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.

User: Patrick Okolo Edeogu

Patrick Okolo Edeogu's wiki page.

Patrick Okolo Edeogu has authored 3 sequences.

A272144 Convolution of A000217 and A001045.

Original entry on oeis.org

0, 0, 1, 4, 12, 30, 69, 150, 316, 652, 1329, 2688, 5412, 10866, 21781, 43618, 87300, 174672, 349425, 698940, 1397980, 2796070, 5592261, 11184654, 22369452, 44739060, 89478289, 178956760, 357913716, 715827642, 1431655509, 2863311258, 5726622772
Offset: 0

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Author

Patrick Okolo Edeogu, Apr 21 2016

Keywords

Examples

			a(4) = 12 = 0*10+1*6+1*3+3*1+5*0 from A000217: 0,1,3,6,10,... and A001045: 0,1,1,3,5,11,...

Links

Crossrefs

Partial Sums of A011377(n-2)=A178420(n-1).

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^2/((1-x)^3*(1+x)*(1-2*x)))); // G. C. Greubel, Oct 26 2018
  • Maple
    seq(coeff(series(x^2/((1-x)^3*(1+x)*(1-2*x)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 26 2018
  • Mathematica
    CoefficientList[Series[x^2/((1 - x)^3 (1 + x) (1 - 2 x)), {x, 0, 30}], x] (* Michael De Vlieger, Apr 21 2016 *)
  • PARI
    concat([0, 0], Vec(x^2/((1-x)^3*(1+x)*(1-2*x)) + O(x^40))) \\ Altug Alkan, Apr 21 2016

Formula

a(n) = Sum{k=0..n} A000217(k) * A001045(n-k). - Joerg Arndt, May 17 2016
a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 5*a(n-4) - 2*a(n-5).
G.f.: x^2/((1-x)^3*(1+x)*(1-2*x)).
a(n+2) = (-105+(-1)^n+2^(7+n)-48*n-6*n^2)/24. - Colin Barker, Apr 21 2016
E.g.f.: (exp(-x) + 32*exp(2*x) - 3*(11 + 10*x + 2*x^2)*exp(x))/24. - Ilya Gutkovskiy, Apr 21 2016

A245548 Number of distinct sum representations of n by Fibonacci numbers with minimal digit sum.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 3, 1, 3, 1, 1, 1, 1, 1, 2, 2, 3, 2, 1, 1, 3, 1, 4, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 5, 1, 1, 1, 3, 4, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 3, 5, 2, 5, 2, 1, 1, 1, 1, 2, 3, 4, 3, 1, 1, 4, 1, 5
Offset: 1

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Author

Patrick Okolo Edeogu, Oct 20 2015

Keywords

Comments

The digits are any nonnegative integers. The value of the minimal sum of digits is given by A007895. The sequence of those numbers where this sequence has value 1 is A256133.

Examples

			a(12) = 3 because 12 = 8 + 3 + 1 = 8 + 2 + 2 = 5 + 5 + 2 has three distinct representations.

Links

Crossrefs

Cf. A000045, A007895, A256133.

Programs

  • Maple
    L:=[1,2,3,5,8,13,21,34,55]; LC:=[1,1,1,2,1,2,1,1,1]:LS:=[1,1,1,2,1,2,2,1,2]: for n from 10 to 88 do: ct:=0: ss:=n: sm:=n: b0:=1: b1:=2: b2:=3: b3:=4: b4:=trunc(n/L[5]): b5:=trunc(n/L[6]): b6:=trunc(n/L[7]):b7:=trunc(n/L[8]):b8:=trunc(n/L[9]):
> for n0 from 0 to b0 do:for n1 from 0 to b1 do: for n2 from 0 to b2 do:for n3 from 0 to b3 do: for n4 from 0 to b4 do: for n5 from 0 to b5 do: for n6 from 0 to b6 do:
> for n7 from 0 to b7 do:for n8 from 0 to b8 do: if n=n0*L[1]+n1*L[2]+n2*L[3]+n3*L[4]+n4*L[5]+n5*L[6]+n6*L[7]+n7*L[8]+n8*L[9] then ss:=n0+n1+n2+n3+n4+n5+n6+n7+n8:fi:
> if sm>ss then sm:=ss: fi: od:od:od:od:od:od:od:od:od:for n0 from 0 to b0 do:for n1 from 0 to b1 do: for n2 from 0 to b2 do:for n3 from 0 to b3 do:
> for n4 from 0 to b4 do:for n5 from 0 to b5 do:for n6 from 0 to b6 do:
> for n7 from 0 to b7 do:for n8 from 0 to b8 do:
> if n=n0*L[1]+n1*L[2]+n2*L[3]+n3*L[4]+n4*L[5]+n5*L[6]+n6*L[7]+n7*L[8]+n8*L[9] then st:=n0+n1+n2+n3+n4+n5+n6+n7+n8: if st=sm then ct:=ct+1: fi:fi: od;od:od:od:od:od:od:od:od: LS:=[op(LS),sm]: LC:=[op(LC),ct]: od: print(LC):

A256133 Numbers that have unique expansion with minimal digit sum in terms of Fibonacci numbers F_k (k > 1).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 18, 20, 21, 22, 23, 24, 29, 30, 32, 34, 35, 36, 37, 39, 41, 47, 48, 49, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 76, 77, 78, 79, 84, 85, 87, 89, 90, 91, 92, 94, 96, 97, 98, 100, 102, 103, 104, 107, 109, 123, 124, 125, 126
Offset: 1

Views

Author

Patrick Okolo Edeogu, Jul 10 2015

Keywords

Comments

This sequence shows that the "proper digital expansion" mentioned in the introduction to the paper by Drmota and Gajdosik (see Links) is not unique.
This sequence consists of all positive integers that have Zeckendorf expansions not containing any ...1001... and not ending in ...101. Example: 20 is in and it has 20 = 13 + 5 + 2 with Zeckendorf expansion 101010, while 19 is not in and has 19 = 13 + 5 + 1 with Zeckendorf expansion 101001. - Thomas Bier, Oct 09 2015

Examples

			7 = 5 + 2 is unique with respect to its minimal digit sum 1 + 1 = 2.
But 10 = 8 + 2 = 5 + 5 is not unique with respect to its minimal digit sum 1 + 1 = 2.

Links

Crossrefs

Cf. A000045.

Programs

  • Maple
    x0:=0: x1:=1: ML:=[]: L:=[]: mes:=0:  for r from 2 to 14 do: z:=x1+x0: x0:=x1: x1:=z:  rj:=12: L:=[op(L),z]: ML:=[z,op(ML)]: od: XL:=[]: for m from 1 to 400 do:  NL:=[]: n:=m: for j from 12 to 1 by -1 do: if L[j+1]>n and L[j]-1 < n then NL:=[op(NL),j]: n:=n-L[j]: fi: od: eps:=0: nx:=nops(NL):  for j from 1 to nx-1 do: if NL[j]-NL[j+1]=3 then eps:=1: fi: if NL[nx-1]-NL[nx]=2 and NL[nx]=1 then eps:=1: fi:od: if eps=0 then XL:=[op(XL),m]: fi: od: print(XL);

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