[QPochhammer[1-t, x^2]/(t*QPochhammer[x]), {x, 0, n}, {t, 0, k}]; Table[DeleteCases[Table[T[n, k], {k, 0, n}], 0], {n, 0, 25}] // Flatten (* Jean-François Alcover, Dec 11 2016 *)

From Emeric Deutsch, Nov 12 2015: (Start)
G.f.: G(t,x) = Product_{j>=1} ((1-(1-t)x^{2j})/(1-x^j)).
T(n,0) = A000009(n).
T(n,1) = A090867(n).
Sum_{k>=0} k*T(n,k) = A024786(n).
(End)

More terms from Alois P. Heinz, Nov 02 2015

A328346 Triangle read by rows: T(n,k) is the coefficient of x^(n - k*(k+1)) in Product_{j=1..k} 1/(1 - x^j) for n >= 0, 0 <= k <= A259361(n).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 3, 0, 1, 4, 1, 0, 1, 4, 1, 0, 1, 5, 2, 0, 1, 5, 3, 0, 1, 6, 4, 0, 1, 6, 5, 0, 1, 7, 7, 0, 1, 7, 8, 0, 1, 8, 10, 1, 0, 1, 8, 12, 1, 0, 1, 9, 14, 2, 0, 1, 9, 16, 3, 0, 1, 10, 19, 5, 0, 1, 10, 21, 6

Offset: 0

Seiichi Manyama, Oct 13 2019

			Triangle begins:
  1;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2;
  0, 1, 3;
  0, 1, 3;
  0, 1, 4,  1;
  0, 1, 4,  1;
  0, 1, 5,  2;
  0, 1, 5,  3;
  0, 1, 6,  4;
  0, 1, 6,  5;
  0, 1, 7,  7;
  0, 1, 7,  8;
  0, 1, 8, 10, 1;

Seiichi Manyama, Rows n = 0..420, flattened
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Row sums give A003106.

Cf. A008284, A259361, A268187.

T(n, k) = polcoef(1/prod(j=1, k, 1-x^j+x*O(x^n)), n-k*(k+1));
tabf(nn) = for(n=0, nn, for(k=0, (-1+sqrt(1+4*n))/2, print1(T(n, k), ", ")); print)

A003106 Number of partitions of n into parts 5k+2 or 5k+3.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 16, 20, 22, 26, 29, 35, 38, 45, 50, 58, 64, 75, 82, 95, 105, 120, 133, 152, 167, 190, 210, 237, 261, 295, 324, 364, 401, 448, 493, 551, 604, 673, 739, 820, 899, 997, 1091, 1207, 1321, 1457, 1593, 1756, 1916, 2108, 2301

Offset: 0

N. J. A. Sloane, Herman P. Robinson

Expansion of Rogers-Ramanujan function H(x) in powers of x.
Also number of partitions of n such that the number of parts is greater by one than the smallest part. - Vladeta Jovovic, Mar 04 2006
Example: a(10)=4 because we have [9, 1], [6, 2, 2], [5, 3, 2] and [4, 4, 2]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then there are exactly k-1 parts equal to k. Example: a(10)=4 because we have [3, 3, 2, 2], [3, 3, 2, 1, 1], [3, 3, 1, 1, 1, 1] and [2, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then k occurs at least k+1 times. Example: a(10)=4 because we have [2, 2, 2, 2, 2], [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that the smallest part is larger than the number of parts. Example: a(10)=4 because we have [10], [7, 3], [6, 4] and [5, 5]. - Emeric Deutsch, Apr 09 2006
Also number of partitions into distinct parts where parts differ by at least 2 and with minimal part >= 2, a(0)=1 because the condition is void for the empty list. - Joerg Arndt, Jan 04 2011
The g.f. is the special case D=2 of Sum_{n>=0} x^(D*n*(n+1)/2) / Product_{k=1..n} (1-x^k), the g.f. or partitions into distinct parts where the difference between successive parts is >= D and the minimal part >= D. - Joerg Arndt, Mar 31 2014
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[2](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109699 and A109698. - Vaclav Kotesovec, Jan 21 2017

			G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + 4*x^11 + ...
G.f. = q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + 3*q^551 + ...
From _Joerg Arndt_, Dec 27 2012: (Start)
The a(18)=15: the partitions of 18 where all parts are 2 or 3 (mod 5) are
[ 1]  [ 2 2 2 2 2 2 2 2 2 ]
[ 2]  [ 3 3 2 2 2 2 2 2 ]
[ 3]  [ 3 3 3 3 2 2 2 ]
[ 4]  [ 3 3 3 3 3 3 ]
[ 5]  [ 7 3 2 2 2 2 ]
[ 6]  [ 7 3 3 3 2 ]
[ 7]  [ 7 7 2 2 ]
[ 8]  [ 8 2 2 2 2 2 ]
[ 9]  [ 8 3 3 2 2 ]
[10]  [ 8 7 3 ]
[11]  [ 8 8 2 ]
[12]  [ 12 2 2 2 ]
[13]  [ 12 3 3 ]
[14]  [ 13 3 2 ]
[15]  [ 18 ]
(End)
From _Wolfdieter Lang_, Oct 29 2016: (Start)
The a(18)=15 partitions of 18 without part 1 and parts differing by at least 2 are:
  [18]; [16,2], [15,3], [14,4], [13,5], [12,6], [11,7], [10,8]; [12,4,2], [11,5,2], [10,6,2], [9,7,2],[10,5,3], [9,6,3], [8,6,4]. The semicolon separates different number of parts. The maximal number of parts is A259361(18) = 3. (End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(f), p. 591.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 290-291.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums, arXiv:hep-th/0505097, 2005.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
I. Martinjak, D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Cf. A003114.
Cf. A047221, A219607.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.

a003106 = p a047221_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 30 2012

g:=1/product((1-x^(5*j-2))*(1-x^(5*j-3)),j=1..15): gser:=series(g,x=0,66): seq(coeff(gser,x,n),n=0..63); # Emeric Deutsch, Apr 09 2006

max = 63; g[x_] := 1/Product[(1-x^(5j-2))*(1-x^(5j-3)), {j, 1, Floor[max/4]}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 17 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] > Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, -1, -1, 0, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
nmax = 63; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 2}~Join~{Range[0, kmax]*5 + 3}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

{a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, (sqrtint(4*n + 1) - 1) \ 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n - k^2 - k))), 1), n))}; /* Michael Somos, Oct 15 2008 */

The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n*(n+1))/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-2))*(1-t^(5*n-3))); this is the g.f. for the sequence.
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2 + 2*k) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - Michael Somos, Oct 15 2008
From Joerg Arndt, Oct 10 2012: (Start)
Bill Gosper gives (message to the math-fun mailing list, Oct 07 2012)
prod(k>=0, [0 , a;  q^k, 1]) = [0, X(a,q);  0, Y(a,q)] where
X(a,q) = a * sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^n) ) and
Y(a,q) = sum(n>=0, a^n*q^(n^2-n) / prod(k=1..n, 1-q^n) ).
Set a=q to obtain prod(k>=0, [0 , a;  q^k, 1]) = [0, q*H(q);  0, G(q)] where
H(q) is the g.f. of A003106 and G(q) is the g.f. of A003114.
Bill Gosper and N. J. A. Sloane give (message to math-fun, Oct 10 2012)
prod(k>=0, [0 , a*q^k;  1, 1]) = [U(a,q), U(a,q);  V(a,q), V(a,q)] where
U(a,q) = a * sum(n>=0, a^n*q^(n^2+n) / prod(k=1..n, 1-q^k) ) and
V(a,q) = sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^k) ).
Set a=1 to obtain prod(k>=0, [0 , q^k;  1, 1]) = [H(q), H(q);  G(q), G(q)].
(End)
Expansion of f(-x^5) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, May 06 2015
Expansion of f(-x, -x^4) / f(-x) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ sqrt((sqrt(5)-1)/5) * exp(2*Pi*sqrt(n/15)) / (2^(3/2) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(60*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 24 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284152(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

A264591 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[4](q).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 26, 28, 33, 36, 42, 46, 53, 58, 67, 73, 83, 91, 104, 113, 128, 140, 158, 173, 194, 212, 238, 260, 290, 317, 353, 385, 428, 467, 517, 564

Offset: 0

N. J. A. Sloane, Nov 18 2015

nonn

It is conjectured that G[i](q) = 1 + O(q^i) for all i.
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[4](x). - N. J. A. Sloane, Nov 22 2015
From Wolfdieter Lang, Nov 02 2016: (Start)
The second g.f. given below leads to a combinatorial partition interpretation from (2 + 4 + ... + 2*m) + 2*m = m*(m+3). Take for the sum term m the special M=m+1 part partition [2m,2m,2*(m-1),...,4,2] together with arbitrary partitions of N with part number m' <= M-1 = m added to the first m' parts.
Summing over m>=1 leads to partitions of n = m*(m+3) + N which have no part 1, only one part 2 except for n=4 and for number of parts M >= 3 the difference of parts except of the first two parts has to be at least 2. See the examples below.
A simpler interpretation uses m*(m+3) = 4 + 6 + ... + 2*(m+1), leading to a(n) as the number of partitions of n with parts >= 4 and parts differing by at least 2.
This is in the spirit of MacMahon's and Schur's interpretation of the sum version of the Rogers-Ramanujan identities. See the Hardy and Hardy-Wright references under A003114. (End)

			From _Wolfdieter Lang_, Nov 02 2016: (Start)
a(0) = 1 from the n=0 sum term (undefined product put to 1),
a(n) = 1 for n=4..9 from the partition [n-2,2],
a(10) = 2 from [8,2] and [4,4,2],
a(11) = 2 from [9,2] and [5,4,2],
a(12) = 3 from [10,2], [6,4,2], [5,5,2],
a(18) = 7 from [16,2], all 1+4=5 partitions of 18-10 = 8 with part number <= 2 added to the first two part of [4,4,2] and the new four part partition [6,6,4,2].
The maximal number of parts needed for n is floor((-1+sqrt(9+4*n))/2) = A259361(n+2).
A simpler interpretation:
a(18) = 7 from the partitions of 18 with parts >=4 and parts differing by at least 2: [18], [14,4], [13,5], [12,6], [11,7], [10,8], [8,6,4].
The maximal number of parts needed for n is floor((-3+sqrt(9+4*n))/2).
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211.

For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+3))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 24 2016 *)

From Wolfdieter Lang, Nov 02 2016: (Start)
G.f.: G[4](q) = (Sum_{n >= 0} (-1)^n*(1 - q^(n+1))*(1 - q^(n+2))*(1 - q^(2*n+3))*q^((5*n+11)*n/2)) / (Product_{j >= 1} (1 - q^j)), from the Andrews-Baxter (AB) reference, eq. (3.7).
G.f.: Sum_{m >= 0} q^(m*(m+3)) / Product_{j=1..m} (1-q^j) from (AB) eq. 51.
This can also be derived from the Hardy (H) or Hardy-Wright reference (see A006141): Put G_4(a,q):= (H_1(a,q) - H_1(a*q,q)) / (a*q) with H_1(a,x) from (H) p. 95, first eq. Then G[4](q) = G_4(q,q). (End)
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(5/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016

A257213 Least d>0 such that floor(n/d) = floor(n/(d+1)).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 4, 4, 3, 5, 4, 4, 5, 5, 5, 4, 6, 6, 5, 5, 7, 6, 6, 6, 5, 7, 7, 7, 6, 6, 8, 8, 7, 7, 7, 6, 8, 8, 8, 8, 7, 7, 9, 9, 9, 8, 8, 8, 7, 10, 9, 9, 9, 9, 8, 8, 10, 10, 10, 10, 9, 9, 9, 8, 11, 11, 10, 10, 10, 10, 9, 9, 11, 11, 11, 11, 11, 10, 10, 10

Offset: 0

M. F. Hasler, Apr 18 2015

For n > 1: a(A043548(n)) = n. - Reinhard Zumkeller, Apr 19 2015

			a(0)=1 because 0/1 = 0/2.
a(1)=2 because [1/1] = 1 > [1/2] = 0 = [1/3], where [x] := floor(x).
a(2)=3 because [2/1] = 2 > [2/2] = 1 > [2/3] = 0 = [2/4].

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A003059, A257212.

Cf. A043548.

a257213 n = head [d | d <- [1..], div n d == div n (d + 1)]
-- Reinhard Zumkeller, Apr 19 2015

f[n_] := Block[{d, k}, Reap@ For[k = 0, k <= n, k++, d = 1; While[Floor[k/d] != Floor[k/(d + 1)], d++]; Sow@ d] // Flatten // Rest]; f@ 79 (* Michael De Vlieger, Apr 18 2015 *)
Table[Module[{d=1},While[Floor[n/d]!=Floor[n/(d+1)],d++];d],{n,0,80}] (* Harvey P. Dale, Aug 16 2025 *)

A257213(n)=for(d=sqrtint(n)+1,n+1,n\d==n\(d+1)&&return(d))

a(n) >= A003059(n+1) = floor(sqrt(n))+1 >= A003059(n) = ceiling(sqrt(n)) >= A257212(n), with strict inequality (in the middle relation) when n is a square.
a(k^2-1) = k for k > 1. Proof: For n=k^2-1=(k-1)*(k+1), floor(n/k) = k-1 = n/(k+1), but n/(k-1)=k+1 and when denominators decrease further, this keeps increasing.
a(k^2) >= k+d when k > d*(d-1). Proof: This follows from k^2/(k+d) = k-d+d^2/(k+d), which shows that a(k) >= d when k > d*(d-1).
a(n) = A259361(n) + 1 + floor(sqrt((A232091(n+1) - 1 - n) + A079813(n+1)) + A079813(n+1)/2) = floor((sqrt(4*n+1)+1)/2) + floor(sqrt(ceiling((n+1) / ceiling(sqrt(n+1)) + 1) * ceiling(sqrt(n+1)) - round(sqrt(n+1)) - n - 1) + (ceiling(sqrt(n+1)) - round(sqrt(n+1)))/2). - Haofen Liang, Aug 25 2021
a(n) = floor(sqrt(p*q - n) + (p + q)/2), where p = floor(sqrt(n)) and q = floor(sqrt(n+1) + 3/2). - Ridouane Oudra, Jan 24 2023

cp's OEIS Frontend

A264052 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A259361(n)) is the number of integer partitions of n having k distinct parts occurring at least twice.

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A328346 Triangle read by rows: T(n,k) is the coefficient of x^(n - k*(k+1)) in Product_{j=1..k} 1/(1 - x^j) for n >= 0, 0 <= k <= A259361(n).

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A003106 Number of partitions of n into parts 5k+2 or 5k+3.

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A264591 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[4](q).

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A257213 Least d>0 such that floor(n/d) = floor(n/(d+1)).

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