A047507 Numbers that are congruent to {0, 4, 6, 7} mod 8.
0, 4, 6, 7, 8, 12, 14, 15, 16, 20, 22, 23, 24, 28, 30, 31, 32, 36, 38, 39, 40, 44, 46, 47, 48, 52, 54, 55, 56, 60, 62, 63, 64, 68, 70, 71, 72, 76, 78, 79, 80, 84, 86, 87, 88, 92, 94, 95, 96, 100, 102, 103, 104, 108, 110, 111, 112, 116, 118, 119, 120, 124
Offset: 1
Examples
G.f. = 4*x^2 + 6*x^3 + 7*x^4 + 8*x^5 + 12*x^6 + 14*x^7 + 15*x^8 + 16*x^9 + ... - _Michael Somos_, Dec 12 2023
Links
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Programs
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Magma
[n : n in [0..150] | n mod 8 in [0, 4, 6, 7]]; // Wesley Ivan Hurt, May 27 2016
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Maple
A047507:=n->(8*n-3+I^(2*n)-(1+2*I)*I^(-n)-(1-2*I)*I^n)/4: seq(A047507(n), n=1..100); # Wesley Ivan Hurt, May 27 2016
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Mathematica
Table[(8n-3+I^(2n)-(1+2*I)*I^(-n)-(1-2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *) a[ n_] := 2*n - Max[0, 2 - Mod[1-n, 4]]; (* Michael Somos, Dec 12 2023 *) -
PARI
{a(n) = 2*n - max(0, 2 - (1-n)%4)}; /* Michael Somos, Dec 12 2023 */
Formula
G.f.: x^2*(4+2*x+x^2+x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Nov 06 2015
From Wesley Ivan Hurt, May 27 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-3+i^(2*n)-(1+2*i)*i^(-n)-(1-2*i)*i^n)/4 where i=sqrt(-1).
E.g.f.: (2 - 2*sin(x) - cos(x) + (4*x - 2)*sinh(x) + (4*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=2} (-1)^n/a(n) = (6-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - sqrt(2)*Pi/16. - Amiram Eldar, Dec 23 2021
a(n) = -A003485(-n) = a(n+4) - 8 for all n in Z. - Michael Somos, Dec 12 2023