A178827 Partial sums of floor(8^n/9)/7.
0, 1, 9, 74, 594, 4755, 38043, 304348, 2434788, 19478309, 155826477, 1246611822, 9972894582, 79783156663, 638265253311, 5106122026496, 40848976211976, 326791809695817, 2614334477566545, 20914675820532370, 167317406564258970, 1338539252514071771, 10708314020112574179, 85666512160900593444, 685332097287204747564
Offset: 1
Examples
a(3) = (1/7)*(floor(8/9) + floor(64/9) + floor(512/9)) = (1/7)*(0 + 7 + 56) = (1/7)*63 = 9.
Links
- Bruno Berselli, Table of n, a(n) for n = 1..1000
- 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 (9,-7,-9,8).
Programs
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GAP
a:=[0,1,9,74];; for n in [5..30] do a[n]:=9*a[n-1]-7*a[n-2]-9*a[n-3] +8*a[n-4]; od; a; # G. C. Greubel, Jan 22 2019
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Magma
[ &+[Floor(8^k/9)/7: k in [1..n]]: n in [1..25] ]; // Bruno Berselli, Apr 28 2011 (Decimal BASIC) FOR n=1 TO 1000 PRINT n; (32*8^n-126*n-81+49*(-1)^n)/1764 NEXT n END ! Bruno Berselli, Apr 28 2011
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Maple
A178827 := proc(n) add( floor(8^i/9)/7,i=0..n) ; end proc:
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Mathematica
Rest[CoefficientList[Series[x^2/((1+x)*(1-8*x)*(1-x)^2), {x,0,30}],x]] (* G. C. Greubel, Jan 22 2019 *) -
PARI
my(x='x+O('x^30)); concat([0], Vec(x^2/((1+x)*(1-8*x)*(1-x)^2))) \\ G. C. Greubel, Jan 22 2019 -
Sage
a=(x^2/((1+x)*(1-8*x)*(1-x)^2)).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 22 2019
Formula
7*a(n) = round((32*8^n - 126*n - 81)/252).
7*a(n) = floor((16*8^n - 63*n - 16)/126).
7*a(n) = ceiling((16*8^n - 63*n - 65)/126).
7*a(n) = round((16*8^n - 63*n - 16)/126).
a(n) = a(n-2) + (8^(n-1) - 1)/7, n > 2.
a(n) = 9*a(n-1) - 7*a(n-2) - 9*a(n-3) + 8*a(n-4), n > 4.
G.f.: x^2/((1+x)*(1-8*x)*(1-x)^2).
7*a(n) = (32*8^n - 126*n - 81 + 49*(-1)^n)/252. - Bruno Berselli, Jan 19 2011
a(n) = (floor(8^(n+1)/63) - floor((n+1)/2))/7. - Seiichi Manyama, Dec 22 2023