A073267 -id:A073267 - cp's OEIS frontend

A342250 Number of ways to write n as an ordered sum of seven powers of 2.

1, 7, 21, 42, 77, 126, 168, 218, 294, 357, 427, 546, 637, 672, 756, 840, 854, 966, 1134, 1218, 1302, 1408, 1484, 1554, 1680, 1827, 1995, 2002, 1925, 2016, 1988, 1904, 2142, 2352, 2282, 2352, 2534, 2520, 2604, 2954, 3080, 3276, 3262, 3234, 3150, 3248, 3164, 3402, 3640

Offset: 7

Ilya Gutkovskiy, Mar 07 2021

Robert Israel, Table of n, a(n) for n = 7..10000

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A342247, A342251, A342252, A342254.

N:= 100:
S:= add(x^(2^j),j=0..ilog2(N-6))^7:
[seq](coeff(S,x,j),j=7..N); # Robert Israel, Feb 26 2023

nmax = 55; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: ( Sum_{k>=0} x^(2^k) )^7.

A342251 Number of ways to write n as an ordered sum of eight powers of 2.

Original entry on oeis.org

1, 8, 28, 64, 126, 224, 336, 464, 645, 840, 1044, 1344, 1666, 1904, 2192, 2528, 2730, 3024, 3528, 3920, 4284, 4768, 5168, 5488, 5965, 6552, 7140, 7616, 7834, 8176, 8400, 8352, 8862, 9632, 9800, 10080, 10788, 10976, 11152, 12208, 13090, 13664, 14392, 14672, 14868, 15008, 15344

Offset: 8

Ilya Gutkovskiy, Mar 07 2021

nonn

Robert Israel, Table of n, a(n) for n = 8..10000

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A342248, A342250, A342252, A342254.

N:= 100:
S:= add(x^(2^j),j=0..ilog2(N-7))^8:
seq(coeff(S,x,j),j=8..N); # Robert Israel, Feb 26 2023

nmax = 54; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: ( Sum_{k>=0} x^(2^k) )^8.

A342252 Number of ways to write n as an ordered sum of nine powers of 2.

Original entry on oeis.org

1, 9, 36, 93, 198, 378, 624, 927, 1341, 1849, 2412, 3159, 4074, 4950, 5904, 7032, 8010, 9018, 10488, 11970, 13356, 15108, 16848, 18315, 20085, 22257, 24444, 26671, 28674, 30510, 32208, 33282, 34974, 37590, 39384, 40986, 43668, 45468, 46512, 49620, 53298, 55890, 59304, 62442

Offset: 9

Ilya Gutkovskiy, Mar 07 2021

nonn

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A342249, A342250, A342251, A342254.

nmax = 52; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: ( Sum_{k>=0} x^(2^k) )^9.

A342254 Number of ways to write n as an ordered sum of ten powers of 2.

Original entry on oeis.org

1, 10, 45, 130, 300, 612, 1095, 1750, 2655, 3850, 5281, 7110, 9460, 12060, 14940, 18352, 21850, 25380, 29790, 34740, 39672, 45480, 51885, 57870, 64375, 72090, 80145, 88630, 97660, 106380, 114736, 122260, 130050, 139740, 148990, 157572, 168240, 178200, 185490, 196200, 210082

Offset: 10

Ilya Gutkovskiy, Mar 07 2021

nonn

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A319922, A342250, A342251, A342252.

nmax = 50; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: ( Sum_{k>=0} x^(2^k) )^10.

A159981 Catalan numbers read modulo 4.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 0, 1, 2, 2, 0, 2, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Apr 28 2009

Essentially the same as A073267. - R. J. Mathar, May 25 2009

Antti Karttunen, Table of n, a(n) for n = 0..10000
Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016.
Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, Catalan and Motzkin numbers modulo 4 and 8, European Journal of Combinatorics, Vol. 29, No. 6 (2008), pp. 1449-1466. [From _R. J. Mathar_, Apr 30 2009]
Eric Rowland and Reem Yassawi, Profinite automata, Advances in Applied Mathematics, Vol. 85 (2017), pp. 60-83; arXiv preprint, arXiv:1403.7659 [math.DS], 2014-2016. See p. 4, 6 and 7.

Cf. A000108, A073267.

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end: A159981 := proc(n) A000108(n) mod 4 ; end: seq(A159981(n),n=0..120) ; # R. J. Mathar, Apr 30 2009

Mod[CatalanNumber[Range[0,110]],4] (* Harvey P. Dale, Oct 05 2011 *)

A159981(n) = (binomial(2*n, n)/(n+1))%4; \\ Antti Karttunen, Jan 17 2017

a(n) = A000108(n) mod 4.

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Extended by R. J. Mathar, Apr 30 2009

A281228 Expansion of (Sum_{k>=0} x^(3^k))^2 [even terms only].

Original entry on oeis.org

0, 1, 2, 1, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 18 2017

nonn

Number of ways to write 2n as an ordered sum of two powers of 3.

First bisection of self-convolution of characteristic function of powers of 3.

			G.f. = x^2 + 2*x^4 + x^6 + 2*x^10 + 2*x^12 + x^18 + 2*x^28 + 2*x^30 + 2*x^36 + ...
a(2) = 2 because we have [3, 1] and [1, 3].

Index entries for sequences related to compositions

Cf. A000244, A055235, A073267, A078932.

Take[CoefficientList[Series[Sum[x^3^k, {k, 0, 15}]^2, {x, 0, 260}], x], {1, -1, 2}]

G.f.: (Sum_{k>=0} x^(3^k))^2 [even terms only].

A289682 Catalan numbers read modulo 16.

Original entry on oeis.org

1, 1, 2, 5, 14, 10, 4, 13, 6, 14, 12, 2, 12, 4, 8, 13, 6, 6, 12, 6, 4, 12, 8, 2, 12, 12, 8, 4, 8, 8, 0, 13, 6, 6, 12, 14, 4, 12, 8, 6, 4, 4, 8, 12, 8, 8, 0, 2, 12, 12, 8, 12, 8, 8, 0, 4, 8, 8, 0, 8, 0, 0, 0, 13, 6, 6, 12, 14, 4, 12, 8, 14, 4, 4, 8, 12, 8, 8, 0, 6, 4, 4, 8, 4, 8, 8, 0, 12

Offset: 0

R. J. Mathar, Jul 09 2017

Conjecture: a(2^n-1) = 13 and a(2^n) = 6 for n >= 3. - Robert Israel, Jul 09 2017

Robert Israel, Table of n, a(n) for n = 0..10000
Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016.
Shu-Chung Liu and Jean C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq., Vol. 13 (2010), Article 10.5.4, Theorem 5.5.

Cf. A000108, A036987 (mod 2), A073267 (mod 4), A159987 (mod 8).

Cf. A048881 (2-adic valuation of A000108).

[Catalan(n) mod 16: n in [0..100]]; // Vincenzo Librandi, Jul 10 2017

Maple
```
seq ( modp(A000108(n),16),n=0..120) ;
```

Table[Mod[CatalanNumber[n], 16], {n, 0, 100}] (* Vincenzo Librandi, Jul 10 2017 *)

a(n) = (binomial(2*n, n)/(n+1)) % 16; \\ Michel Marcus, Jul 09 2017

a(n) = A000108(n) mod 16.

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - Amiram Eldar, Jan 26 2021

cp's OEIS Frontend

A342250 Number of ways to write n as an ordered sum of seven powers of 2.

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A342251 Number of ways to write n as an ordered sum of eight powers of 2.

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A342252 Number of ways to write n as an ordered sum of nine powers of 2.

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A342254 Number of ways to write n as an ordered sum of ten powers of 2.

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A159981 Catalan numbers read modulo 4.

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A281228 Expansion of (Sum_{k>=0} x^(3^k))^2 [even terms only].

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A289682 Catalan numbers read modulo 16.

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