A000123 -id:A000123 - cp's OEIS frontend

A008804 Expansion of 1/((1-x)^2(1-x^2)(1-x^4)).

1, 2, 4, 6, 10, 14, 20, 26, 35, 44, 56, 68, 84, 100, 120, 140, 165, 190, 220, 250, 286, 322, 364, 406, 455, 504, 560, 616, 680, 744, 816, 888, 969, 1050, 1140, 1230, 1330, 1430, 1540, 1650, 1771, 1892, 2024, 2156, 2300, 2444, 2600, 2756, 2925, 3094, 3276, 3458

Offset: 0

N. J. A. Sloane

b(n)=a(n-3) is the number of asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to n, under action of dihedral group D_4(b(0)=b(1)=b(2)=0). G.f. for b(n) is x^3/((1-x)^2*(1-x^2)*(1-x^4)). - Vladeta Jovovic, May 07 2000
If the offset is changed to 5, this is the 2nd Witt transform of A004526 [Moree]. - R. J. Mathar, Nov 08 2008
a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^3. First differs from A000123 at n=8. - Alois P. Heinz, Apr 02 2012
a(n) is the number of bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. For n=1 we have for example 2 such bracelets with 4 black beads and 4 white beads: BBBWBWWW and BBWBWBWW. - Herbert Kociemba, Nov 27 2016
a(n) is the also number of aperiodic bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. This is equivalent to saying that a(n) is the (n+7)th element of the DHK[4] (bracelet, identity, unlabeled, 4 parts) transform of 1, 1, 1, ... (see Bower's link about transforms). Thus, for n >= 1 , a(n) = (DHK[4] c){n+7}, where c = (1 : n >= 1). This is because every bracelet with 4 black beads and n+3 white beads which has no reflection symmetry must also be aperiodic. This statement is not true anymore if we have k black beads where k is even >= 6. - _Petros Hadjicostas, Feb 24 2019

			G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 14*x^5 + 20*x^6 + 26*x^7 + 35*x^8 + ...
There are 10 asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to 7 under action of D_4:
[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]
[1 6] [2 5] [3 4] [2 4] [3 3] [4 2] [5 1] [3 2] [4 1] [2 3]

T. D. Noe, Table of n, a(n) for n = 0..1000
C. G. Bower, Transforms (2)
Petros Hadjicostas, The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry, 2019.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 197
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From _R. J. Mathar_, Nov 08 2008]
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Cf. A000123, A001400, A002620, A004526, A005232, A014557, A032246, A032248, A053307.

Column k=3 of A181322. Column k = 4 of A180472 (but with different offset).

a:=[1,2,4,6,10,14,20,26];; for n in [9..60] do a[n]:=2*a[n-1] -2*a[n-3]+2*a[n-4]-2*a[n-5]+2*a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)^2*(1-x^2)*(1-x^4)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series(1/((1-x)^2*(1-x^2)*(1-x^4)), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 12 2019

LinearRecurrence[{2,0,-2,2,-2,0,2,-1}, {1,2,4,6,10,14,20,26}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
gf[x_,k_]:=x^k/2 (1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])-(1+x)/(1-x^2)^Floor[k/2+1]); CoefficientList[Series[gf[x,4]/x^7,{x,0,60}],x] (* Herbert Kociemba, Nov 27 2016 *)
Table[(84 +12*(-1)^n +85*n +3*(-1)^n*n +24*n^2 +2*n^3 +12*Sin[n Pi/2])/96, {n,0,60}] (* Eric W. Weisstein, Oct 12 2017 *)
CoefficientList[Series[1/((1-x)^4*(1+x)^2*(1+x^2)), {x,0,60}], x] (* Eric W. Weisstein, Oct 12 2017 *)

a(n)=(84+12*(-1)^n+6*I*((-I)^n-I^n)+(85+3*(-1)^n)*n+24*n^2 +2*n^3)/96 \\ Jaume Oliver Lafont, Sep 20 2009

{a(n) = my(s = 1); if( n<-7, n = -8 - n; s = -1); if( n<0, 0, s * polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; /* Michael Somos, Feb 02 2011 */

def A008804_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^2)*(1-x^4))).list()
A008804_list(60) # G. C. Greubel, Sep 12 2019

For a formula for a(n) see A014557.
a(n) = (84 +85*n +24*n^2 +2*n^3 +12*A056594(n+3) +3*(-1)^n*(n+4))/96. - R. J. Mathar, Nov 08 2008
a(n) = 2*(Sum_{k=0..floor(n/2)} A002620(k+2)) - A002620(n/2+2)*(1+(-1)^n)/2. - Paul Barry, Mar 05 2009
G.f.: 1/((1-x)^4*(1+x)^2*(1+x^2)). - Jaume Oliver Lafont, Sep 20 2009
Euler transform of length 4 sequence [2, 1, 0, 1]. - Michael Somos, Feb 05 2011
a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Feb 05 2011
From Herbert Kociemba, Nov 27 2016: (Start)
More generally gf(k) is the g.f. for the number of bracelets without reflection symmetry with k black beads and n-k white beads.
gf(k): x^k/2 * ( (1/k)*Sum_{n|k} phi(n)/(1 - x^n)^(k/n) - (1 + x)/(1 -x^2)^floor(k/2 + 1) ). The g.f. here is gf(4)/x^7 because of the different offset. (End)
E.g.f.: ((48 + 54*x + 15*x^2 + x^3)*cosh(x) + 6*sin(x) + (36 + 57*x + 15*x^2 + x^3)*sinh(x))/48. - Stefano Spezia, May 15 2023
a(n) = A001400(n) + A001400(n-1) + A001400(n-2). - David García Herrero, Aug 26 2024
a(n) = floor((2*n^3 + 24*n^2 + n*(85+3*(-1)^n) + 96) / 96). - Hoang Xuan Thanh, May 24 2025

A050377 Number of ways to factor n into "Fermi-Dirac primes" (members of A050376).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 2, 4, 1, 1, 1, 2, 1

Offset: 1

Christian G. Bower, Nov 15 1999

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Antti Karttunen, Table of n, a(n) for n = 1..100000 (the first 10000 terms from Reinhard Zumkeller)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000123, A001055, A002110, A005117, A018819, A050376-A050380, A025487.

Cf. A108951, A330687 (positions of records), A330688 (record values), A330689, A330690, A382295.

A018819:= proc(n) option remember;
  if n::odd then procname(n-1)
  else procname(n-1) + procname(n/2)
  fi
end proc:
A018819(0):= 1:
f:= n -> mul(A018819(s[2]),s=ifactors(n)[2]):
seq(f(n),n=1..100); # Robert Israel, Jan 14 2016

b[0] = 1; b[n_] := b[n] = b[n - 1] + If[EvenQ[n], b[n/2], 0];
a[n_] := Times @@ (b /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 27 2018 *)

A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
A050377(n) = factorback(apply(e -> A018819(e), factor(n)[, 2])); \\ Antti Karttunen, Dec 28 2019

Dirichlet g.f.: Product_{n in A050376} (1/(1-1/n^s)).
a(p^k) = A000123([k/2]) for all primes p.
a(A002110(n)) = 1.
Multiplicative with a(p^e) = A018819(e). - Christian G. Bower and David W. Wilson, May 22 2005
a(n) = Sum{a(d): d^2 divides n}, a(1) = 1. - Reinhard Zumkeller, Jul 12 2007
a(A108951(n)) = A330690(n). - Antti Karttunen, Dec 28 2019
a(n) = 1 for all squarefree values of n (A005117). - Eric Fox, Feb 03 2020
G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Nov 25 2020
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.7876368001694456669... (A382295), where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)). - Amiram Eldar, Oct 03 2023

More terms from Antti Karttunen, Dec 28 2019

A101417 Number of partitions of n into parts without powers of 2.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 3, 6, 5, 6, 10, 9, 12, 17, 17, 22, 28, 30, 37, 48, 52, 62, 78, 86, 103, 127, 141, 166, 201, 227, 266, 317, 358, 417, 492, 560, 647, 757, 860, 991, 1153, 1309, 1503, 1738, 1971, 2257, 2594, 2941, 3356, 3843, 4351, 4948, 5644, 6382, 7240

Offset: 0

Reinhard Zumkeller, Jan 16 2005

nonn

			a(12) = #{3+3+3+3, 6+3+3, 6+6, 7+5, 9+3, 12} = 6.
From _Gus Wiseman_, Jan 07 2019: (Start)
The a(3) = 1 through a(14) = 5 integer partitions (A = 10, ..., E = 14):
  (3)  (5)  (6)   (7)  (53)  (9)    (A)   (B)    (C)     (D)    (E)
            (33)             (63)   (55)  (65)   (66)    (76)   (77)
                             (333)  (73)  (533)  (75)    (A3)   (95)
                                                 (93)    (553)  (B3)
                                                 (633)   (733)  (653)
                                                 (3333)         (5333)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A018819, A000123.

Cf. A087897, A102430, A276431, A321346, A323053, A323092, A323093.

g:= product(1-x^(2^j),j=0..15)/product(1-x^i,i=1..75): gser:= series(g, x=0,62): seq(coeff(gser,x,n),n=0..59); # Emeric Deutsch, Mar 29 2006

Table[Length[Select[IntegerPartitions[n],And@@Not/@IntegerQ/@Log[2,#]&]],{n,20}] (* Gus Wiseman, Jan 07 2019 *)

G.f.: Product_{j>=1} (1-x^(2^j)) / Product_{i>=2} (1-x^i). - Emeric Deutsch, Mar 29 2006

A088567 Number of "non-squashing" partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19, 24, 25, 31, 32, 40, 41, 50, 51, 63, 64, 77, 78, 95, 96, 114, 115, 138, 139, 163, 164, 194, 195, 226, 227, 266, 267, 307, 308, 357, 358, 408, 409, 471, 472, 535, 536, 612, 613, 690, 691, 785, 786, 881, 882, 995, 996, 1110, 1111

Offset: 0

N. J. A. Sloane, Nov 30 2003

nonn

"Non-squashing" refers to the property that p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k: if the parts are stacked in increasing size, at no point does the sum of the parts above a certain part exceed the size of that part.

			The partitions of n = 1 through 6 are: 1; 2; 3=1+2; 4=1+3; 5=1+4=2+3; 6=1+5=2+4=1+2+3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Amanda Folsom et al., On a general class of non-squashing partitions, Discrete Mathematics 339.5 (2016): 1482-1506.
Y. Homma, J. H. Ryu, and B. Tong, Sequence non-squashing partitions, Slides from a talk, 2014.
O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Cf. A000123, A088575, A088585, A088931, A089054. A090678 gives sequence mod 2.

Cf. A187821 (non-squashing partitions of n into odd parts).

import Data.List (transpose)
a088567 n = a088567_list !! n
a088567_list = 1 : tail xs where
   xs = 0 : 1 : zipWith (+) xs (tail $ concat $ transpose [xs, tail xs])
-- Reinhard Zumkeller, Nov 15 2012

f := proc(n) option remember; local t1,i; if n <= 2 then RETURN(1); fi; t1 := add(f(i),i=0..floor(n/2)); if n mod 2 = 0 then RETURN(t1-1); fi; t1; end;
t1 := 1 + x/(1-x); t2 := add( x^(3*2^(k-1))/ mul( (1-x^(2^j)),j=0..k), k=1..10); series(t1+t2, x, 256); # increase 10 to get more terms

max = 63; f = 1 + x/(1-x) + Sum[x^(3*2^(k-1))/Product[(1-x^(2^j)), {j, 0, k}], {k, 1, Log[2, max]}]; s = Series[f, {x, 0, max}] // Normal; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 06 2014 *)

a(0)=1, a(1)=1; and for m >= 1, a(2m) = a(2m-1) + a(m) - 1, a(2m+1) = a(2m) + 1.
Or, a(0)=1, a(1)=1; and for m >= 1, a(2m) = a(0)+a(1)+...+a(m)-1; a(2m+1) = a(0)+a(1)+...+a(m).
G.f.: 1 + x/(1-x) + Sum_{k>=1} x^(3*2^(k-1))/Product_{j=0..k} (1-x^(2^j)).
G.f.: Product_{n>=0} 1/(1-x^(2^n)) - Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ). (The two terms correspond to A000123 and A088931 respectively.)

A072170 Square array T(n,k) (n >= 0, k >= 2) read by antidiagonals, where T(n,k) is the number of ways of writing n as Sum_{i>=0} c_i 2^i, c_i in {0,1,...,k-1}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 3, 3, 4, 2, 2, 1, 1, 1, 1, 4, 3, 4, 2, 2, 1, 1, 1, 4, 4, 5, 4, 4, 2, 2, 1, 1, 1, 3, 5, 4, 5, 4, 4, 2, 2, 1, 1, 1, 5, 5, 8, 5, 6, 4, 4, 2, 2, 1, 1, 1, 2, 6, 6, 8, 5, 6, 4, 4, 2, 2, 1, 1, 1, 5, 6, 9, 8, 9, 6, 6, 4, 4, 2, 2, 1, 1

Offset: 0

N. J. A. Sloane, Jun 29 2002

k-th column is k-th binary partition function.

The sequence data corresponds (via the table link) to the transpose of the array shown in example and given by the definition. - M. F. Hasler, Feb 14 2019

			Array begins: (rows n >= 0, columns k >= 2)
1 1 1 1 1 1 1 1 ...
1 1 1 1 1 1 1 1 ...
1 2 2 2 2 2 2 2 ...
1 1 2 2 2 2 2 2 ...
1 3 3 4 4 4 4 4 ...
1 2 3 3 4 4 4 4 ...
1 3 4 5 5 6 6 6 ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.

k=3 column is A002487, k=4 is A008619 (positive integers repeated), k = 5, 6, 7 are A007728, A007729, A007730, limiting (infinity) column is A000123 doubled up.

b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<0, 0, add(`if`(n-j*2^i<0, 0,
         b(n-j*2^i, i-1, k)), j=0..k-1)))
    end:
T:= (n, k)-> b(n, ilog2(n), k):
seq(seq(T(d+2-k, k), k=2..d+2), d=0..14); # Alois P. Heinz, Jun 21 2012

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 0, 0, Sum[If[n-j*2^i < 0, 0, b[n-j*2^i, i-1, k]], {j, 0, k-1}]]];
t[n_, k_] := b[n, Length[IntegerDigits[n, 2]] - 1, k];
Table[Table[t[d+2-k, k], {k, 2, d+2}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

M72170=[[]]; A072170(n,k,i=logint(n+!n,2),r=1)={if( !i, k>n, r&&(k<5||k>=n),if(k>4, A000123(n\2)-(k==n), k<3, 1, k<4, A002487(n), n\2+1), M72170[r=setsearch(M72170,[n,k,i,""],1)-1][^-1]==[n,k,i], M72170[r][4], M72170=setunion(M72170,[[n,k,i,r=sum(j=0,min(k-1,n>>i),A072170(n-j*2^i,k,i-1,0))]]);r)} \\ Code for k<5 (using A002487 for k=3) and k>=n (using A000123) is optional but makes it about 3x faster. - M. F. Hasler, Feb 14 2019

T(n,k) = T(n,n+1) = T(n,n)+1 = A000123(floor(n/2)) for all k >= n+1. - M. F. Hasler, Feb 14 2019

A152977 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^n into powers of 2 less than or equal to 2^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 9, 9, 1, 1, 2, 4, 10, 25, 17, 1, 1, 2, 4, 10, 35, 81, 33, 1, 1, 2, 4, 10, 36, 165, 289, 65, 1, 1, 2, 4, 10, 36, 201, 969, 1089, 129, 1, 1, 2, 4, 10, 36, 202, 1625, 6545, 4225, 257, 1, 1, 2, 4, 10, 36, 202, 1827, 17361, 47905, 16641, 513, 1

Offset: 0

Alois P. Heinz, Jan 26 2011

Column sequences converge towards A002577.

			A(3,2) = 9, because there are 9 partitions of 2^3=8 into powers of 2 less than or equal to 2^2=4: [4,4], [4,2,2], [4,2,1,1], [4,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,   1,   1,   1,  ...
  1,  2,  2,   2,   2,   2,  ...
  1,  3,  4,   4,   4,   4,  ...
  1,  5,  9,  10,  10,  10,  ...
  1,  9, 25,  35,  36,  36,  ...
  1, 17, 81, 165, 201, 202,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000012, A094373, A028400(n-2) for n>1, A210772, A210773, A210774, A210775, A210776, A210777, A210778, A210779.
Cf. A002577, A000123, A181322, A145515.
Main diagonal and lower diagonals give: A002577, A125792, A125794.

b:= proc(n,j) local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n `if`(n=0, 1, b(2^(n-k), k)):
seq(seq(A(n, d-n), n=0..d), d=0..11);

b[n_, j_] := Module[{nn, r}, Which[n < 0, 0, j == 0, 1, j == 1, n+1, n < j, b[n, j] = b[n-1, j]+b[2*n, j-1], True, nn = 1+Floor[n]; r := n-nn; (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)*b[j-h+r, j]*(-1)^h, {h, 0, j-1}]]]; a[n_, k_] := If[n == 0, 1, b[2^(n-k), k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A(n,k) = [x^2^(n-1)] 1/(1-x) * 1/Product_{j=0..k-1} (1-x^(2^j)) for n>0; A(0,k) = 1.

A281487 a(n+1) = -Sum_{d|n} a(d), a(1) = 1.

Original entry on oeis.org

1, -1, 0, -1, 1, -2, 2, -3, 4, -5, 4, -5, 8, -9, 7, -9, 13, -14, 12, -13, 18, -21, 17, -18, 29, -31, 23, -28, 36, -37, 36, -37, 50, -55, 42, -46, 64, -65, 53, -62, 83, -84, 75, -76, 94, -107, 90, -91, 129, -132, 107, -121, 145, -146, 135, -141, 180, -193, 157

Offset: 1

Andrey Zabolotskiy, Jan 22 2017

a(1) = 1, any other choice simply adds a factor to all terms.
Observations: sign of a(n) is -(-1)^n, the subsequences |a(n)| with n = 1, 2 mod 4 and |a(n)| with n = 3, 0 mod 4 both grow at n>5. Both these subsequences seem to share the asymptotics with A003238 (and hence A000123): log(|a(n)|) is approximately proportional to (log(n/log(n)))^2; however, the factor is much less than log(4).
There is a family of sequences with the formula a(n) = s*Sum_{d|(n-k), 1<=dA002033, A003238, A007439. For s=-1 and k = 0,1,2, these are the Möbius function A008683, this sequence, and A281488.

			a(9) = -(a(1)+a(2)+a(4)+a(8)) = -(1-1-1-3) = 4.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..20000

Cf. A003238, A281488, A000123.

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n-1, d, va[d]);); va;} \\ Michel Marcus, Apr 29 2019

a = [1]
for n in range(1, 100):
   a.append(-sum(a[d-1] for d in range(1, n+1) if n%d == 0))
print(a)

a(1) = 1.
a(n+1) = -Sum_{d|n} a(d) for n>=1.
a(n+1) = Sum_{d|n} |a(d)|*(-1)^(d+n) for n>=1.
From Ilya Gutkovskiy, Apr 29 2019: (Start)
G.f.: x * (1 - Sum_{n>=1} a(n)*x^n/(1 - x^n)).
L.g.f.: log(Product_{n>=1} (1 - x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n. (End)

A005706 Number of partitions of 5n into powers of 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 82, 89, 96, 103, 110, 119, 128, 137, 146, 155, 166, 177, 188, 199, 210, 223, 236, 249, 262, 275, 290, 305, 320, 335, 350, 368, 386, 404, 422, 440, 461, 482, 503, 524, 545

Offset: 0

N. J. A. Sloane

Euler transform of [2,0,0,0,1,0,0,0,0,...] with 1's at 5^n. - Michael Somos, Mar 16 2004

Partial sums of number of partitions of n into powers of 5. - Michael Somos, Mar 16 2004

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, preprint 2012; SIAM J. Discrete Math., 28(1), 342-371, 2014. - _N. J. A. Sloane_, Dec 23 2012
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, Australasian J. Combin., 30 (2004), 193-196.
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]
O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

Column k=5 of A292477.

Cf. A000041, A000123, A005704, A005705.

a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/5]]; Table[a@ n, {n, 0, 60}] (* Michael De Vlieger, Mar 25 2016 *)

PARI
```
a(n)=if(n<1,n==0,a(n-1)+a(n\5))
```

a(n) = a(n-1) + a([n/5]).

a(n) = [x^(5*n)] Product_{k>=0} 1/(1 - x^(5^k)). - Ilya Gutkovskiy, Jun 05 2017

Formula and more terms from Henry Bottomley, Apr 30 2001

A102378 a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 25, 35, 45, 59, 73, 93, 113, 139, 165, 201, 237, 283, 329, 389, 449, 523, 597, 691, 785, 899, 1013, 1153, 1293, 1459, 1625, 1827, 2029, 2267, 2505, 2789, 3073, 3403, 3733, 4123, 4513, 4963, 5413, 5937, 6461, 7059, 7657, 8349

Offset: 1

Mitch Harris, Jan 05 2005

nonn

From Gus Wiseman, Mar 23 2019: (Start)
The offset could safely be changed to zero by setting the boundary condition to a(0) = 0.
Also the number of integer partitions of 2n into powers of 2 with at least one part > 1. The Heinz numbers of these partitions are given by A324927. For example, the a(1) = 1 through a(5) = 13 integer partitions are:
  (2)  (4)    (42)     (8)        (82)
       (22)   (222)    (44)       (442)
       (211)  (411)    (422)      (811)
              (2211)   (2222)     (4222)
              (21111)  (4211)     (4411)
                       (22211)    (22222)
                       (41111)    (42211)
                       (221111)   (222211)
                       (2111111)  (421111)
                                  (2221111)
                                  (4111111)
                                  (22111111)
                                  (211111111)
(End)

Cf. A000123, A033485.

Cf. A018819, A318400, A324927, A324928.

Table[Length[Select[IntegerPartitions[n],And[Max@@#>1,And@@IntegerQ/@Log[2,#]]&]],{n,0,30,2}] (* Gus Wiseman, Mar 23 2019 *)

from itertools import islice
from collections import deque
def A102378_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3)
    while True:
        a += b
        yield 2*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A102378_list = list(islice(A102378_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) - a(n-1) = A018819(n+1)
G.f. A(x) satisfies (1-x)*A(x) = 2(1 + x)*B(x^2), where B(x) is the gf of A033485
a(n) = A000123(n) - 1. - Gus Wiseman, Mar 23 2019
G.f. A(x) satisfies: A(x) = (x + (1 - x^2) * A(x^2)) / (1 - x)^2. - Ilya Gutkovskiy, Aug 11 2021

A181322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2*n into powers of 2 less than or equal to 2^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 2, 4, 6, 5, 1, 1, 2, 4, 6, 9, 6, 1, 1, 2, 4, 6, 10, 12, 7, 1, 1, 2, 4, 6, 10, 14, 16, 8, 1, 1, 2, 4, 6, 10, 14, 20, 20, 9, 1, 1, 2, 4, 6, 10, 14, 20, 26, 25, 10, 1, 1, 2, 4, 6, 10, 14, 20, 26, 35, 30, 11, 1, 1, 2, 4, 6, 10, 14, 20, 26, 36, 44, 36, 12, 1, 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 56, 42, 13, 1

Offset: 0

Alois P. Heinz, Jan 26 2011

Column sequences converge towards A000123.

			A(3,2) = 6, because there are 6 partitions of 2*3=6 into powers of 2 less than or equal to 2^2=4: [4,2], [4,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1], [1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  ...
  1,  2,  2,  2,  2,  2,  ...
  1,  3,  4,  4,  4,  4,  ...
  1,  4,  6,  6,  6,  6,  ...
  1,  5,  9, 10, 10, 10,  ...
  1,  6, 12, 14, 14, 14,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 4.

Columns k=0-5 give: A000012, A000027(n+1), A002620(n+2), A008804, A088932, A088954.
Main diagonal gives A000123.
Cf. A145515.
See A262553 for another version of this array.
See A072170 for a related array (having the same limiting column).

b:= proc(n, j) local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(n/2^(k-1), k):
seq(seq(A(n, d-n), n=0..d), d=0..13);

b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n<0, 0, j == 0, 1, j == 1, n+1, nJean-François Alcover, Jan 15 2014, translated from Maple *)

A181322(n,k,r=1)={if(nA181322(n-1,k,0)+A181322(2*n,k-1,0),n-=r=1+n\1,(r-k)*binomial(r,k)*sum(i=0,min(k-1,k+n), binomial(k,i)/(r-k+i)*A181322(k-i+n,k,0) *(-1)^i))} \\ From Maple. - M. F. Hasler, Feb 19 2019

G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(2^j)).
A(n,k) = Sum_{i=0..k} A089177(n,i).
For n < 2^k, T(n,k) = A000123(k). T(n,0) = 1, T(n,1) = n+1. - M. F. Hasler, Feb 19 2019

cp's OEIS Frontend

A008804 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)).

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A050377 Number of ways to factor n into "Fermi-Dirac primes" (members of A050376).

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A101417 Number of partitions of n into parts without powers of 2.

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A088567 Number of "non-squashing" partitions of n into distinct parts.

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A072170 Square array T(n,k) (n >= 0, k >= 2) read by antidiagonals, where T(n,k) is the number of ways of writing n as Sum_{i>=0} c_i 2^i, c_i in {0,1,...,k-1}.

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A152977 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^n into powers of 2 less than or equal to 2^k.

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A008804 Expansion of 1/((1-x)^2(1-x^2)(1-x^4)).