cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A177116 Partial sums of round(n^2/11).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 11, 17, 24, 33, 44, 57, 72, 90, 110, 133, 159, 188, 221, 257, 297, 341, 389, 441, 498, 559, 625, 696, 772, 854, 941, 1034, 1133, 1238, 1349, 1467, 1591, 1722, 1860, 2005, 2158, 2318, 2486, 2662, 2846, 3038, 3239, 3448, 3666, 3893
Offset: 0

Views

Author

Mircea Merca, Dec 09 2010

Keywords

Comments

The round function, also called the nearest integer 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).

Examples

			a(11) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 6 + 7 + 9 + 11 = 44.

Links

Crossrefs

Cf. A173690 (k=5), A173691 (k=6), A173722 (k=8), A177100 (k=9), A181120 (k=12).

Programs

  • Maple
    seq(round((2*n^3+3*n^2-11*n)/66),n=0..50)
  • Mathematica
    Accumulate[Round[Range[0,50]^2/11]] (* or *) LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,1,-3,3,-1},{0,0,0,1,2,4,7,11,17,24,33,44,57,72},60] (* Harvey P. Dale, Dec 10 2014 *)
  • PARI
    a(n)=(2*n^3+3*n^2-11*n+18)\66 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = round((n-2)*(n+3)*(2*n+1)/66).
a(n) = floor((2*n^3 + 3*n^2 - 11*n + 18)/66).
a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 30)/66).
a(n) = round(n*(2*n^2 + 3*n - 11)/66).
a(n) = a(n-11) + (n+1)*(n-11) + 44, n > 10.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14). - R. J. Mathar, Dec 10 2010
G.f.: x^3 *(1+x) *(x^2-x+1) *(x^4-x^3+x^2-x+1) / ( (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) *(x-1)^4 ). - R. J. Mathar, Dec 10 2010 [Typo fixed by Colin Barker, Oct 10 2012]

A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 0, 2, 1, 4, 2, 7, 4, 11, 7, 16, 11, 23, 16, 31, 23, 41, 31, 53, 41, 67, 53, 83, 67, 102, 83, 123, 102, 147, 123, 174, 147, 204, 174, 237, 204, 274, 237, 314, 274, 358, 314, 406, 358, 458, 406, 514, 458, 575, 514, 640, 575, 710, 640, 785, 710, 865, 785, 950
Offset: 0

Views

Author

Michael Somos, Feb 02 2015

Keywords

Comments

Partitions of n into parts of size 3 and size 4 and two kinds of parts of size 2.
The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, u+v >= x+w, and x+u+v+w is even.
Euler transform of length 4 sequence [ 0, 2, 1, 1].

Examples

			G.f. = 1 + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + 7*x^6 + 4*x^7 + 11*x^8 + 7*x^9 + ...

Links

Crossrefs

Cf. A001400, A002620, A005044, A115264, A165188.
Cf. A241526, A000601, A181120.

Programs

  • Magma
    I:=[1,0,2,1,4,2,7,4,11,7,16]; [n le 11 select I[n] else 2*Self(n-2)+Self(n-3)-2*Self(n-5)-2*Self(n-6)+Self(n-8)+2*Self(n-9)-Self(n-11): n in [1..60]]; // Vincenzo Librandi, Feb 03 2015
  • Mathematica
    a[ n_] := Quotient[ n^3 + If[ OddQ[n], 12 n^2 + 33 n + 54, 21 n^2 + 132 n + 288], 288];
a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -11 - n]; s SeriesCoefficient[ 1 / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, u + v >= x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];
CoefficientList[Series[1 / (1 - 2 x^2 - x^3 + 2 x^5 + 2 x^6 - x^8 - 2 x^9 + x^11), {x, 0, 60}], x] (* Vincenzo Librandi, Feb 03 2015 *)
  • PARI
    {a(n) = (n^3 + if(n%2, 12*n^2 + 33*n + 54, 21*n^2 + 132*n + 288)) \ 288};
  • PARI
    {a(n) = my(s=1); if( n<0, s=-1; n=-11-n); s * polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

Formula

G.f.: 1 / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
a(n) = -a(-11-n) for all n in Z.
a(n+3) - a(n) = 0 if n even else floor((n+7)^2 / 16).
0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
a(n) - a(n-2) = A005044(n+3) for all n in Z.
a(n) + a(n-1) = A001400(n) for all n in Z.
a(n) + a(n-2) = A165188(n+1) for all n in Z.
a(n) = A115264(n) - A115264(n-1) for all n in Z.
a(2*n) - a(2*n-6) = a(2*n+3) - a(2*n-3) = A002620(n+2) for all n in Z. - Michael Somos, Feb 11 2015
a(n) = (2*n^3+33*n^2+181*n+234+3*(3*n^2+33*n+86)*(-1)^n+84*(-1)^((2*n+1-(-1)^n)/4)-96*((1+(-1)^n)*floor(((2*n+9+(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24))+(1-(-1)^n)*floor(((2*n+5+(-1)^n-6*(-1)^((2*n-1+(-1)^n)/4))/24))))/576. - Luce ETIENNE, May 22 2015

A177176 Partial sums of round(n^2/13).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 11, 16, 22, 30, 39, 50, 63, 78, 95, 115, 137, 162, 190, 221, 255, 292, 333, 377, 425, 477, 533, 593, 658, 727, 801, 880, 964, 1053, 1147, 1247, 1352, 1463, 1580, 1703, 1832, 1968, 2110, 2259, 2415, 2578, 2748, 2925, 3110, 3302
Offset: 0

Views

Author

Mircea Merca, Dec 10 2010

Keywords

Comments

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

Examples

			a(13) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 = 63.

Links

Crossrefs

Cf. A177100, A177116, A181120.

Programs

  • Magma
    [Round(n*(n+1)*(2*n+1)/78): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011
  • Maple
    seq(round(n*(n+1)*(2*n+1)/78),n=0..50)
  • PARI
    s=0;vector(90,n,s+=n^2\13)

Formula

a(n) = round(n*(n+1)*(2*n+1)/78).
a(n) = floor((n+3)*(2*n^2 - 3*n + 10)/78).
a(n) = ceiling((n-2)*(2*n^2 + 7*n + 15)/78).
a(n) = a(n-13) + (n+1)*(n-13) + 63, n > 12.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-13) - 3*a(n-14) + 3*a(n-15) - a(n-16) with g.f. x^3*(1+x)*(x^2 - x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) / ( (x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x-1)^4 ). - R. J. Mathar, Dec 13 2010
Showing 1-3 of 3 results.

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