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A346375
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a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.
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2
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6, 16, 37, 92, 263, 858, 3069, 11584, 44995, 177350, 704201, 2806476, 11205327, 44780242, 179038933, 715991768, 2863639259, 11453901534, 45814295265, 183254559460, 733012994791, 2932041493226, 11728145001197, 46912538061552, 187650068359923, 750600105667318, 3002400087124729
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.
a(n) = (2^(n+1) + 7) * (2^(n+1) + 8)/6 - 9 + 3*n.
More generally: let f(n, b) = Sum_{k=0..n} (2^k + b) * (2^k + b + 1)/2 then f(n, b) = (2^(n+1) + 3*b + 1) * (2^(n+1) + 3*b + 2) / 6 - (b + 1)^2 + b*(b + 1)*n/2.
G.f.: ((b^2+3*b+2)/2 - (3*b^2+8*b+4)*x + (4*b^2+8*b+3)*x^2) / ((4*x-1) * (2*x-1) * (x-1)^2).
E.g.f.: exp(x)*((6*b+3)*exp(x) + 2*exp(3*x) + 3*(b^2+b)*x/2 +(3*b^2-3*b-4) / 2) / 3.
a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) with n > 3.
This recurrence is valid for all sequences f(n, b).
G.f.: (35*x^2 - 32*x + 6) / ((4*x - 1) * (2*x - 1) * (x - 1)^2).
E.g.f.: exp(x) * (1 + 15*exp(x) + 2*exp(3*x) + 9*x)/3. - Stefano Spezia, Aug 15 2021
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MAPLE
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a:= proc(n) option remember:
if n=0 then 6 else procname(n-1)+(2^n+3)*(2^n+2)/2 fi:
end proc:
seq(a(n), n=0..26);
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MATHEMATICA
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PROG
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(PARI) a(n) = sum(k=0, n, (2^k+2)*(2^k+3)/2); \\ Michel Marcus, Jul 28 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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