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 1-5 of 5 results.

A110202 a(n) = sum of squares of numbers < 2^n having exactly 2 ones in their binary representation.

Original entry on oeis.org

0, 9, 70, 395, 1984, 9429, 43434, 196095, 872788, 3842729, 16773118, 72693075, 313158312, 1342144509, 5726557522, 24338016935, 103078952956, 435222828369, 1832518331046, 7696579297275, 32252336887120, 134873417951909
Offset: 1

Views

Author

Paul D. Hanna, Jul 16 2005

Keywords

Comments

Equals column 2 of triangle A110200.

Examples

			For n=4, the sum of the squares of numbers < 2^4
having exactly 2 ones in their binary digits is:
a(4) = 3^2 + 5^2 + 6^2 + 9^2 + 10^2 + 12^2 = 395.

Links

Crossrefs

Cf. A110200 (triangle), A110201 (central terms), A002450 (column 1), A110203 (column 3), A110204 (column 4), A018900.

Programs

  • Mathematica
    nn=30;With[{c=Union[FromDigits[#,2]&/@(Flatten[Table[Join[ {1},#]&/@ Permutations[Join[{1},PadRight[{},n,0]]],{n,0,nn}],1])]}, Table[ Total[ Select[c,#<2^n&]^2],{n,nn}]] (* Harvey P. Dale, Jan 27 2013 *)
  • PARI
    a(n)=polcoeff(x^2*(9-38*x+32*x^2)/((1-x)^2*(1-2*x)*(1-4*x)^2+x*O(x^n)),n)

Formula

G.f.: x^2*(9-38*x+32*x^2)/( (1-x)^2*(1-2*x)*(1-4*x)^2 ). a(n) = Sum_{k|A018900(k)<2^n} A018900(k)^2.

A110201 a(n) = sum of squares of numbers < 2^n having exactly [n/2]+1 ones in their binary expansion.

Original entry on oeis.org

1, 9, 70, 535, 3906, 29274, 215900, 1628175, 12197570, 92830430, 704127060, 5400199350, 41331491124, 318871044756, 2456608834680, 19039140186495, 147401590706370, 1146463189301430, 8909683732878500, 69495629981713650
Offset: 1

Views

Author

Paul D. Hanna, Jul 16 2005

Keywords

Comments

a(n) equals the largest term in row n of triangle A110200.

Crossrefs

Cf. A110200 (triangle), A002450 (column 1), A110202 (column 2), A110203 (column 3), A110204 (column 4).

Programs

  • Mathematica
    Join[{1},Table[Total[Select[Range[2^n],DigitCount[#,2,1]==Floor[ n/2]+ 1&]^2],{n,2,20}]] (* Harvey P. Dale, Aug 22 2021 *)
  • PARI
    a(n)=(4^n-1)/3*binomial(n-2,n\2)+(2^n-1)^2*binomial(n-2,n\2-1)

Formula

a(n) = (4^n-1)/3*C(n-2, n\2) + (2^n-1)^2*C(n-2, n\2-1).

A110203 a(n) = sum of squares of numbers < 2^n having exactly 3 ones in their binary representation.

Original entry on oeis.org

0, 0, 49, 535, 3906, 24066, 135255, 717825, 3662848, 18158932, 88043517, 419348475, 1968346446, 9126412278, 41875079155, 190408381765, 858989527020, 3848282308584, 17134038373689, 75866264567775, 334251455152090
Offset: 1

Views

Author

Paul D. Hanna, Jul 16 2005

Keywords

Comments

Equals column 3 of triangle A110200.

Examples

			For n=4, the sum of the squares of numbers < 2^4
having exactly 3 ones in their binary digits is:
a(4) = 7^2 + 11^2 + 13^2 + 14^2 = 535.

Crossrefs

Cf. A110200 (triangle), A110201 (central terms), A002450 (column 1), A110202 (column 2), A110204 (column 4).

Programs

  • PARI
    {a(n)=polcoeff(x^3*(49-396*x+1140*x^2-1360*x^3+576*x^4)/ ((1-x)^3*(1-2*x)^2*(1-4*x)^3+x*O(x^n)),n)}

Formula

G.f.: x^3*(49-396*x+1140*x^2-1360*x^3+576*x^4)/((1-x)^3*(1-2*x)^2*(1-4*x)^3).

A110204 a(n) = sum of squares of numbers < 2^n having exactly 4 ones in their binary representation.

Original entry on oeis.org

0, 0, 0, 225, 3224, 29274, 215900, 1412275, 8541876, 48876212, 268288008, 1425694725, 7381073920, 37399844174, 186110137668, 911952794935, 4409472232060, 21073909951080, 99688911645264, 467292120940425
Offset: 1

Views

Author

Paul D. Hanna, Jul 16 2005

Keywords

Comments

Equals column 4 of triangle A110200.

Crossrefs

Cf. A110200 (triangle), A110201 (central terms), A002450 (column 1), A110202 (column 2), A110203 (column 3).

Programs

  • PARI
    {a(n)=polcoeff(x^4*(225-2626*x+12500*x^2-30872*x^3+41536*x^4-28928*x^5+8192*x^6)/ ((1-x)^4*(1-2*x)^3*(1-4*x)^4+x*O(x^n)),n)}

Formula

G.f.: x^4*(225-2626*x+12500*x^2-30872*x^3+41536*x^4-28928*x^5+8192*x^6)/ ((1-x)^4*(1-2*x)^3*(1-4*x)^4).

A110205 Triangle, read by rows, where T(n,k) equals the sum of cubes of numbers < 2^n having exactly k ones in their binary expansion.

Original entry on oeis.org

1, 9, 27, 73, 368, 343, 585, 3825, 6615, 3375, 4681, 36394, 88536, 86614, 29791, 37449, 332883, 1024002, 1449198, 970677, 250047, 299593, 2979420, 10970133, 20078192, 19714083, 9974580, 2048383, 2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375
Offset: 1

Views

Author

Paul D. Hanna, Jul 16 2005

Keywords

Comments

Compare to triangle A110200 (sum of squares).

Examples

			Row 4 is formed by sums of cubes of numbers < 2^4:
  T(4,1) = 1^3 + 2^3 + 4^3 + 8^3 = 585;
  T(4,2) = 3^3 + 5^3 + 6^3 + 9^3 + 10^3 + 12^3 = 3825;
  T(4,3) = 7^3 + 11^3 + 13^3 + 14^3 = 6615;
  T(4,4) = 15^3 = 3375.
Triangle begins:
        1;
        9,       27;
       73,      368,       343;
      585,     3825,      6615,      3375;
     4681,    36394,     88536,     86614,     29791;
    37449,   332883,   1024002,   1449198,    970677,    250047;
   299593,  2979420,  10970133,  20078192,  19714083,   9974580,  2048383;
  2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375; ...
Row g.f.s are:
  row 1: (1 + 2*x + 1*x^2)/(1+x)^2;
  row 2: (9 + 36*x + 27*x^2)/(1+x);
  row 3: (73 + 368*x + 343*x^2);
  row 4: (585 + 3240*x + 3375*x^2)*(1+x).
G.f. for row n is:
  ((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).

Links

Crossrefs

Cf. A110206 (row sums), A110207 (central terms), A023001 (column 1).

Programs

  • Magma
    b:= func< n,k | Binomial(n-3, k) >;
A110205:= func< n,k | (8^n-1)/7*(b(n,k-1) -b(n,k-2)) +(2^n-1)^2*((2^n+1)*b(n,k-2) +(2^n-1)*b(n,k-3)) >;
[A110205(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2024
  • Mathematica
    b[n_, k_]= Binomial[n-3, k];
T[n_, k_]:= (8^n-1)/7*(b[n,k-1] -b[n,k-2]) + (2^n-1)^2*((2^n+1)*b[n,k-2] + (2^n-1)*b[n,k-3]);
A110205[n_, k_]:= If[n<3, T[n,k]/2, T[n,k]];
Table[A110205[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2024 *)
  • PARI
    T(n,k)=(8^n-1)/7*binomial(n-3,k-1)+((2^n-1)*(4^n-1)-(8^n-1)/7)*binomial(n-3,k-2) +(2^n-1)^3*binomial(n-3,k-3)
  • PARI
    /* Sum of cubes of numbers<2^n with k 1-bits: */
T(n,k)=local(B=vector(n+1));if(n
  • SageMath
    def b(n,k): return binomial(n-3, k)
def A110205(n,k): return (8^n-1)/7*(b(n,k-1) - b(n,k-2)) + (2^n-1)^2*((2^n+1)*b(n,k-2) + (2^n-1)*b(n,k-3))
flatten([[A110205(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 03 2024

Formula

T(n, k) = (8^n-1)/7*C(n-3, k-1) + ((2^n-1)*(4^n-1)-(8^n-1)/7)*C(n-3, k-2) + (2^n-1)^3*C(n-3, k-3).
G.f. for row n: ((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).
Showing 1-5 of 5 results.

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