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 A116157 #27 Sep 08 2022 08:45:24
%S A116157 1,1,3,3,7,8,17,22,43,60,110,161,283,428,732,1132,1901,2984,4950,7848,
%T A116157 12912,20609,33721,54065,88137,141737,230490,371411,602982,972961,
%U A116157 1577840,2548288,4129457,6673335,10808634,17474230,28293116,45753765,74064872,119794804
%N A116157 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) + a(n-5).
%C A116157 A Fibonacci-Padovan sequence.
%C A116157 The summation over some naturally chosen planes in the pyramid composed of MacWilliams transform matrices yields this sequence, which is the convolution of the Fibonacci numbers and the (alternating) Padovan numbers. Namely, the formula F(n) = Sum_{i+k=n, i>0, k>0} binomial(k,i) = Sum_{i+k=n, i>0, k>0} Krawtchouk[{k,i},0] where Krawtchouk[{k,i},x] is the i-th Krawtchouk polynomial of order k has a natural generalization as G(n) = Sum_{i+j+k=n, i>0,j>0, k>0} Krawtchouk[{k,i},j].
%H A116157 G. C. Greubel, <a href="/A116157/b116157.txt">Table of n, a(n) for n = 0..1000</a>
%H A116157 N. Gogin and A. Myllari, <a href="https://doi.org/10.1134/S0361768807020041">The Fibonacci-Padovan sequence and MacWilliams transform matrices</a>, Programming and Computing Software, Vol. 33, Issue 2 (March 2007), pp. 74-79.
%H A116157 Sait Tas, Omur Deveci and Erdal Karaduman, <a href="http://www.mijst.mju.ac.th/vol8/279-287.pdf">The Fibonacci-Padovan sequences in finite groups</a>, Maejo Int. J. Sci. Technol. 8 (2014), no. 03, pp. 279-287.
%H A116157 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,0,1).
%F A116157 G.f.: 1/((1-x-x^2)*(1-x^2+x^3)).
%t A116157 a[0]=1; a[1]=1; a[2]=3; a[3]=3; a[4]=7; a[n_]:=a[n]=a[n-1]+2a[n-2]-2a[n-3]+a[n-5]; Table[a[n], {n, 0, 50}]
%t A116157 LinearRecurrence[{1,2,-2,0,1},{1,1,3,3,7},50] (* _Harvey P. Dale_, Mar 07 2015 *)
%o A116157 (PARI) my(x='x+O('x^50)); Vec(1/((1-x-x^2)*(1-x^2+x^3))) \\ _G. C. Greubel_, May 10 2019
%o A116157 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x-x^2)*(1-x^2+x^3)) )); // _G. C. Greubel_, May 10 2019
%o A116157 (Sage) (1/((1-x-x^2)*(1-x^2+x^3))).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, May 10 2019
%o A116157 (GAP) a:=[1,1,3,3,7];; for n in [6..50] do a[n]:=a[n-1]+2*a[n-2]- 2*a[n-3]+a[n-5]; od; a; # _G. C. Greubel_, May 10 2019
%Y A116157 Cf. A000045, A000931.
%K A116157 easy,nonn
%O A116157 0,3
%A A116157 Nikita Gogin & Aleksandr Myllari (alemio(AT)utu.fi), Apr 15 2007