This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A122931 #44 Jul 08 2025 07:49:45
%S A122931 1,2,7,18,50,132,351,924,2431,6380,16732,43848,114869,300846,787815,
%T A122931 2062830,5401054,14140940,37022755,96928920,253766591,664375032,
%U A122931 1739365272,4553731728,11921847625,31211839802,81713718151,213929389674
%N A122931 Row sums of triangular array A122930.
%C A122931 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.
%H A122931 Harvey P. Dale, <a href="/A122931/b122931.txt">Table of n, a(n) for n = 1..1000</a>
%H A122931 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1,1).
%F A122931 From _R. J. Mathar_, Oct 07 2006: (Start)
%F A122931 a(n) = Sum_{i=A000071(n+1)+1..A000071(n+2)} i.
%F A122931 a(n) = A000045(n)*floor(A000071(n+1) + (A000045(n)+1)/2). (End)
%F A122931 a(n) = Sum_{k=1..n} A000045(k)^2*A000045(n-k+1). - _Gerald McGarvey_, Nov 08 2007
%F A122931 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
%F A122931 G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)*(1-x-x^2)). - _Colin Barker_, Mar 12 2012
%F A122931 a(n) = F(n)*(F(n+3)-1)/2. - _J. M. Bergot_, Mar 16 2013
%F A122931 a(n) = (F(n+1) - 1)*(F(n+2) + 1)/2 + (n mod 2). - _Greg Dresden_, Sep 25 2021
%F A122931 a(n) = A301809(n+1) - A000045(n) = A191797(n+2) - A191797(n+1). - _J.S. Seneschal_, Jul 07 2025
%p A122931 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
%t A122931 (#[[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 *)
%Y A122931 Cf. A000027, A000045, A001654, A003714, A010049, A035514, A122930.
%Y A122931 Cf. A000217, A191797, A301809.
%K A122931 easy,nonn
%O A122931 1,2
%A A122931 _Alford Arnold_, Sep 20 2006
%E A122931 More terms from _R. J. Mathar_, Oct 07 2006