A316224 a(n) = n*(2*n + 1)*(4*n + 1).
0, 15, 90, 273, 612, 1155, 1950, 3045, 4488, 6327, 8610, 11385, 14700, 18603, 23142, 28365, 34320, 41055, 48618, 57057, 66420, 76755, 88110, 100533, 114072, 128775, 144690, 161865, 180348, 200187, 221430, 244125, 268320, 294063, 321402, 350385, 381060, 413475, 447678, 483717
Offset: 0
Examples
Row sums of the triangle: | 0 | ................................................................. 0 | 1 | 2 3 4 5 .................................................... 15 | 6 | 7 8 9 10 11 12 13 14 ........................................ 90 | 15 | 16 17 18 19 20 21 22 23 24 25 26 27 ........................... 273 | 28 | 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 ............... 612 | 45 | 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 .. 1155 ... where: . first column is A000384, . second column is A130883 (without 1), . third column is A033816, . diagonal is A014106, . 0, 2, 8, 18, 32, 50, ... are in A001105.
Links
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Crossrefs
Programs
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GAP
List([0..40], n -> n*(2*n+1)*(4*n+1));
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Julia
[n*(2*n+1)*(4*n+1) for n in 0:40] |> println
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Magma
[n*(2*n+1)*(4*n+1): n in [0..40]];
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Maple
seq(n*(2*n+1)*(4*n+1),n=0..40); # Muniru A Asiru, Jun 27 2018
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Mathematica
Table[n (2 n + 1) (4 n + 1), {n, 0, 40}] -
Maxima
makelist(n*(2*n+1)*(4*n+1), n, 0, 40);
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PARI
vector(40, n, n--; n*(2*n+1)*(4*n+1))
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Python
[n*(2*n+1)*(4*n+1) for n in range(40)]
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Sage
[n*(2*n+1)*(4*n+1) for n in (0..40)]
Formula
O.g.f.: 3*x*(5 + 10*x + x^2)/(1 - x)^4.
E.g.f.: x*(15 + 30*x + 8*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*A258582(n).
a(n) = -3*A100157(-n).
Sum_{n>0} 1/a(n) = 2*(3 - log(4)) - Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 2*sqrt(2)*log(1+sqrt(2)) + (sqrt(2)-1/2)*Pi - 6. - Amiram Eldar, Sep 17 2022
Comments