A001304 -id:A001304 - cp's OEIS frontend

A000115 Denumerants: Expansion of 1/((1-x)(1-x^2)(1-x^5)).

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 18, 20, 22, 24, 26, 29, 31, 34, 36, 39, 42, 45, 48, 51, 54, 58, 61, 65, 68, 72, 76, 80, 84, 88, 92, 97, 101, 106, 110, 115, 120, 125, 130, 135, 140, 146, 151, 157, 162, 168, 174, 180, 186, 192, 198, 205, 211, 218, 224, 231, 238

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 2, or 5.

First differences are in A008616. First differences of A001304. Pairwise sums of A008720.

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

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,2,5).
M. Jeger, Ein partitions problem ..., Elemente de Math., 13 (1958), 97-120.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1,-1,-1,1)

Cf. A001304, A008616, A008720.

[Round((n+4)^2/20): n in [0..70]]; // Vincenzo Librandi, Jun 23 2011

1/((1-x)*(1-x^2)*(1-x^5)): seq(coeff(series(%, x, n+1), x, n), n=0..65);
# next Maple program:
s:=proc(n) if n mod 5 = 0 then RETURN(1); fi; if n mod 5 = 1 then RETURN(0); fi; if n mod 5 = 2 then RETURN(1); fi; if n mod 5 = 3 then RETURN(-1); fi; if n mod 5 = 4 then RETURN(-1); fi; end: f:=n->(2*n^2+16*n+27+5*(-1)^n+8*s(n))/40: seq(f(n), n=0..65);  # from Jeger's paper

nn=50;CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^5),{x,0,nn}],x]  (* Geoffrey Critzer, Jan 20 2013 *)
LinearRecurrence[{1,1,-1,0,1,-1,-1,1},{1,1,2,2,3,4,5,6},70] (* Harvey P. Dale, Sep 27 2019 *)

a(n)=(n^2+8*n+26)\20 \\ Charles R Greathouse IV, Jun 23 2011

a(n) = round((n+4)^2/20).

a(n) = a(-8 - n) for all n in Z. - Michael Somos, May 28 2014

A001362 Number of ways of making change for n cents using coins of 1, 2, 4, 10 cents.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 13, 13, 18, 18, 24, 24, 31, 31, 39, 39, 49, 49, 60, 60, 73, 73, 87, 87, 103, 103, 121, 121, 141, 141, 163, 163, 187, 187, 213, 213, 242, 242, 273, 273, 307, 307, 343, 343, 382, 382, 424, 424, 469, 469, 517, 517, 568, 568, 622, 622

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n into parts 1, 2, 4, and 10. - Joerg Arndt, Sep 05 2014

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 186
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 1, -1, -1, 1, 0, 0, 1, -1, -1, 1, -1, 1, 1, -1).

Twice A001304.

1/(1-x)/(1-x^2)/(1-x^4)/(1-x^10): seq(coeff(series(%,x,n+1),x,n), n=0..80);

nn = 1000; CoefficientList[Series[1/((1 - x^1) (1 - x^2) (1 - x^4) (1 - x^10)), {x, 0, nn}], x] (* T. D. Noe, Jun 28 2012 *)
Table[Length[FrobeniusSolve[{1,2,4,10},n]],{n,0,60}] (* Harvey P. Dale, May 20 2021 *)

a(n)=floor((n\2+8)*(2*(n\2)^2+11*(n\2)+18)/120) \\ Tani Akinari, May 14 2014

G.f.: 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^10)).

A028291 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)(1-x^5)) in powers of x.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 25, 35, 48, 64, 84, 108, 137, 171, 211, 258, 312, 374, 445, 525, 616, 718, 832, 959, 1100, 1256, 1428, 1617, 1824, 2050, 2297, 2565, 2856, 3171, 3511, 3878, 4273, 4697, 5152, 5639, 6160, 6716, 7309, 7940, 8611, 9324, 10080, 10881, 11729

Offset: 0

N. J. A. Sloane, Dec 11 1999

Partitions of n into parts 1, 2, 3, and 5. - Joerg Arndt, Jun 05 2014

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 17*x^5 + 25*x^6 + 35*x^7 + ...

Susan Elle, Ore extensions of global dimension 5, Abstract 1110-17-204, Abstracts Amer. Math. Soc., 36 (No. 2, 2015), p. 822.

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

Cf. A001304, A008669.

a[ n_] := Quotient[n (n + 12) (n^2 + 12 n + 52), 720] + 1; (* Michael Somos, Jun 05 2014 *)
a[ n_] := With[{m = If[ n < 0, -12 - n, n]}, SeriesCoefficient[ 1 / ((1 - x)^2*(1 - x^2)*(1 - x^3)*(1 - x^5)), {x, 0, m}]]; (* Michael Somos, Jun 05 2014 *)
Table[Round[(n + 1)*(n^3 + 23*n^2 + 173*n + 451)/720], {n, 0, 40}] (* Wesley Ivan Hurt, Jun 05 2014 *)
LinearRecurrence[{2,0,-1,-1,1,0,-1,1,1,0,-2,1},{1,2,4,7,11,17,25,35,48,64,84,108},50] (* Harvey P. Dale, Sep 06 2022 *)

Vec(1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

{a(n) = n * (n+12) * (n^2 + 12*n + 52) \ 720 + 1}; /* Michael Somos, Jun 05 2014 */

{a(n) = if( n<0, n = -12 - n); polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^3) * (1 - x^5)) + x * O(x^n), n)}; /* Michael Somos, Jun 05 2014 */

a(n) = round((n+1)*(n^3+23*n^2+173*n+451)/720). - Tani Akinari, Jun 05 2014
a(n) - 2*a(n-1) + a(n+3) + a(n+4) - 2*a(n+6) + a(n+7) = 1 if n == 3 (mod 5) else 0. - Michael Somos, Jun 05 2014
a(n) = a(-12 - n) for all n in Z. - Michael Somos, May 14 2015
a(n) - a(n-1) = A008669(n), a(n) - a(n-3) = A001304(n) for all n in Z. - Michael Somos, May 14 2015
Euler transform of length 5 sequence [ 2, 1, 1, 0, 1]. - Michael Somos, May 14 2015

A177239 Partial sums of round(n^2/20).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 6, 9, 13, 18, 24, 31, 39, 49, 60, 73, 87, 103, 121, 141, 163, 187, 213, 242, 273, 307, 343, 382, 424, 469, 517, 568, 622, 680, 741, 806, 874, 946, 1022, 1102, 1186, 1274, 1366, 1463, 1564, 1670, 1780, 1895, 2015, 2140

Offset: 0

Mircea Merca, Dec 10 2010

The round function is defined here by round(x) = floor(x + 1/2).

There are several sequences of integers of the form round(n^2/k) for whose partial sums we can establish identities as following (only for k = 2, ..., 9, 11, 12, 13, 16, 17, 19, 20, 28, 29, 36, 44).

			a(20) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 11 + 13 + 14 + 16 + 18 + 20 = 141.

Vincenzo Librandi, Table of n, a(n) for n = 0..895
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1,1,-2,0,2,-1).

Cf. A001304, A177100, A177116.

[Floor((n+4)*(2*n^2-5*n+6)/120): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Maple
```
seq(round(n*(n-2)*(2*n+7)/120),n=0..50)
```

f[n_] := Round[n^2/20]; Accumulate@ Array[f, 51, 0] (* Robert G. Wilson v, Dec 20 2010 *)

[(n+4)*(2*n^2 -5*n +6)//120 for n in range(56)] # G. C. Greubel, Apr 27 2024

a(n) = A001304(n-4).
a(n) = round((2*n+1)*(2*n^2 + 2*n - 15)/240).
a(n) = floor((n+4)*(2*n^2 - 5*n + 6)/120).
a(n) = ceiling((n-3)*(2*n^2 + 9*n + 13)120).
a(n) = round(n*(n-2)*(2*n+7)/120).
a(n) = a(n-20) + (n+1)*(n-20) + 141, n > 19.
From R. J. Mathar, Dec 12 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-8) - a(n-9).
G.f.: x^4 / ( (1+x)*(1+x+x^2+x^3+x^4)*(1-x)^4 ). (End)

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A000115 Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).

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A001362 Number of ways of making change for n cents using coins of 1, 2, 4, 10 cents.

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A028291 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)(1-x^5)) in powers of x.

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A177239 Partial sums of round(n^2/20).

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A000115 Denumerants: Expansion of 1/((1-x)(1-x^2)(1-x^5)).