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URL: https://oeis.org/A122931

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A122931
Row sums of triangular array A122930.
4
1, 2, 7, 18, 50, 132, 351, 924, 2431, 6380, 16732, 43848, 114869, 300846, 787815, 2062830, 5401054, 14140940, 37022755, 96928920, 253766591, 664375032, 1739365272, 4553731728, 11921847625, 31211839802, 81713718151, 213929389674
OFFSET
1,2
COMMENTS
Also sums of the natural numbers with A000045 entries per row: for example, 1 2 3+4 5+6+7 8+9+10+11+12.
FORMULA
From R. J. Mathar, Oct 07 2006: (Start)
a(n) = Sum_{i=A000071(n+1)+1..A000071(n+2)} i.
a(n) = A000045(n)*floor(A000071(n+1) + (A000045(n)+1)/2). (End)
a(n) = Sum_{k=1..n} A000045(k)^2*A000045(n-k+1). - Gerald McGarvey, Nov 08 2007
a(n) = (F(n+2)^2 - F(n+1)^2 - F(n+2) + F(n+1))/2 where F(n)=Fibonacci(n). - Gary Detlefs, Mar 10 2011
G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)*(1-x-x^2)). - Colin Barker, Mar 12 2012
a(n) = F(n)*(F(n+3)-1)/2. - J. M. Bergot, Mar 16 2013
a(n) = (F(n+1) - 1)*(F(n+2) + 1)/2 + (n mod 2). - Greg Dresden, Sep 25 2021
a(n) = A301809(n+1) - A000045(n) = A191797(n+2) - A191797(n+1). - J.S. Seneschal, Jul 07 2025
MAPLE
A000045 := proc(n) if n <= 1 then RETURN(n) ; else RETURN( A000045(n-1)+A000045(n-2)) ; fi ; end: A000071 := proc(n) RETURN(A000045(n)-1) ; end: A122931 := proc(n) local a45 ; a45 := A000045(n) ; RETURN (a45*(A000071(n+1)+(a45+1)/2)) ; end: for n from 1 to 30 do printf("%d, ", A122931(n)) ; od ; # R. J. Mathar, Oct 07 2006
MATHEMATICA
(#[[2]]^2-#[[1]]^2-#[[2]]+#[[1]])/2&/@Partition[Fibonacci[ Range[ 2, 30]], 2, 1] (* or *) Module[{nn=30, fib}, fib=Fibonacci[Range[nn]]; Total/@ TakeList[ Range[Total[ fib]], fib]](* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Nov 19 2018 *)
KEYWORD
easy,nonn
AUTHOR
Alford Arnold, Sep 20 2006
EXTENSIONS
More terms from R. J. Mathar, Oct 07 2006
STATUS
approved